Framed DDF operators and the general solution to Virasoro constraints

Abstract

We define the framed DDF operators by introducing the concept of local frames in the usual formulation of DDF operators. In doing so it is possible to completely decouple the DDF operators from the associated tachyon and show that they are good zero-dimensional conformal operators. These framed DDFs allow for an explicit covariant formulation of the general solution of the Virasoro constraints both on-shell and off-shell which depends on a unique set of tangent space lightcone polarizations. This is possible since the frame allows us to embed the $SO(D-2)$ lightcone physical polarizations into the $SO(1,D-1)$ covariant ones in the most general way. The solution to the Virasoro constraints is not in the gauge that is usually used since the states obtained from DDF operators are generically the sum of terms which are partially transverse due to the presence of a projector but not traceless and terms which are partially traceless but not transverse. We then show that different embeddings are connected by states generated by improved Brower operators both on shell and off shell. Brower states are null on-shell and equivalent to BRST exact states. Therefore on shell one embedding is sufficient to compute all amplitudes except the ones with vanishing momentum. Off-shell a change of frame cannot be obtained by null states but only by Brower states. Improved Brower operators generate a gauge invariance associated with local frame rotations, which replaces the usual gauge invariance generated by Virasoro operators or BRST. To check the identification, we verify the matching of the expectation value of the second Casimir of the Poincaré group for some lightcone states with the corresponding covariant states built using the framed DDFs.

1 Introduction

String theory is probably the best candidate for quantum gravity and, as such, it should be able to tell something about both spacelike and timelike singularities in General Relativity.

Spacelike singularities involve time dependent backgrounds and therefore are very hard and also less studied. Already at the QFT level time-dependent backgrounds are difficult, since the concept of a particle is lost because they are created by the interaction with the background during the time evolution. String theory has been studied in toy models capable of reproducing a space-like (or null) singularity, which appears in space at a specific value of the time coordinate and then disappears. The easiest way to do so is by generating singularities by quotienting Minkowski with a discrete group with fixed points, i.e. orbifolding Minkowski. In this way, it is possible to produce both space-like singularities and supersymmetric null singularities [1,2,3,4,5,6,7,8,9,10,11,12,13,14] (see also [15,16,17] for some reviews). Another possible way which is a generalization of the previous orbifolds with null singularity is to consider gravitational shock wave backgrounds [18,19,20,21,22,23,24]. It happens that in these orbifolds the four tachyon closed string amplitude diverges in some kinematical ranges, more explicitly for the Null Boost orbifold (which may be made supersymmetric and has a null singularity) we have

$$\begin{aligned} \mathcal{A} ^{(closed)}_{4 T} \sim \int _{q\sim \infty } \frac{d q}{|q|} q^{ 4- \alpha ' \vec p^2_{\perp \, t}}, \end{aligned}$$

(1.1)

so the amplitude diverges for $\alpha ' \vec p^2_{\perp \, t}<4$ where $\vec p_{\perp \, t}$ is the orbifold transverse momentum in t channel. Until recently, this pathological behavior has been interpreted in the literature as “the result of a large gravitational backreaction of the incoming matter into the singularity due to the exchange of a single graviton”. This is not very promising for a theory which should tame quantum gravity. However, if we perform an analogous computation for the four point open string function, we find

$$\begin{aligned} \mathcal{A} ^{(open)}_{4 T} \sim \int _{q\sim \infty } \frac{d q}{|q|} q^{ 1- \alpha ' \vec p^2_{\perp \, t}} tr\left( \{T_1, T_2\} \{T_3, T_4\}\right) , \end{aligned}$$

(1.2)

which is also divergent when $\alpha ' \vec p^2_{\perp \, t}<1$ [25, 26]. This casts doubts on the backreaction as the main explanation since we are dealing with open strings at tree level. This is further strengthened by the fact that three point amplitudes with massive states may diverge [25] when appropriate polarizations are chosen. For example, for the three point function of two tachyons and the first level massive state, we find that, for an appropriate massive string polarization

$$\begin{aligned} \mathcal{A} ^{(open)}_{T T M} \sim \int _{u\sim 0} \frac{d u}{|u|^{5/2}} tr\left( \{T_1, T_2\} T_3\right) . \end{aligned}$$

(1.3)

This stresses the importance of massive states and can be interpreted as a complete breakdown of perturbation theory, i.e. perturbation theory does not exist. Although it is somewhat obvious that we do not expect to find a well-behaved perturbation theory because of the singularity, one could expect some kind of pathology like the series being asymptotic, but we find a much worse behavior since perturbation theory does not exist while it may exist as a complete theory [\(L_0\) constraint. Moreover, we find that in the appropriate meaning, the tangent space lightcone polarizations are Lorentz invariant and that all Lorentz covariant physical states may expressed using them in a covariant way since the Lorentz covariance may be encapsulated in the global rotations of the local frame E.

Another application of this way of going off-shell is to possibly solve an issue in lightcone string field theory recently pointed out in [52]: lightcone off-shell amplitudes seem to be badly behaved when $k^+\rightarrow 0$. From the perspective of this paper, when going off-shell, Brower states are necessary for Lorentz invariance and replace the use of null (BRST exact) states. Therefore it should be enough to add the Brower states (even if technically not so simple) in such a way as to be able to rotate the lightcone so that in the new lightcone the states would not have $k^+\rightarrow 0$.

A further application of this off-shell extension could be the computation of form factors which require off-shell amplitudes.

The paper is organized as follows.

In Sect. 2 we introduce the notation, conventions and framed DDF operators.

Then in Sect. 3 we discuss how framed DDFs differ from the usual ones and their algebra. We also introduce the Brower and the “improved” Brower operators. Brower operators and states play an important role off-shell as discussed in Sect. 6.

Given this, in Sect. 4 we discuss how to go minimally off-shell while preserving all the other Virasoro conditions and how to get the general Lorentz covariant solution to all the Virasoro conditions.

The solution is covariant and the general one but not in the simplest gauge, i.e. the obvious states are a mixture of irreps. The discussion is performed without technical details for the first two string levels in order to be able to pinpoint the ideas. However, the basic idea is quite simple – Framed DDFs are an explicit way of embedding in the most general way lightcone states into the covariant ones. The difference w.r.t. the existing literature is that the relation between lightcone polarizations and covariant ones is mediated by the frame. In the usual approach, one lightcone is chosen and the relation between lightcone polarizations and covariant ones is rigid, and it is not really possible to get all covariant ones. In contrast, using the local frame it is possible to keep fixed the tangent space lightcone polarizations while getting all possible covariant ones.

In Sect. 5 we further discuss how framed DDFs realize the idea of embedding lightcone states into covariant ones. This is obtained by comparing the Lorentz algebra restricted to DDFs operators to the one written on the lightcone.

In Sect. 6 we explicitly examine the details of the embedding in the simplest cases, i.e., for levels $N=1$ and $N=2$. When we consider on-shell states an infinitesimal change of frame or an infinitesimal boost involving the lightcone we recover the old result [48] that this is equivalent to performing a shift of the original state by a null state, i.e. a BRST exact state. On-shell this is the same as a Brower state. Because any frame can be reached with null states this means that on-shell only one frame is sufficient to compute all the on-shell amplitudes by “analytical continuations” since null states decouple. This is no longer true for off-shell amplitudes. In this case, an infinitesimal change of frame or an infinitesimal boost involving the lightcone is equivalent to performing a shift of the original state by an off-shell Brower state. Off-shell Brower states are not null and therefore are not BRST exact. This can be interpreted as the fact that off-shell, the Brower operators replace the gauge symmetry generated by Virasoro operators, i.e. BRST invariance.

Finally, to further strengthen the match in Sect. 7 we compare the second order Casimir for the Poincaré group for some states when computed on lightcone or in covariant formalism after the embedding of lightcone states into the covariant ones using framed DDFs.

2 Framed DDF operators and local frames

In this section, we present the reformulation of the standard DDF and Brower operators in terms of a local frame, i.e., a vielbein. We call them framed DDF. We start with a brief section on our conventions and then introduce the framed DDF operators. We then discuss their conformal properties and their algebra. It is worth stressing that the original DDF operators are not good conformal operators for computing off-shell amplitudes, since they have cuts (see Sect. 3.4).

2.1 The bosonic string

We write the string action in the conformal gauge

$$\begin{aligned} S = -\frac{1}{ 2\pi \alpha ' } \int d\tau \int _0^ {\pi } d \sigma \frac{1}{2}g_{\mu \nu } \left( \dot{X}^\mu \dot{X}^\nu - X^{' \mu } X^{' \nu } \right) ,\nonumber \\ \end{aligned}$$

(2.1)

using $g_{\mu \nu }$ in place of $\eta _{\mu \nu }$ in order to make more evident the role of the local frames. The solution for the open string then reads

$$\begin{aligned}&X^\mu (u, {\bar{u}}) = L^\mu (u)+ R^\mu ({\bar{u}}) , \nonumber \\&L^\mu (u) = \frac{1}{2}x^\mu _0 - i \alpha 'p^\mu _0 \ln (u) +i \sqrt{\frac{\alpha '}{2}} \sum _{n\ne 0} \frac{\alpha ^\mu _n}{n} u^{-n} , \end{aligned}$$

(2.2)

$$\begin{aligned}&R^\mu ({\bar{u}}) = L^\mu ({\bar{u}}) , \end{aligned}$$

(2.3)

where

$$\begin{aligned} u=e^{\tau _E+i \sigma } = e^{i (\tau +\sigma )} \quad \text{ and } \quad \arg (\ln (z)) \in (-\pi , \pi ], \end{aligned}$$

(2.4)

along with the canonical commutation relations,

$$\begin{aligned} {[}x^\mu _0, p_0^\nu ]&= i\, g^ {\mu \nu }, \nonumber \\ {[}\alpha ^\mu _m, \alpha _n^\nu ]&= m\, \delta _{m+n,0}\, g^ {\mu \nu } . \end{aligned}$$

(2.5)

As usual we also define $\alpha _0^\mu = \sqrt{2\alpha '} p_0^\mu$.

One can introduce the radial ordering R on the complex plane and compute the OPE,

$$\begin{aligned} R\left[ L^\mu (u) L^\nu (v) \right]&= :L^\mu (u) L^\nu (v): - \frac{\alpha '}{2} g^{\mu \nu } \Bigg [ \theta (|u|-|v|) \ln (u-v) \nonumber \\&\quad + \theta (-|u|+|v|) \ln (-u+v) \Bigg ]. \end{aligned}$$

(2.6)

It follows from (2.6) the usual useful relations

(2.7)

For writing down the integrated DDF operator, we consider the tachyon and photon vertices in the $-1$ ghost sector as,

$$\begin{aligned} V_T(x; k_T)&= c(x) : e^{i k_{T \mu } X^\mu (x, {\bar{x}}) }: \text{ with } \alpha 'g^{\mu \nu } k_{T \mu } k_{T \nu } = 1 \nonumber \\&= c(x) : e^{2 i k_{T \mu } L^\mu (x)} : \quad \text{ when } x>0 , \nonumber \\ V_A(x; k, \epsilon )&= c(x) : \epsilon _\mu \partial _x X^\mu (x, \bar{x}) e^{i k_{\mu } X^\mu (x, {\bar{x}}) }: \nonumber \\ {}&\quad \text{ with } g^{\mu \nu } k_{\mu } k_{\nu } = g^{\mu \nu } k_{\mu } \epsilon _\nu = 0 \nonumber \\&= c(x) : 2 \epsilon _\mu \partial _x L^\mu e^{2 i k_{ \mu } L^\mu (x)} : \quad \text{ when } {x>0} , \end{aligned}$$

(2.8)

where we made explicit the dependence on the chiral part of the string coordinates L(u).

2.2 Framed DDF operators

To reformulate the standard DDF and Brower operators, leading to the framed definitions in Eqs. (2.10) and (2.11) we introduce a local frame, i.e., a vielbein. While for the flat space we are working in, this may seem like overkill, it is actually useful to clearly distinguish the symmetries of the DDF and Brower constructions and the action of Lorentz transformations on the physical states.

We introduce the vielbein, i.e. the local frame by $E_\mu ^{\underline{\mu }}$ and its inverse $E^\mu _{\underline{\mu }}$ This dual local frame is such that

$$\begin{aligned} E_\mu ^{\underline{\mu }} E_\nu ^{\underline{\nu }} \eta _{\underline{\mu } \underline{\nu }} = g_{\mu \nu } , \end{aligned}$$

(2.9)

and has two symmetries: the local and global Lorentz transformations. As we discuss later in detail in Sect. 6.1.3 for the level $N=1$ states, the global Lorentz transformations associated with the curved index $\mu$ act in a very simple way on the DDF states, while the local Lorentz transformations discussed in detail in Sect. 6.1.4 act on ${\underline{\mu }}$ in a complicated way.

The framed DDF and Brower operators are then expressed using the flat left-moving string operators in the ghost number sector 0 by

$$\begin{aligned} {{\underline{A}}}^i_n(E)&= i \sqrt{ \frac{ 2 }{ \alpha '} } \oint _{z=0} \frac{d z}{ 2\pi i} : \partial _z {{\underline{L}}}^i(z) e^{i n \frac{ {{\underline{L}}}^+(z)}{\alpha ' {{\underline{p}}}^+_0}} : . \end{aligned}$$

(2.10)

$$\begin{aligned} {{\underline{A}}}^-_n(E)&= i \sqrt{ \frac{ 2 }{ \alpha '} } \oint _{z=0} \frac{d z}{ 2\pi i} : \left[ \partial _z {{\underline{L}}}^-(z) - i \frac{n}{4 {{\underline{p}}}_0^+} \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+} \right] e^{i n \frac{ {{\underline{L}}}^+(z) }{\alpha ' {{\underline{p}}}^+_0} } : . \end{aligned}$$

(2.11)

For later convenience we introduce also the improved Brower operators as¹

$$\begin{aligned} {{\tilde{ {{\underline{A}}}}}}^-_n(E)&= {{\underline{A}}}^-_n(E) - \frac{1}{ {{\underline{\alpha }}}_0^+} \mathcal{L} _n(E) + \frac{D-2}{24} \frac{1}{ {{\underline{\alpha }}}_0^+}\, \delta _{n, 0} , \end{aligned}$$

(2.12)

$$\begin{aligned} \mathcal{L} _m(E)&= \frac{1}{2} \sum _{j=2}^{D-1} \sum _{l\in {\mathbb {Z}} }: {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^j_{m-l}(E) : \end{aligned}$$

(2.13)

where the last term is present only for $n=0$ and it is fundamental to ensure that the states built from the improved Brower operators are null on-shell and in the critical dimension $D=26$ as shown in Eq. (6.16) for the $N=1$ level state and in Eq. (6.47) for the levels $N=2$. In the previous expressions we have introduced the notation

$$\begin{aligned} {{\underline{L}}}^{{\mu }}(z) = L^{\underline{\mu }}(z) = E^{\underline{\mu }}_\mu \, L^\mu (z), \end{aligned}$$

(2.14)

for the flat chiral string coordinates extended to the whole complex plane (with a cut on the negative real axis). Henceforth, we usually denote the transformation of any quantity onto the local frame defined by $E^\mu _{\underline{\mu }}$ by a line under their respective symbol or indexes. Where necessary, we explicitly denote the local frame.

Notice the explicit appearance of the operator $1/ {{\underline{p}}}_0^+$ in the exponent [53]. This allows one to derive and discuss many properties independently of the associated tachyonic operator. In particular only this form of DDF operators are true zero-dimensional conformal operators since they have no cuts when inserted in correlators.

The expression for ${{\underline{A}}}^i_n(E)$ using the curved coordinates is

$$\begin{aligned} {{\underline{A}}}^i_n(E)&= \sqrt{ \frac{ 2 }{ \alpha '} } \oint _{z=0} \frac{d z}{ 2\pi i} g_{\mu \nu }\, E^{\mu \underline{i} }\, : \partial _z L^\nu (z) \nonumber \\&\quad \exp \left\{ i n \frac{ g_{\rho \sigma }\, E^{\rho {\underline{+}}}\, L^\sigma (z) }{ \alpha 'g_{\mu \nu }\, E^{\mu {\underline{+}}}\, p^\nu _0 } \right\} : . \end{aligned}$$

(2.15)

The connection between the usual expression and the one with explicit frame can be obtained by choosing a null vector $n^\mu$, an auxiliary null vector $\bar{n}^\mu$ and the usual transverse polarization $\epsilon _\mu ^{(i)}$, such that

$$\begin{aligned} E^{ {\underline{+}}}_\mu = n_\mu ,\quad E^{\underline{-}}_\mu = {\bar{n}}_\mu ,\quad E^{\underline{i}}_\mu = \epsilon _\mu ^{(i)}, \end{aligned}$$

(2.16)

and it is discussed in more details in Sect. 3.1.

2.3 A better expression for the framed DDF operators

The previous framed DDF operators admit a representation that makes the transversality property of DDF states clearer. In particular, there is a null vector $E^\mu _{ {\underline{+}}}$ in the flat basis, such that,

$$\begin{aligned} {{\underline{x}}}^+_0 = g_{\mu \nu } x^\mu _0 E^{\nu \underline{+}} ,\quad {{\underline{p}}}^+_0 = - {{\underline{p}}}_{0 -} = g_{\mu \nu } p^\mu _0 E^{\nu \underline{+}} . \end{aligned}$$

(2.17)

If we define the part of the chiral coordinate without the momentum ${{\underline{L}}}^\mu _{(\ne )}(z)$ as

$$\begin{aligned} {{\underline{L}}}^\mu (z) = {{\underline{L}}}^\mu _{(\ne )}(z) -i \alpha ' {{\underline{p}}}^\mu _0 \ln (z), \end{aligned}$$

(2.18)

then we can write the DDF creator as

$$\begin{aligned} {{\underline{A}}}^i_{-m}(E)&= i \sqrt{\frac{2}{\alpha '}} \oint _{z=0} \frac{d z}{ 2\pi i} \frac{1}{z^m} :\nonumber \\ {}&\quad \left( \partial _z {{\underline{L}}}_{(\ne )}^i(z) - \frac{1}{z} i \alpha ' {{\underline{p}}}^i_0 \right) e^{-i m \frac{ {{\underline{L}}}_{(\ne )}^+(z) }{\alpha ' {{\underline{p}}}^+_0} } : , \end{aligned}$$

(2.19)

which allows for a better form upon integration by parts of the $\frac{1}{z^{m+1}}$ term

$$\begin{aligned}&{{\underline{A}}}^i_{-m}(E) \nonumber \\&\quad = i \sqrt{\frac{2}{\alpha '}} \oint _{z=0} \frac{d z}{ 2\pi i} \frac{1}{z^m} : \left( \partial _z {{\underline{L}}}_{(\ne )}^i(z) - \frac{ {{\underline{p}}}^i_0 }{ {{\underline{p}}}^+_0}\, \partial _z {{\underline{L}}}_{(\ne )}^+(z) \right) \nonumber \\&\qquad \times e^{-i m \frac{ {{\underline{L}}}_{(\ne )}^+(z) }{\alpha ' {{\underline{p}}}^+_0} } : \nonumber \\&\quad = i \sqrt{\frac{2}{\alpha '}} \oint _{z=0} \frac{d z}{ 2\pi i} \frac{1}{z^m} : \Pi ^{\underline{i}}_\mu \partial _z L_{(\ne )}^\mu (z) e^{-i m \frac{ E^{\underline{+}}_\mu L_{(\ne )}^\mu (z) }{ \alpha 'E^{\underline{+}}_\mu p^\mu _0} } : \nonumber \\&\quad = i \sqrt{\frac{2}{\alpha '}} \frac{1}{(m-1)!} \frac{ d^{m-1} }{d z^{m-1} }\Bigg |_{z=0} : \Pi ^{\underline{i}}_\mu \partial _z L_{(\ne )}^\mu (z) e^{-i m \frac{ E^{\underline{+}}_\mu L_{(\ne )}^\mu (z) }{ \alpha 'E^{\underline{+}}_\mu p^\mu _0} } : , \end{aligned}$$

(2.20)

where we have defined transverse projector

$$\begin{aligned} \Pi ^{\underline{i}}_\mu = E^{\underline{i}} _\mu - \frac{ {{\underline{p}}}_0^i }{ {{\underline{p}}}_0^+}\, E^{\underline{+}}_\mu = E^{\underline{i}} _\mu - \frac{ E^{\underline{i}}_\nu \, p_0^\nu }{ E^{\underline{+}}_\gamma \,p_0^\gamma } \, E^{\underline{+}}_\mu ,\quad p_0^\mu \, \Pi ^{\underline{i}}_\mu =0 . \end{aligned}$$

(2.21)

This shows clearly that the “leading order” of a DDF operator is

$$\begin{aligned} {{\underline{A}}}^i_{-m}(E) \sim e^{-i \frac{m}{2\alpha ' {{\underline{p}}}_0^+} {{\underline{x}}}_0^+} (\Pi \, {{\underline{\alpha }}})^i_{-m}, \end{aligned}$$

(2.22)

where the $e^{-i \frac{m}{2\alpha ' {{\underline{p}}}_0^+} {{\underline{x}}}_0^+}$ plays a fundamental role in the fact that ${{\underline{A}}}^i_m$ has zero conformal dimension.

This expression could give the wrong impression that states obtained by using DDF are transverse; unfortunately, it is not so because the contour integration becomes a multiple derivative as in the last line of Eq. (2.20) and for $m \ge 2$ derivatives can act on the exponent part which gives a non-transverse contribution as explicitly shown in Eq. (6.30) for the simplest case of a level $N=2$ state.

3 Important properties of the framed DDF operators

In this section we first discuss the differences between the framed DDF operators and the usual formulation. Then we discuss the algebra and the hermiticity properties, and finally we derive and discuss the fact that only the framed DDF operators are really zero conformal dimension operators (see also [52]). The derivation of the algebra and the hermiticity properties is made in the Appendices B and C.

3.1 Differences with the usual formulation

The framed DDF construction described above has two significant differences compared to the standard DDF formulation: decoupling from the associated tachyonic momentum and being conformal operators.

Here we discuss the former, and in Sect. 3.4 we discuss the conformal properties.

In particular, the decoupling means that we could even act on a physical state different from the tachyon!

Usually, starting from the on-shell tachyon vertex operator $V_T(x; p_T) = c(x)\,:e^{i p_T\cdot X(x, {\bar{x}})}: = c(x)\,:e^{2 i p_T\cdot L(x)}:$ with

$$\begin{aligned} p_T^2 = 1/\alpha ' \end{aligned}$$

(3.1)

and imposing

$$\begin{aligned} 2\alpha ' p_T \cdot q = 1, \end{aligned}$$

(3.2)

for some null vector q (eventually also choosing $p_T^+ = -p_T^- = \frac{1}{ \sqrt{\alpha '} }$, $p_T^i = 0$ and $q^+ = q^i = q^2 = 0,~ q^- \ne 0$ ), one can successively construct not exact BRST invariant states with momenta given by,

$$\begin{aligned} p_N = p_T - N\, q, \end{aligned}$$

(3.3)

such that the mass is given by,

$$\begin{aligned} -\alpha ' p_N^2 = \alpha ' M_N^2 = N-1, \end{aligned}$$

(3.4)

by applying a string of DDF operators $A^i_n$ ($n<0$) on the tachyonic state as

$$\begin{aligned} \prod _{n=1}^\infty \prod _{i=2}^{D-2} \left( A^i_{-n}(q, p_T) \right) ^{N^i_n} | p_T \rangle , \end{aligned}$$

(3.5)

with

$$\begin{aligned} N= \sum _{n=1}^\infty \sum _{i=2}^{D-2} N^i_n. \end{aligned}$$

(3.6)

The bosonic DDF operators are defined by:

$$\begin{aligned} A^i_n(q, p_T) = i \sqrt{ \frac{ 2 }{ \alpha '} } \oint \, \frac{d z}{2\pi i}\, \epsilon ^{(i)}(q, p_T)\cdot \partial _z L \, e^{i 2\, n \,q\cdot L(z)},\nonumber \\ \end{aligned}$$

(3.7)

where $\epsilon _\mu ^{(i)}(q, p_T)$ is a polarization vector with

$$\begin{aligned} \epsilon ^{(i)}(q, p_T)\cdot q = 0, \end{aligned}$$

(3.8)

which transforms as a vector under $SO(D-2)$ - the little group of massless states and it is implicitly dependent on $p_T$ through q.

Therefore, the associated tachyonic momentum $p_T$ plays a direct role in ensuring that the usual DDF construction satisfies the harmonic oscillator algebra

$$\begin{aligned} {[}A^i_m(q, p_T),\, A^j_n(q, p_T)] = m\, \delta ^{i j}\, \delta _{m+n,0}. \end{aligned}$$

(3.9)

In contrast the flat space DDF operators ${{\underline{A}}}^i_m(E)$ as constructed in the previous section automatically satisfy the algebra (3.9) without any reference to the tachyonic momentum associated to the photon vertex, provided that, ${{\underline{p}}}^+_0 \ne 0$ in (2.10), i.e. we have decoupled the tachyonic momentum from the definition of the (flat space) DDF operators.

Formally one can recover the usual formulation from the framed one by letting

$$\begin{aligned} \frac{ E^{ {\underline{+}}}_\mu }{ \alpha ' {{\underline{p}}}_0^+} = \frac{ n_\mu }{ \alpha ' {{\underline{p}}}_0^+} \rightarrow 2 {q_\mu }, \end{aligned}$$

(3.10)

which amounts to replacing an operator with a ${\mathbb {C}}$-number.

3.2 Algebra of the operators

The algebra of the framed operators read

$$\begin{aligned} {[} {{\underline{A}}}^i_m(E),\, {{\underline{A}}}^j_n(E)]&= m\, \delta _{m+n,0} \delta ^{i j} , \end{aligned}$$

(3.11)

$$\begin{aligned} {[} {{\underline{A}}}^i_m(E),\, {{\underline{\alpha }}}_0^+ {{\underline{A}}}^-_n(E)]&= m \, {{\underline{A}}}^i_{m+n}(E) \end{aligned}$$

(3.12)

$$\begin{aligned} {[} {{\underline{\alpha }}}_0^+ {{\underline{A}}}^-_m(E),\, {{\underline{\alpha }}}_0^+ {{\underline{A}}}^-_n(E)]&= (m-n)\, {{\underline{\alpha }}}_0^+ {{\underline{A}}}^-_{m+n}(E) + 2 m^3\, \delta _{m+n, 0} , \end{aligned}$$

(3.13)

and

$$\begin{aligned} {[} {{\underline{A}}}^i_m(E),\, {{\underline{\alpha }}}_0^+ {\tilde{ {{\underline{A}}}}}^-_n(E)]&= 0 \nonumber \\ {[} {{\underline{\alpha }}}_0^+ {\tilde{ {{\underline{A}}}}}^-_m(E),\, {{\underline{\alpha }}}_0^+ {\tilde{ {{\underline{A}}}}}^-_n(E)]&= (m-n)\, {{\underline{\alpha }}}_0^+ {\tilde{ {{\underline{A}}}}}^-_{m+n}(E) \nonumber \\ {}&\quad + \frac{26 - D }{12} m^3\, \delta _{m+n, 0} . \end{aligned}$$

(3.14)

Again we observe that this algebra is independent of the tachyon momentum as well as the choice of the local frame by construction. The derivation is fairly straightforward and is given in Appendix B. The only subtlety is in computing the ${\tilde{ {{\underline{A}}}}}^-_m(E)$ algebra, where it is necessary to start with the explicitly normal ordered expression for $\sum _{l\in {\mathbb {Z}} }: {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^j_{m-l}(E):$ in order to get a well-defined expression for the ${\mathbb {C}}$-number.

3.3 Hermiticity properties

The hermiticity properties of the framed operators are the expected ones

$$\begin{aligned}{} & {} \left[ {{\underline{A}}}^i_m(E) \right] ^\dagger = {{\underline{A}}}^i_{-m}(E) ,~~ \left[ {{\underline{A}}}^-_m(E) \right] ^\dagger = {{\underline{A}}}^-_{-m}(E) ,\nonumber \\{} & {} \left[ \tilde{{\underline{A}}}^-_m(E) \right] ^\dagger = {\tilde{ {{\underline{A}}}}}^-_{-m}(E). \end{aligned}$$

(3.15)

However in the case of ${{\underline{A}}}^-_{-m}(E)$, although straightforward, it is not trivial because of the normal ordering of $e^{i \frac{m}{2 \alpha ' {{\underline{p}}}_0^+} {{\underline{x}}}_0^+} {{\underline{p}}}_0^-$ as discussed in Appendix 1.

3.4 Conformal properties

As already stated, only the framed DDF operators are true conformal operators of dimension zero. The usual ones are not when a rigorous approach is adopted. To understand this statement we consider the basic commutator

$$\begin{aligned}&\left[ \oint _{z=0} \frac{d z}{2 \pi i} z^{n+1} \frac{-2}{\alpha '} e^{\delta \cdot \partial {{\underline{L}}}(z)} \right| _{\delta ^2} , i \sqrt{\frac{2}{\alpha '}} \nonumber \\&\quad \left. \left. \oint _{w=0} \frac{d w}{2 \pi i} e^{{\underline{\epsilon }} \cdot \partial {{\underline{L}}}(w) + i {{\underline{k}}}\cdot {{\underline{L}}}(w)} \right| _{{\underline{\epsilon }}_{(i)}, {{\underline{k}}}_+} \right] , \end{aligned}$$

(3.16)

where we restrict ourselves to ${\mathcal {O}}(\delta ^2)$ and ${\mathcal {O}}({\underline{\epsilon }})$ above, to extract the forms of $L_n$ and ${{\underline{A}}}^i_m$ respectively, from the exponentials in (3.16).

This commutator does not depend on the fact that ${{\underline{k}}}_+$ is proportional to $\frac{1}{\alpha ' {{\underline{p}}}_0^+}$ since there is no ${{\underline{x}}}_0^-$ in $L_n$ and therefore this computation also applies to the usual DDF, when the flat quantities are replaced by the “curved” ones.

Then we get,

$$\begin{aligned} {[}L_n, {{\underline{A}}}^i_m]&= i \sqrt{\frac{2}{\alpha '}} \frac{-2}{\alpha '} \oint _{w=0} \oint _{z=w} z^{n+1} :\nonumber \\ {}&\quad \times e^{\delta \cdot \partial {{\underline{L}}}(z)} e^{{\underline{\epsilon }} \cdot \partial {{\underline{L}}}(w) + i {{\underline{k}}}\cdot {{\underline{L}}}(w)} : e^{ - \frac{\alpha '}{2} \frac{\delta \cdot \epsilon }{(z-w)^2} - \frac{\alpha '}{2} \frac{i \delta \cdot {{\underline{k}}}}{z-w} } \nonumber \\&= i \sqrt{\frac{2}{\alpha '}} \oint _{w=0} \oint _{z=w} z^{n+1} : \Bigg [ - \frac{\alpha '}{2} \frac{\delta \cdot {\underline{\epsilon }}}{(z-w)^2} \frac{i \delta \cdot {{\underline{k}}}}{z-w}\nonumber \\&\quad + \frac{\delta \cdot {\underline{\epsilon }}}{(z-w)^2} \delta \cdot \partial {{\underline{L}}}(z) \nonumber \\&\quad +{\underline{\epsilon }} \cdot \partial {{\underline{L}}}(w) \delta \cdot \partial L(z) \frac{i \delta \cdot {{\underline{k}}}}{z-w} \Bigg ] e^{ i {{\underline{k}}}\cdot {{\underline{L}}}(w)} :, \end{aligned}$$

(3.17)

using the substitution $\delta _\mu \delta _\nu \rightarrow \eta _{\mu \nu }$ and ${\underline{\epsilon }}\cdot {{\underline{k}}}=0$ we get

$$\begin{aligned} {[}L_n, {{\underline{A}}}^i_m]&= i \sqrt{\frac{2}{\alpha '}} \oint _{w=0} : \left[ \epsilon \cdot \partial ( w^{n+1} \partial {{\underline{L}}}(w) )\right. \nonumber \\&\quad \left. + w^{n+1} {\underline{\epsilon }} \cdot \partial {{\underline{L}}}(w) i {{\underline{k}}}\cdot \partial {{\underline{L}}}(w) \right] e^{ i {{\underline{k}}}\cdot {{\underline{L}}}(w)} : \nonumber \\&= i \sqrt{\frac{2}{\alpha '}} \oint _{w=0} : \partial \left[ w^{n+1} {\underline{\epsilon }} \cdot \partial {{\underline{L}}}(w) e^{ i {{\underline{k}}}\cdot {{\underline{L}}}(w)} \right] : \nonumber \\&= i \sqrt{\frac{2}{\alpha '}} \oint _{w=0} : \partial \left[ w^{ \alpha ' {{\underline{k}}}\cdot {{\underline{p}}}_0 + n + 1} \cdots \right] . \end{aligned}$$

(3.18)

We observe that in the framed DDF definition we have $\alpha ' {{\underline{k}}}\cdot {{\underline{p}}}_0 \in {\mathbb {Z}}$ by construction hence the integrand in (3.18 has no cuts and the integral is zero proving (the operator relation) that,

$$\begin{aligned} {[} L_n, {{\underline{A}}}^i_m]=0. \end{aligned}$$

(3.19)

On the other hand, in the usual DDF formulation, using the map** in Eq. (3.10) we get a result which is generically different from zero and depends on the initial and final point $x=-r$ on the negative real axis where we must place the branch cut, explicitly

$$\begin{aligned} {[} L_n, A^i_m(q, p_T)]&= ~i \sqrt{\frac{2}{\alpha '}} \, \frac{1}{\pi } \, \sin \left( 2 \pi \alpha 'm q \cdot p_0 \right) \, (-r)^{n+1} \nonumber \\&\quad \times \epsilon \cdot \partial L( -r ) \, r^{ \alpha 'm q \cdot p_0 } e^{ i m q \cdot L_{(\ne )}(-r) } . \end{aligned}$$

(3.20)

As far as the ${{\underline{A}}}^-$ operators are concerned, we need to be more careful, since the previous computation fails if we only consider the term with $\partial {{\underline{L}}}^-$ since it has a cubic pole. This issue is discussed in Appendix 1. However, the result is still that the framed ${{\underline{A}}}^-(E)$ and ${\tilde{ {{\underline{A}}}}}^-(E)$ are zero-dimensional conformal operators in the ${{\underline{p}}}_0^+$ sector, i.e.

$$\begin{aligned} {[} L_n, {\tilde{ {{\underline{A}}}}}^-_m]=0. \end{aligned}$$

(3.21)

4 The general covariant solution to Virasoro constraints and on shell and off shell physical states

We are now ready to discuss how the framed DDF operators offer a way of finding the general covariant solution of the Virasoro constraints. This is obtained by explicitly implementing, in an appropriate way, the idea already present in the literature [34, 35, 49,50,51] that DDF operators correspond to the physical lightcone operators.

In order to obtain the most general covariant state, we use the local frame E to add to the game the dependence on the little group of the embedding of the lightcone. This is the main difference w.r.t. the usual point of view. In fact, usually the embedding is performed by choosing one specific lightcone and this means that not all covariant states can be described, namely, the states with $k^+=0$. Introducing the frame E allows one to keep fixed the “tangent space” lightcone while rotating the “curved space” lightcone. This in turn means that the “tangent space” lightcone polarizations can be seen as the true d.o.f. .

In the next section, we examine in more detail how the lightcone states can be embedded into the covariant ones, here we simply take inspiration from this idea.

In this section we lay down the general ideas but for a better understanding of the details, we explicitly carry out the computations for the first two levels in Sect. 6.

We start with a discussion on how the framed DDFs allow us to define on-shell and (minimally) off-shell states, i.e. states which satisfy all Virasoro constraints except the mass shell condition.

4.1 On-shell BRST exact and not exact states

The most general physical state with ghost number $-1$ which is not BRST exact is given by²

$$\begin{aligned}&|k_-, k_i, \{N_{i\, n}, N_{c_{-1}}=1 \}_{n\ge 1}; E \rangle \nonumber \\&\quad =\left[ \prod _{i=2}^{D-1} \prod _{m=1}^\infty \frac{ \left( A^i_m{}^\dagger (E) \right) ^{N_{i\,m}}}{ \sqrt{ m!\, N_{i\,m}! }} \right] c_{-1}\, |k_{T \rho } = E_\rho ^{\underline{\mu }} {{\underline{k}}}_{T \mu }, 0_a\rangle , \end{aligned}$$

(4.1)

with

$$\begin{aligned} k_\rho = E_\rho ^{\underline{\mu }} {{\underline{k}}}_{T+N\, \mu } = k_{T \rho } -N \frac{ E_\rho ^{ {\underline{+}}} }{ 2 \alpha 'g^{\mu \nu } E_\nu ^{ {\underline{+}}}\, k_{T \mu } }, \end{aligned}$$

(4.2)

where we have defined the “shifted” ${{\underline{k}}}_{T+N}$ momentum as

$$\begin{aligned} {{\underline{k}}}^-_{T+N}= {{\underline{k}}}_T^- + \frac{ N }{ 2\alpha ' {{\underline{k}}}^+},~~ N=\sum _{i=2}^{D-1}\sum _{m=1}^\infty m N_{i\, m}, \end{aligned}$$

(4.3)

and where we have shown the explicit frame dependence.

Notice that the tachyonic momentum ${{\underline{k}}}_T$ is on-shell and fixed. All possible physical momenta $k_\rho$ ( $k^2<0$ for $N\ge 2$ ) of a particle at level N can be obtained from the fixed momentum ${{\underline{k}}}_{T+N}$ by means of the global Lorentz transformation, i.e., the Lorentz transformation on curved indexes associated with E. This means that it is not a restriction to take the tachyonic rest frame ${{\underline{k}}}^-_T = {{\underline{k}}}^+_T = - \frac{1}{ \sqrt{2\alpha '} }$ and ${{\underline{k}}}^i_T=0$.

On the other hand the states

$$\begin{aligned} \prod _{m=1}^\infty (A^i_{-m}(E))^{N^i_m} ({\tilde{A}}^-_{-m}(E))^{N_m} c_{-1} |k_T; 0_a\rangle , \end{aligned}$$

(4.4)

with at least one $N_m \ne 0$ satisfy the Virasoro constraints but are null, i.e. are BRST exact since in $D=26$ the ${\tilde{A}}^-(E)$ satisfy a $c=0$ Virasoro algebra so that their norm is proportional to

$$\begin{aligned}&\langle k_T | {\tilde{A}}^-_{0} | k_T \rangle \propto \sqrt{2\alpha '} {{\underline{k}}}_T^- + \frac{D-2}{24} \frac{ 1}{ \sqrt{2\alpha '} {{\underline{k}}}_T^+}\nonumber \\&\quad = \frac{ 1}{ \sqrt{2\alpha '} {{\underline{k}}}_T^+} \left( -\alpha ' {{\underline{k}}}_T^2 + \frac{D-2}{24} \right) , \end{aligned}$$

(4.5)

which vanishes when the tachyon $\alpha '(2 {{\underline{k}}}^+_T {{\underline{k}}}^-_T - \vec {{\underline{k}}}_T^2 ) = -1$ is on-shell. They play, however, a role when going minimally off-shell as we discuss now.

4.2 Going minimally off-shell

As elaborated in Sect. 3 an advantage of this formulation is that the ${{\underline{A}}}^i_m(E)$ and ${\tilde{A}}^-_m(E)$ do not refer to the tachyon momentum and always satisfy the same algebra Eqs. (3.11) and (3.14) independently of it. This allows us to easily go minimally off-shell by setting the tachyon momentum to be off-shell.³ In doing so, all the Virasoro conditions are satisfied, except the mass shell one, in fact using Eqs. (3.19) and (3.21), we have for $n\ge 1$

$$\begin{aligned}{} & {} L_n \prod _{m=1}^\infty (A^i_{-m}(E))^{N^i_m} (\tilde{A}^-_{-m}(E))^{N_m}|k_T\rangle \nonumber \\{} & {} \quad = \prod _{m=1}^\infty (A^i_{-m}(E))^{N^i_m} ({\tilde{A}}^-_{-m}(E))^{N_m} L_n |k_T\rangle = 0, \end{aligned}$$

(4.6)

since for a tachyonic off-shell state we still have $L_n |k_T\rangle = 0$ for $n\ge 1$. Notice, however, that these off-shell states may have a negative norm since they are off-shell. This is better discussed in Sect. 6 with the use of examples.

4.3 The general covariant solution to Virasoro constraints

Let us turn to the main point: how to obtain the general covariant solution to the Virasoro constraints using the framed DDF and the tangent space lightcone polarizations. Instead of writing the more general expression we discuss the first non-trivial levels.

We start with the $N=1$ level where the generic on-shell framed DDF state is

$$\begin{aligned} |G_1(E)\rangle = \underline{\epsilon }_i\, {{\underline{A}}}^i_{-1}(E) | {{\underline{k}}}_T\rangle + \underline{\epsilon }_-\, {\tilde{ {{\underline{A}}}}}^-_{-1}(E) | {{\underline{k}}}_T\rangle , \end{aligned}$$

(4.7)

which shows the true independent d.o.f.s associated to the lightcone polarizations $\underline{\epsilon }_i$ since the Brower state ${\tilde{ {{\underline{A}}}}}^-_{-1}(E) | {{\underline{k}}}_T\rangle$ is null on-shell and decouples from everything. and the associated polarization $\underline{\epsilon }_{-}$ is irrelevant. The number of independent $\underline{\epsilon }_i$ is obviously independent of the frame E and can be chosen at will. In principle, they depend on the frame E, but the two polarizations $\underline{\epsilon }_i(E)$ and $\underline{\epsilon }_i({\hat{E}})$ chosen in two different reference frames related by a global Lorentz rotation are the same, since $\underline{\epsilon }_i$ do not transform. The sets $\{\underline{\epsilon }_i(E),\, \underline{\epsilon }_-(E)\}$ and $\{\underline{\epsilon }_i({\hat{E}}),\, \underline{\epsilon }_-({\hat{E}})\}$ which are related by a local Lorentz transformation can be mapped into each other. Their relation is non-trivial and requires the use of gauge transformations generated by a Brower state as detailed in Sect. 6.1.5, i.e. a change in $\underline{\epsilon }_-(E)$.

Therefore, to obtain the general solution we can keep the polarizations $\underline{\epsilon }_i$ fixed and change the frame E by a global Lorentz rotation which leaves $\underline{\epsilon }_i$ invariant.

Having in mind the goal of obtaining the general solution by changing the frame E while kee** the polarization fixed, we have not made the dependence on E explicit.

The general solution to the Virasoro constraints is obtained by expressing the previous non-null state in a covariant basis as (see Eq. (6.26) for details)

$$\begin{aligned}&|G_1(E)\rangle \nonumber \\&\quad = \left[ \underline{\epsilon }_i\, \left( E_{\mu }^{\underline{i}} - \frac{ {{\underline{k}}}^i_T}{ {{\underline{k}}}^+_T} E_{\mu }^{\underline{+}} \right) \right. \nonumber \\&\qquad \left. + \underline{\epsilon }_-\, \left( E_{\mu }^{\underline{-}} + \frac{ -2 {{\underline{k}}}^-_{T+1} {{\underline{k}}}^+_T + 2 \vec {{\underline{k}}}^2_T}{ 2 ( {{\underline{k}}}^+_T)^2} E_{\mu }^{\underline{+}} - \frac{ {{\underline{k}}}^j_T}{ {{\underline{k}}}^+_T} E_{\mu }^{\underline{j}} \right) \right] \, \alpha ^\mu _{-1} |k\rangle \nonumber \\&\quad = \left[ \underline{\epsilon }_i\, \left( E^{\underline{i}}_\mu - \frac{E^{\underline{i}}_\rho k^\rho }{ E^{\underline{+}}_\rho k^\rho } E_{\mu }^{\underline{+}} \right) \right. \nonumber \\&\qquad \left. + \underline{\epsilon }_-\, \left( E_{\mu }^{\underline{-}} + \frac{ ( -2 E_{\rho }^{\underline{-}} E_{\sigma }^{\underline{+}} + E_{\rho }^{\underline{j}} E_{\sigma }^{\underline{j}} ) k^\rho \, k^\sigma }{ 2 ( E_{\rho }^{\underline{+}} k^\rho )^2 } E_{\mu }^{\underline{+}} - \frac{ {{\underline{k}}}^j}{ {{\underline{k}}}^+} E_{\mu }^{\underline{j}} \right) \right] \nonumber \\&\qquad \times \alpha ^\mu _{-1} |k\rangle , \text{ with } k_\mu = E_\mu ^{\underline{\nu }}\, {{\underline{k}}}_{T+1\, \nu } = E_\mu ^{\underline{\nu }}\, \left( {{\underline{k}}}_{T\, \nu } - \frac{\delta ^-_\nu }{ 2 \alpha ' {{\underline{k}}}_T^+} \right) , \end{aligned}$$

(4.8)

where $| {{\underline{k}}}_{T+1}\rangle$ is the level $N=1$ physical state momentum obtained from the tachyon momentum $| {{\underline{k}}}_T\rangle$.

From the previous expression we read off the polarizations of the general solution to the Virasoro constraints for a $N=1$ state as a function of the physical momentum k as

$$\begin{aligned} \epsilon _\mu (k)&= \left( \underline{\epsilon }_i - \underline{\epsilon }_-\, \frac{ {{\underline{k}}}^i}{ {{\underline{k}}}^+} \right) E^{\underline{i}}_\mu + \underline{\epsilon }_-\, E_{\mu }^{\underline{-}} + \left( - \underline{\epsilon }_i \frac{E^{\underline{i}}_\rho k^\rho }{ E^{\underline{+}}_\rho k^\rho }\right. \nonumber \\&\quad \left. + \underline{\epsilon }_- \frac{ ( -2 E_{\rho }^{\underline{-}} E_{\sigma }^{\underline{+}} + E_{\rho }^{\underline{j}} E_{\sigma }^{\underline{j}} ) k^\rho \, k^\sigma }{ 2 ( E_{\rho }^{\underline{+}} k^\rho )^2 } \right) E_{\mu }^{\underline{+}}, \nonumber \\ \text{ with }&k_\mu = E_\mu ^{\underline{\nu }}\, {{\underline{k}}}_{T+1\, \nu } ,\quad k^2=0 , \end{aligned}$$

(4.9)

in which it is clear that the momentum k can point in any direction due to the action of the local frame E on the fixed ${{\underline{k}}}_{T+1}$. The polarization is rotated accordingly while still depending on the right number of d.o.f.s . This is the reason why this is the general solution. Actually, only the little group of ${{\underline{k}}}_{T+1}$ matters.

Obviously, it is by far simpler to consider the general solution up to null states which reads

$$\begin{aligned} \epsilon _\mu (k) = \underline{\epsilon }_i\, \left( E_{\mu }^{\underline{i}} - \frac{E^{\underline{i}}_\rho k^\rho }{ E^{\underline{+}}_\rho k^\rho } E_{\mu }^{\underline{+}} \right) ,\quad k_\mu = E_\mu ^{\underline{\nu }}\, {{\underline{k}}}_{T+1\, \nu } .\nonumber \\ \end{aligned}$$

(4.10)

Let us now consider the general on shell-level $N=2$ DDF state. This can be written using the framed DDFs as

$$\begin{aligned} |G_2(E)\rangle&= \underline{T}_i\, {{\underline{A}}}^i_{-2}(E)\, | {{\underline{k}}}_T\rangle + \underline{S}_{i j}\, {{\underline{A}}}^i_{-1}(E)\, {{\underline{A}}}^j_{-1}(E)\, | {{\underline{k}}}_T\rangle \nonumber \\&\quad + \underline{g}_{i -}\, {{\underline{A}}}^i_{-1}(E)\, {\tilde{ {{\underline{A}}}}}^-_{-1}(E)\, | {{\underline{k}}}_T\rangle \nonumber \\&\quad + \underline{g}_{- -}\, \left( {\tilde{ {{\underline{A}}}}}^-_{-1}(E) \right) ^2\, | {{\underline{k}}}_T\rangle \nonumber \\&\quad + \underline{g}_-\, {\tilde{ {{\underline{A}}}}}^-_{-2}(E)\, | {{\underline{k}}}_T\rangle , \end{aligned}$$

(4.11)

which shows the true independent d.o.f.s associated to the lightcone polarizations $\underline{T}_i$ and $\underline{S}_{i j}$ since the other polarizations $\underline{g}_{i -}$, $\underline{g}_{- -}$ and $\underline{g}_{-}$ are associated to null states (when on-shell). The polarizations $\underline{T}_i$ and $\underline{S}_{i j}$ are independent and can be chosen at will in a given frame E. The number of polarizations is obviously independent of the frame E. These polarizations are invariant under a global Lorentz rotation, but the polarizations in different frames related by a tangent Lorentz are not. They can be mapped into each other by a transformation which is non-trivial and requires the use of gauge transformations generated by Brower null states.

We can then rewrite the previous state using the usual “curved index” operators. From this expression we can read the general covariant solution to the Virasoro constraints for level $N=2$. As in the $N=1$ case this happens because the proper number of d.o.f.s is embedded into the covariant polarization in a momentum-dependent way. The arbitrary direction of the physical momentum k is achieved by the local frame E which allows the momentum to point in any direction.

We first write the generic non-null part of the previous state in an oscillator (flat) basis ${{\underline{\alpha }}}^\mu _{-n}$. We could write this expression using the projector $\underline{\Pi }(E)$ but it is also instructive to write the full expanded expression down as

$$\begin{aligned} |G_2\rangle&= \left( \underline{T}_i \underline{T}^{(i)}_l \right) {{\underline{\alpha }}}^l_{-2} | {{\underline{k}}}_{T+2}\rangle + \left( \underline{T}_i \underline{T}^{(i)}_+ + \underline{S}_{i j} \underline{T}^{(i j)}_+ \right) \nonumber \\&\quad \times {{\underline{\alpha }}}^+_{-2} | {{\underline{k}}}_{T+2}\rangle + \left( \underline{S}_{i j} \underline{S}^{(i j)}_{l m} \right) {{\underline{\alpha }}}^l_{-1} {{\underline{\alpha }}}^m_{-1} | {{\underline{k}}}_{T+2}\rangle \nonumber \\&\quad + \left( \underline{T}_i \underline{S}^{(i)}_{+ m} + \underline{S}_{i j} \underline{S}^{(i j)}_{+ m} \right) {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^m_{-1} | {{\underline{k}}}_{T+2}\rangle \nonumber \\&\quad + \left( \underline{T}_i \underline{S}^{(i)}_{+ +} + \underline{S}_{i j} \underline{S}^{(i j)}_{+ +} \right) {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^+_{-1} | {{\underline{k}}}_{T+2}\rangle , \end{aligned}$$

(4.12)

where we used

$$\begin{aligned} {{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle&= \left( \underline{T}^{(i)}_l {{\underline{\alpha }}}^l_{-2} + \underline{T}^{(i)}_+ {{\underline{\alpha }}}^+_{-2} \right) | {{\underline{k}}}_{T+2}\rangle \nonumber \\&\quad + \left( \underline{S}^{(i)}_{+ m} {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^m_{-1} + \underline{S}^{(i)}_{+ +} {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^+_{-1} \right) | {{\underline{k}}}_{T+2}\rangle , \nonumber \\ {{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1} | {{\underline{k}}}_T\rangle =&\left( \underline{T}^{(i j)}_+ {{\underline{\alpha }}}^+_{-2} \right) | {{\underline{k}}}_{T+2}\rangle + \left( \underline{S}^{(i j)}_{l m} {{\underline{\alpha }}}^l_{-1} {{\underline{\alpha }}}^m_{-1}\right. \nonumber \\&\quad \left. +\underline{S}^{(i j)}_{+ m} {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^m_{-1} + \underline{S}^{(i j)}_{+ +} {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^+_{-1} \right) | {{\underline{k}}}_{T+2}\rangle , \end{aligned}$$

(4.13)

and $| {{\underline{k}}}_{T+2}\rangle$ is the level $N=2$ physical state momentum obtained from the tachyon momentum $| {{\underline{k}}}_T\rangle$. There is no $\underline{S}^{(i)}_{l m}$ due to the little group transformation property of ${{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle$. Similarly for $\underline{T}^{(i j)}_l$. The expressions for $\underline{T}^{(i)}_l$, $\underline{T}^{(i j)}_l$, $\underline{S}^{(i)}_{l m}$, $\underline{S}^{(i j)}_{l m}$ can be obtained in an algorithmically straightforward way and in the present case are given in Eqs. (6.29) and (6.34).

We then change to the usual “curved” coordinates and obtain the general covariant solution for the $N=2$ on-shell states up to null states (see Eqs. (6.30) and (6.36) for the explicit expressions)

$$\begin{aligned} T_\mu&= \left( \underline{T}_i \underline{T}^{(i)}_l \right) E^{\underline{l}}_\mu + \left( \underline{T}_i \underline{T}^{(i)}_+ + \underline{S}_{i j} \underline{T}^{(i j)}_+ \right) E^{\underline{+}}_\mu , \nonumber \\ S_{\mu \nu }&= \left( \underline{S}_{i j} \underline{S}^{(i j)}_{l m} \right) E^{\underline{l}}_\mu E^{\underline{m}}_\nu + \left( \underline{T}_i \underline{S}^{(i)}_{+ m} + \underline{S}_{i j} \underline{S}^{(i j)}_{+ m} \right) E^{\underline{+}}_\mu E^{\underline{m}}_\nu \nonumber \\&\quad + \left( \underline{T}_i \underline{S}^{(i)}_{+ +} + \underline{S}_{i j} \underline{S}^{(i j)}_{+ +} \right) E^{\underline{+}}_\mu E^{\underline{+}}_\nu . \end{aligned}$$

(4.14)

This is the general solution of the level $N=2$ Virasoro constraints up to null states. It is given by the linear superposition of the proper number of independent solutions, for example associated to the polarization $\underline{T}_i$ we have the covariant solution

$$\begin{aligned} T_\mu ^{(i)}&= \underline{T}^{(i)}_l\, E^{\underline{l}}_\mu + \underline{T}^{(i)}_+\, E^{\underline{+}}_\mu , \nonumber \\ S_{\mu \nu }^{(i)}&= \underline{S}^{(i)}_{+ m} E^{\underline{+}}_\mu E^{\underline{m}}_\nu + \underline{S}^{(i)}_{+ +} E^{\underline{+}}_\mu E^{\underline{+}}_\nu , \end{aligned}$$

(4.15)

which depends on the frame E or said differently on the on-shell momentum $k = E {{\underline{k}}}_{T+2}$ which is related to the fixed ${{\underline{k}}}_{T+2}$ but it is anyhow arbitrary because of the action of E.

Notice how this solution is not in the usual gauge, where the polarizations are transverse and traceless (see (6.29) for the explicit form).

Some steps along this line of thought (but using the covariant formalism) were taken in [54] where a projector was introduced.

4.4 The lightcone string amplitudes point of view

Kee** ${{\underline{k}}}_T$ and therefore ${{\underline{k}}}_{T+N}$ fixed and varying E to discuss the spectrum is perfectly fine. It also implies that lightcone polarizations have an “absolute” meaning since they are the polarizations seen in a fixed reference frame, for example, the rest frame of massive particles. For example ${{\underline{\epsilon }}}_i$ is the polarization of a photon as seen in a well-defined frame.

However, it is not the best choice for dealing with amplitudes. In fact, in the presence of M particles with momenta $k_{ { [r] } }= E_{ { [r] } }\, {{\underline{k}}}_{T+N_{ { [r] } }}$ ($r=1,2, \cdots M$) we should be obliged to consider different frames $E_{ { [r] } }$. This implies the use of DDF operators ${{\underline{A}}}^i_n( E_{ { [r] } })$ each of which contains different flat coordinates left-moving string coordinates ${{\underline{L}}}_{ { [r] } }^\mu (z)$ then

$$\begin{aligned} e^{i n \frac{ {{\underline{L}}}_{ { [r] } }(z) }{ \alpha ' {{\underline{p}}}^+_{0 { [r] } }} } = e^{i n \frac{ E_{ { [r] } \mu }^{\underline{+}} L^\mu (z) }{ \alpha 'E_{ { [r] } \mu }^{\underline{+}} p^\mu _{0}} }. \end{aligned}$$

(4.16)

This complicates the evaluation of DDF amplitudes. The simplest point of view for evaluating amplitudes is to keep $E=E_0$ fixed and vary ${{\underline{k}}}_T$. This means we have many $k_{ { [r] } }$, explicitly we write

$$\begin{aligned} k_{ { [r] } } = E_{ { [r] } }\, {{\underline{k}}}_{T+N_{ { [r] } }} = E_0\, {{\underline{k}}}_{T+N_{ { [r] } } { [r] } }. \end{aligned}$$

(4.17)

This is always possible since for any $k_{ { [r] } }$ we can determine the corresponding momentum at fixed $E_0$ as ${{\underline{k}}}_{T+N_{ { [r] } } { [r] } } = E_0^{-1}\,k_{ { [r] } } = E_0^{-1}\, E_{ { [r] } }\, {{\underline{k}}}_{T+N_{ { [r] } }}$.

However, this implies that polarizations are not anymore “absolute” but they represent quantities measured in different frames, so they cannot be compared directly.

These two points of view can be seen explicitly in the case of M photons where we write

$$\begin{aligned}&\epsilon _{ { [r] } \mu }(k_{ { [r] } }) = \underline{\epsilon }_i\, \left( E_{ { [r] } \mu }^{\underline{i}} - \frac{E^{\underline{i}}_{ { [r] } \rho } k^\rho _{ { [r] } }}{ E^{\underline{+}}_{ { [r] } \rho } k^\rho _{ { [r] } }} E_{ { [r] } \mu }^{\underline{+}} \right) ,\nonumber \\&k_{ { [r] } \mu } = E_{\mu { [r] } }^{\underline{\nu }}\, {{\underline{k}}}_{T+1\, \nu } = \underline{\epsilon }_{ { [r] } i}\, \left( E_{0 \mu }^{\underline{i}} - \frac{E^{\underline{i}}_{0 \rho } k^\rho _{ { [r] } }}{ E^{\underline{+}}_{0 \rho } k^\rho _{ { [r] } }} E_{0 \mu }^{\underline{+}} \right) , \nonumber \\&k_{ { [r] } \mu } = E_{0 \mu }^{\underline{\nu }}\, {{\underline{k}}}_{ { [r] } T+1\, \nu } , \end{aligned}$$

(4.18)

where the transversality condition reads in the first case

$$\begin{aligned} \epsilon _{ { [r] } \mu }(k_{ { [r] } })\, k_{ { [r] } \mu } = \underline{\epsilon }_i\, {{\underline{k}}}_{T+1}^i, \end{aligned}$$

(4.19)

so that it is always satisfied because of the same flat coordinate condition and in the second case

$$\begin{aligned} \epsilon _{ { [r] } \mu }(k_{ { [r] } })\, k_{ { [r] } \mu } = \underline{\epsilon }_{ { [r] } i}\, {{\underline{k}}}_{ { [r] } T+1}^i, \end{aligned}$$

(4.20)

so that it is satisfied because we have adapted the flat polarization to the different flat frames.

5 Map** lightcone operators into framed DDFs

The results of the previous section are naturally understood as the embedding of lightcone Fock space into covariant Fock space. The local frame E allows the original lightcone to be rotated to point in any possible lightcone direction and therefore yields the general solution to the Virasoro constraints. This is the main difference w.r.t. the usual point of view where a fixed lightcone, i.e. a fixed E is used.

Let us make this more formal. We have a family of injective algebra homomorphisms i(E) parameterized by the local frame E of the lightcone operators into the covariant ones given by

$$\begin{aligned} \alpha ^i_{n ({lc} )}\, {\mathop { \rightarrow }\limits ^{i(E)}} {{\underline{A}}}^i_n(E) , \end{aligned}$$

(5.1)

with inverse

$$\begin{aligned} {{\underline{A}}}^i_n(E),\, {\tilde{ {{\underline{A}}}}}^-_n(E) {\mathop { \rightarrow }\limits ^{i^{-1}(E)}} \alpha ^i_{n ({lc} )},\, 0 , \end{aligned}$$

(5.2)

which shows that ${\tilde{ {{\underline{A}}}}}^-_n(E)$ are the proper operators to consider since they are in the kernel and are the fundamental operators to consider in this map, since otherwise we would have the map** between lightcone composite operators and the Brower operators as ${\hat{\alpha }}^-_{n (lightcone)} \leftrightarrow {{\underline{A}}}^-_n(E)$. The fact that the improved Brower operators are the fundamental ones is also seen in the fact that they are the correct operators needed to get the off-shell gauge invariance in the space of off-shell DDF states as shown in Sect. 6.1.5.

We can also reformulate the previous paragraph by saying that when we also consider the null states generated by improved Brower operators, we can span the whole momentum space, i.e also ${{\underline{k}}}^+=0$ but not the point $k_\mu =0$.

One could think of extending the previous homomorphism to the zero mode sector as

$$\begin{aligned}&\alpha ^i_{n ({lc} )},\, x_{0 ({lc} )}^i,\, x_{0 ({lc} )}^-,\, p_{0 ({lc} )}^i,\, p_{0 ({lc} )}^-\, {\mathop { \rightarrow }\limits ^{{\hat{i}}(E)}} {{\underline{A}}}^i_n(E),\, {{\underline{x}}}_{0}^i(E),\, \nonumber \\ {}&\quad {{\underline{x}}}_{0}^-(E),\, {{\underline{p}}}_{0}^i(E),\, {{\underline{p}}}_{0}^-(E)\, , \end{aligned}$$

(5.3)

but this is tricky since the zero modes are connected to the Fock space on which we represent them and so we cannot get a clear map between algebras.

Instead we can get an homomorphism $\phi$ between the vector spaces as

$$\begin{aligned}{} & {} \phi \left( \prod _{i\, n}\, \left[ \alpha ^i_{n ({lc} )} \right] ^{N^i_n}\, |k_{- ({lc} )}, k_{i ({lc} )}\rangle \right) \nonumber \\{} & {} \quad =\prod _{i\, n}\, \left[ {{\underline{A}}}^i_{n} (E) \right] ^{N_{i\,n}}\, \Bigg | {{\underline{k}}}_{-},\, {{\underline{k}}}_{i},\, {{\underline{k}}}_{+}= \frac{ {{\underline{k}}}_i^2+\sum N_{i\,n}-1}{2 {{\underline{k}}}_-}\Bigg \rangle .\nonumber \\ \end{aligned}$$

(5.4)

Explicitly from the generic level $N=2$ lightcone state (in lightcone there is no $k_+$ dependence),

$$\begin{aligned} |G_{2 ({lc} )}\rangle&= {T}_i\, \alpha ^i_{-2 ({lc} )} |k_{- ({lc} )}, k_{i ({lc} )}\rangle \nonumber \\&\quad + {S}_{i j}\, \alpha ^i_{-1 ({lc} )} \alpha ^j_{-1 ({lc} )} |k_{-({lc} )}, k_{i ({lc} )}\rangle , \end{aligned}$$

(5.5)

we can get the covariant state (4.11) which has the “queue” of ${{\underline{\alpha }}}^+(E)$ operators. These ${{\underline{\alpha }}}^+(E)$ operators vanish on the lightcone and are therefore not present in the lightcone state.

To complete the map**, we need to compute the covariant momentum. The tachyon momentum associated with the DDF construction, can be reverse engineered and has a well-defined and computable ${{\underline{k}}}_{T +}$. It can be calculated from ${{\underline{k}}}_{T -}$, ${{\underline{k}}}_{T i}$, the level N and the mass shell condition of the tachyon.

Using the previous idea of embedding, we can now connect the physical covariant polarizations with the ones used in lightcone formulation, thus making explicit the natural idea already put forward in [4 into the lightcone Lorentz algebra (this is very similar to what was done in [48]) and by the explicit computation of the second Casimir of the Poincaré group in the lightcone formalism and in the covariant one.

As detailed in Appendix E the action of Lorentz generators on DDF ${{\underline{A}}}^i_n$ and Brower $\tilde{A}^-_n$ ($n\ge 1$) operators can be written as

$$\begin{aligned} \underline{M}^{i j}|_{DDF}&= i \sum _{n\ne 0} \frac{1}{n} {{\underline{A}}}_n^{i}(E)\, {{\underline{A}}}^j_{-n}(E) , \nonumber \\ \underline{M}^{+ i}|_{DDF}&= 0 , \nonumber \\ \underline{M}^{+ -}|_{DDF}&= \text{ not } \text{ possible } , \nonumber \\ \underline{M}^{- i}|_{DDF}&= i \sum _{m\ne 0} \frac{1}{m} {{\underline{A}}}_m^-(E) {{\underline{A}}}^i_{-m}(E) \nonumber \\&= i \sum _{m\ne 0} \frac{1}{m} {{\tilde{ {{\underline{A}}}}}}_m^-(E) {{\underline{A}}}^i_{-m}(E)\nonumber \\&\quad +\frac{i}{ 2 {{\underline{\alpha }}}_0^+} \sum _{m\ne 0} \frac{1}{m} \mathcal{L} _m(E) {{\underline{A}}}^i_{-m}(E) . \end{aligned}$$

(5.6)

Few comments are needed. The generator $\underline{M}^{+ -}$ acts only on ${{\underline{A}}}^-$ and requires the use of zero modes. In the same way the generator $\underline{M}^{- i}|_{DDF}$ also has a part which depends on zero modes. These two points are discussed in the Appendix E.

Finally, note that the generator $\underline{M}^{- i}|_{DDF}$ is not normal-ordered – the difference from the normal-ordered version is a term that gives no change on the variations of DDFs.

5.1 Comparison with the lightcone expressions

If we use the lightcone gauge where $\alpha ^+_{0 ({lc} )}$ is the only non vanishing operator, from the Virasoro conditions we get ($n\ne 0$)

$$\begin{aligned} L_n&= \frac{1}{2}\sum _m \alpha _{n-m} \cdot \alpha _{m} ~\Rightarrow ~ -\alpha ^+_{0 ({lc} )} {{\hat{\alpha }}}^-_n \nonumber \\ {}&\quad + \frac{1}{2}\sum _m \alpha _{n-m ({lc} )}^i \alpha _{m ({lc} )}^i = 0 \nonumber \\&\Longrightarrow {{\hat{\alpha }}}^-_n = \frac{1}{2 \alpha ^+_{0 ({lc} )} } \sum _m \alpha _{n-m ({lc} )}^i \alpha _{m ({lc} )}^i . \end{aligned}$$

(5.7)

We can then compute for $n\ne 0$

$$\begin{aligned} {[}\alpha ^+_{0 ({lc} )} \alpha ^j_{n ({lc} )},\, \alpha ^-_{m ({lc} )}] = n\, \alpha ^j_{n+m ({lc} )} , \end{aligned}$$

(5.8)

and then the lightcone Lorentz generators

$$\begin{aligned} M^{+ i}_{ ({lc} ) }&= x_0^+ p_0^i - x_0^i p_0^+ , \nonumber \\ M^{- +}_{ ({lc} ) }&= x_0^- p_0^+ - x_0^+ p_0^- , \nonumber \\ M^{- i}_{ ({lc} ) }&= x_0^- p_0^i - x_0^i p_0^- \nonumber \\&\quad - i \sum _{n=1}^\infty \frac{ {\hat{\alpha }}^-_{-n ({lc} )} \alpha ^i_{n ({lc} )} - \alpha ^i_{-n ({lc} )} {\hat{\alpha }}^-_{n ({lc} )} }{n} , \end{aligned}$$

(5.9)

which matches the expression for the DDF in (5.6) when this is restricted to non-zero modes, and we make the map between DDF and Brower operators and the lightcone ones for $n\ne 0$ as

$$\begin{aligned} {{\underline{A}}}^i_n(E) \leftrightarrow \alpha ^i_{n ({lc} )} ,~~ {{\underline{A}}}^-_n(E) \leftrightarrow {\hat{\alpha }}^-_{n ({lc} )} ,~~ {\tilde{ {{\underline{A}}}}}^-_n(E) \leftrightarrow 0 ,\nonumber \\ \end{aligned}$$

(5.10)

where the map** for the improved Brower operators is zero since they contain the transverse Virasoro generators $\mathcal{L} _n \leftrightarrow L_{n ({lc} )}^{(transverse)}\sim {\hat{\alpha }}^-_{n ({lc} )}$ so it cancels, as follows from the first map** rule.

6 Examples of framed DDF states

In this section we discuss the lowest levels in order to clarify the ideas and observations put forward in the previous section. We start with the massless $N=1$ level and then discuss the $N=2$ level. A similar discussion was done in the early days of string theory in [46] where the physical and null states were discussed in $D=4$ in the covariant formalism.

In particular, we are interested in discussing the gauge chosen by DDF states, the local and global Lorentz transformations, and the role of the Brower states, also in connection with the off-shell extension of the states. On shell we recover the old results [48,49,50,51]. However it turns out that off-shell an infinitesimal change of frame or an infinitesimal boost involving the lightcone is equivalent to performing a shift of the original state by an off-shell Brower state. Off-shell Brower states are not anymore null and therefore they are not BRST exact. This can be interpreted as the fact that off-shell Brower states replace the gauge symmetry generated by Virasoro operators, i.e. BRST invariance.

6.1 Level 1 states

We start with the simplest case of the photon where computations can also be done on the back of an envelope.

6.1.1 Level 1 DDF states

The $N=1$ DDF reads

$$\begin{aligned} {{\underline{A}}}^i_{-1} |k_T\rangle = \left[ {{\underline{\alpha }}}^i_{-1} - \frac{ {{\underline{k}}}^i }{ {{\underline{k}}}^+ } {{\underline{\alpha }}}^+_{-1} \right] | {{\underline{k}}}_{T + 1}^-, {{\underline{k}}}_{T}^+, {{\underline{k}}}_{T }^i \rangle , \end{aligned}$$

(6.1)

so that the physical momentum is related to the tachyon momentum as and the only momentum component which differs from ${{\underline{k}}}_T$ is ${{\underline{k}}}_{T}^-$, explicitly

$$\begin{aligned}{} & {} {{\underline{k}}}^-_{T+1} = - {{\underline{k}}}_{T+1\,+} = {{\underline{k}}}_{T +}^- + \frac{1}{2\alpha ' {{\underline{k}}}^+}, \nonumber \\{} & {} {{\underline{k}}}_- = - {{\underline{k}}}^+ = {{\underline{k}}}_{T -},\quad {{\underline{k}}}_i = {{\underline{k}}}^i = {{\underline{k}}}_{T i}. \end{aligned}$$

(6.2)

Comparing with the most general covariant state

$$\begin{aligned} |\epsilon , k\rangle = {{\underline{\epsilon }}}_\mu {{\underline{\alpha }}}^\mu _{-1} | {{\underline{k}}}_\nu \rangle = \epsilon _\mu \alpha ^\mu _{-1} | k_\nu \rangle , \end{aligned}$$

(6.3)

we get that the components of the flat polarization are

$$\begin{aligned} \underline{\epsilon }^{(i)}_j= & {} \delta ^i_j,\quad \underline{\epsilon }^{(i)}_+=- \frac{ {{\underline{k}}}^i }{ {{\underline{k}}}^+},\nonumber \\ \underline{\epsilon }^{(i)}_-= & {} 0 ~~ \leftrightarrow ~~ \underline{\epsilon }^{(i)}_\mu = \delta ^i_\mu - \frac{ {{\underline{k}}}^i }{ {{\underline{k}}}^+} \delta ^+_\mu , \end{aligned}$$

(6.4)

and in curved ones are

$$\begin{aligned} {\epsilon }^{(i)}_\mu = E^{\underline{i}}_\mu - \frac{E^{\underline{i}}_\rho k^\rho }{ E^{\underline{+}}_\sigma k^\sigma } E^{\underline{+}}_\mu . \end{aligned}$$

(6.5)

This form reveals the projector discussed in (2.20).

The only Virasoro condition reads

$$\begin{aligned} L_1 |\epsilon , k\rangle = 0 ~~\Rightarrow ~~ k^\mu \epsilon _\mu =0, \end{aligned}$$

(6.6)

and it is trivially satisfied because of the projector.

It can also be verified in flat coordinates as

$$\begin{aligned} k^\mu \epsilon _\mu = {{\underline{k}}}^i_T \cdot 1 + {{\underline{k}}}^+_T \cdot \frac{ {{\underline{k}}}^i_T }{ {{\underline{k}}}_{ T -}} \equiv 0. \end{aligned}$$

(6.7)

6.1.2 Level 1 Brower state

In the discussion on the gauge transformation below we use the “Brower state”

$$\begin{aligned} {{\underline{A}}}^-_{-1} | {{\underline{k}}}_T \rangle&= \left\{ \left[ e^{-i \frac{ {{\underline{x}}}_0^+}{2\alpha ' {{\underline{p}}}_0^+} } \left( {{\underline{\alpha }}}^-_{-1} - \frac{ {{\underline{\alpha }}}_0^-}{ {{\underline{\alpha }}}_0^+} {{\underline{\alpha }}}_{-1}^+ \right) \right] \right. \nonumber \\&\quad \left. -\frac{1}{2 {{\underline{\alpha }}}_0^+} \left[ 2\, e^{-i \frac{ {{\underline{x}}}_0^+}{2\alpha ' {{\underline{p}}}_0^+} } \frac{ {{\underline{\alpha }}}_{-1}^+}{ {{\underline{\alpha }}}_0^+} \right] \right\} | {{\underline{k}}}_T \rangle , \end{aligned}$$

(6.8)

where the $[ \dots ]$ are the contributions from the two terms in ${{\underline{A}}}^-$. Notice the effect of normal ordering in the first contribution. Evaluating explicitly the previous expression we get

$$\begin{aligned}&= \left[ {{\underline{\alpha }}}^-_{-1} - \frac{1}{ {{\underline{k}}}^+} \left( {{\underline{k}}}_T^- + \frac{1}{2\alpha ' {{\underline{k}}}^+} \right) {{\underline{\alpha }}}_{-1}^+ \right] | {{\underline{k}}}^-\nonumber \\&= {{\underline{k}}}_T^- + \frac{1}{2\alpha ' {{\underline{k}}}^+}\, {{\underline{k}}}^+ = {{\underline{k}}}_T^+ \rangle \nonumber \\&= \left[ {{\underline{\alpha }}}^-_{-1} - \frac{ {{\underline{k}}}^-_{T+1}}{ {{\underline{k}}}^+} {{\underline{\alpha }}}_{-1}^+ \right] | {{\underline{k}}}_{T+1}\rangle . \end{aligned}$$

(6.9)

Again the flat polarization is given by

$$\begin{aligned} \underline{\epsilon }^{(-)}_\mu = \delta ^-_\mu - \frac{ {{\underline{k}}}^-_{T+1} }{ {{\underline{k}}}^+} \delta ^+_\mu , \end{aligned}$$

(6.10)

and in curved one is

$$\begin{aligned} {\epsilon }^{(-)}_\mu = E^{\underline{-}}_\mu - \frac{E^{\underline{-}}_\rho k^\rho }{ E^{\underline{+}}_\sigma k^\sigma } E^{\underline{+}}_\mu . \end{aligned}$$

(6.11)

This expression satisfies the Virasoro conditions

$$\begin{aligned} L_1\, {{\underline{A}}}^-_{-1} | {{\underline{k}}}_T \rangle =&(- {{\underline{\alpha }}}_0^+ {{\underline{\alpha }}}_{-1}^- - {{\underline{\alpha }}}_0^- {{\underline{\alpha }}}_{-1}^+ ) {{\underline{A}}}^-_{-1} | {{\underline{k}}}_T \rangle =0 , \end{aligned}$$

(6.12)

in an obvious way because of the projector $\Pi ^-$.

It is also important to compute the norm of this state, which may also be negative when off-shell. This can be achieved in two ways - either by direct computation on the explicit expression

$$\begin{aligned}&\langle {{\underline{k}}}_{T+1} | \left[ {{\underline{\alpha }}}^-_{1} - \frac{ {{\underline{k}}}^-_{T+1}}{ {{\underline{k}}}^+} {{\underline{\alpha }}}_{1}^+ \right] \, \left[ {{\underline{\alpha }}}^-_{-1} - \frac{ {{\underline{l}}}^-_{T+1}}{ {{\underline{l}}}^+} {{\underline{\alpha }}}_{-1}^+ \right] | {{\underline{l}}}_{T+1}\rangle \nonumber \\&\quad = 2 \frac{ {{\underline{k}}}^-_{T+1}}{ {{\underline{k}}}^+} \, \delta ( {{\underline{k}}}- {{\underline{l}}})\, , \end{aligned}$$

(6.13)

or by using the ${{\underline{A}}}^-_m$ algebra (see (3.13)) as,

$$\begin{aligned} \langle {{\underline{k}}}_T | {{\underline{A}}}_{1}^-\, {{\underline{A}}}_{-1}^- | {{\underline{l}}}_T\rangle&= \langle {{\underline{k}}}_T | \left( 2 \frac{ {{\underline{A}}}_{0}^-}{ {{\underline{\alpha }}}_0^+} + 2 \frac{(1)^3}{( {{\underline{\alpha }}}_0^+)^2} \right) | {{\underline{l}}}_T\rangle \nonumber \\&= 2 \frac{ {{\underline{k}}}^-_{T+1}}{ {{\underline{k}}}^+} \, \delta ( {{\underline{k}}}_T - {{\underline{l}}}_T)\, . \end{aligned}$$

(6.14)

Notice the different bras and kets in the two computations.

Actually we are also interested in the “improved Brower state”

$$\begin{aligned} {\tilde{ {{\underline{A}}}}}^-_{-1} | {{\underline{k}}}_T \rangle&= \left( {{\underline{A}}}^-_{-1} - \frac{1}{2 {{\underline{\alpha }}}_0^+} 2 \vec {{\underline{A}}}_{-1} \cdot \vec {{\underline{A}}}_{0} \right) | {{\underline{k}}}_T \rangle \nonumber \\&= \left[ {{\underline{\alpha }}}_{-1}^- + \frac{ -2 {{\underline{k}}}^-_{T+1} {{\underline{k}}}^+ + 2 \vec {{\underline{k}}}^2}{ 2 ( {{\underline{k}}}^+)^2} {{\underline{\alpha }}}_{-1}^+ - \frac{ {{\underline{k}}}^j}{ {{\underline{k}}}^+} {{\underline{\alpha }}}_{-1}^j \right] \nonumber \\&\quad \times | {{\underline{k}}}_{T+1} \rangle , \end{aligned}$$

(6.15)

with norm

$$\begin{aligned} \langle {{\underline{k}}}_T| {\tilde{ {{\underline{A}}}}}^-_{1} {\tilde{ {{\underline{A}}}}}^-_{-1} | {{\underline{l}}}_T \rangle&= \langle {{\underline{k}}}_T| \left[ 2 \frac{1}{ {{\underline{\alpha }}}^+_0 } {\tilde{ {{\underline{A}}}}}^-_{0} + \frac{26-D}{12} \frac{1}{ ( {{\underline{\alpha }}}^+_0)^2 } \right] | {{\underline{l}}}_T \rangle \nonumber \\&= - \frac{-2 {{\underline{k}}}^-_{T+1} {{\underline{k}}}^+ + \vec {{\underline{k}}}^2}{ 2 ( {{\underline{k}}}^+)^2} \delta ( {{\underline{k}}}- {{\underline{l}}}) , \end{aligned}$$

(6.16)

which vanishes on-shell only.

6.1.3 Global Lorentz rotations

As we stated at the beginning the local frame $E^\mu _{\underline{\mu }}$ is acted upon by two independent groups: the local Lorentz group, i.e. the group of tangent space rotations, and the global Lorentz group. Any global Lorentz transformation can be reabsorbed into E without changing ${{\underline{A}}}$. More explicitly we have

$$\begin{aligned} {{\underline{\epsilon }}}^{(i)}_j\, {{\underline{A}}}^j_{-1}(E) {|{ {{\underline{k}}}_T}\rangle }&= \epsilon ^{(i)}_\mu (E)\, \alpha _{-1}^\mu |k\rangle ~~ \Rightarrow ~~ \epsilon ^{(i)}_\mu (E)\nonumber \\&= {{\underline{\epsilon }}}^{(i)}_j \left( E^{\underline{j}}_\mu - \frac{E^{\underline{j}}_\nu k^\nu }{ E^ {\underline{+}}_\nu k^\nu } E^ {\underline{+}}_\mu \right) , \end{aligned}$$

(6.17)

so that a global Lorentz transformation acts trivially on framed DDF polarization, or said in a different way, framed DDF polarizations are Lorentz invariants with respect to global rotations. This means that the generic solution of the Virasoro conditions can be expressed using a set of parameters, the DDF polarizations which are invariant under global Lorentz transformations at the cost of changing E.

However, when two or more states are present we generically need as many DDF polarizations in a given frame E as states, but these polarizations describe the states in all possible global coordinates which are related by a global Lorentz transformation. In fact, for two vectors the scalar product is invariant

$$\begin{aligned} \epsilon _{ { [1] } }^{(i_1)}(k_{ { [1] } }) \cdot \epsilon _{ { [2] } }^{(i_2)}(k_{ { [2] } })&= {{\underline{\epsilon }}}_{ { [1] } j}^{(i_1)}\, \delta ^{j l}\, {{\underline{\epsilon }}}_{ { [2] } l}^{(i_2)} , \end{aligned}$$

(6.18)

and can be reproduced using two fixed DDF polarizations.

6.1.4 Local Lorentz rotations 1: on-shell gauge equivalence of different frames

Suppose that we want to describe the very same physical state using two different frames E and ${\hat{E}}$. Since the physical state is the same these two frames are related by a tangent space Lorentz rotation. Doing so, we actually get two different states that describe the same physical state and must therefore be gauge equivalent, i.e.

$$\begin{aligned} | k, \epsilon _{{\hat{E}}}\rangle = | k, \epsilon _{E}\rangle + L_{-1} \xi | k\rangle , \end{aligned}$$

(6.19)

for some number $\xi$.

Notice that the previous equality is valid for states with both ${{\underline{k}}}^+= E^{\underline{+}}_\mu k^\mu \ne 0$ and ${\widehat{ {{\underline{k}}}}}^+= {{\hat{E}}}^{\underline{+}}_\mu k^\mu \ne 0$ and therefore can be understood as a transition between two charts in the space of momenta. In particular, as discussed in the next section, this means that on-shell the “analytic continuation” of one frame only can be used to compute all the on-shell amplitudes.

When we rewrite the previous equivalence using DDF states we get

$$\begin{aligned} {{\underline{\epsilon }}}_{({\hat{E}}) i} {{\underline{A}}}^i_{-1}({\hat{E}}) | {{\underline{k}}}_{T ({\hat{E}})}\rangle = {{\underline{\epsilon }}}_{(E) i} {{\underline{A}}}^i_{-1}(E) | {{\underline{k}}}_{T (E)}\rangle + L_{-1}\, \xi | k\rangle , \end{aligned}$$

(6.20)

and we require that both DDF states have the same physical momentum $|k\rangle$. This means that

$$\begin{aligned}&E^\mu _{\underline{\mu }} k_\mu = \left( {{\underline{k}}}_{T (E) +} + \frac{1}{2\alpha ' {{\underline{k}}}_{ (E) -}}, {{\underline{k}}}_{ (E) -}, {{\underline{k}}}_{ (E) i} \right) ,~~~~\nonumber \\&{\hat{E}}^\mu _{\underline{\mu }} k_\mu = \left( {{\underline{k}}}_{T ({\hat{E}}) +} + \frac{1}{2\alpha ' {{\underline{k}}}_{({\hat{E}}) -}}, {{\underline{k}}}_{({\hat{E}}) -}, {{\underline{k}}}_{({\hat{E}}) i} \right) , \end{aligned}$$

(6.21)

or

$$\begin{aligned}&\left( {{\underline{k}}}_{T ({\hat{E}}) +} + \frac{1}{2\alpha ' {{\underline{k}}}_{({\hat{E}}) -}}, {{\underline{k}}}_{({\hat{E}}) -}, {{\underline{k}}}_{({\hat{E}}) i} \right) \nonumber \\&\quad = E^{-1} {\hat{E}}\, \left( {{\underline{k}}}_{T (E) +} + \frac{1}{2\alpha ' {{\underline{k}}}_{(E) -}}, {{\underline{k}}}_{(E) -}, {{\underline{k}}}_{(E) i} \right) . \end{aligned}$$

(6.22)

Then for the polarization we get

$$\begin{aligned} {{\underline{\epsilon }}}_{({\hat{E}}) i} {\hat{E}}^{\underline{i}}_\mu = {{\underline{\epsilon }}}_{(E) i} E^{\underline{i}}_\mu + \xi k_\mu . \end{aligned}$$

(6.23)

This is a set of D equations in $D-1$ unknown, i.e. ${{\underline{\epsilon }}}_{(E) i}$ and $\xi$. The system is soluble on-shell where $k^2=0$ and $k\cdot \epsilon =0$. In fact contracting with k and (an auxiliary null vector) ${\bar{k}}$ ($k\cdot {\bar{k}} =-1$) we get

$$\begin{aligned} {{\underline{\epsilon }}}_{({\hat{E}}) i} {{\underline{k}}}^i_{{\hat{E}}}&= {{\underline{\epsilon }}}_{(E) i} {{\underline{k}}}^i_{E} + \xi k^2 ~~\Rightarrow ~~ 0= \xi k^2 , \nonumber \\ {{\underline{\epsilon }}}_{({\hat{E}}) i} {{\bar{ {{\underline{k}}}}}}^i_{{\hat{E}}}&= {{\underline{\epsilon }}}_{(E) i} {{\bar{ {{\underline{k}}}}}}^i_{E} - \xi , \end{aligned}$$

(6.24)

where the first equation is identically zero on-shell and the second yields $\xi$.

6.1.5 Local Lorentz rotations 2: off-shell gauge equivalence of different local frames

The result of the previous section is annoying since it is valid only when the states are on-shell, while we would like to be able to go off-shell.

There is a way out using the Brower states generated by ${\tilde{{\underline{A}}}}^-$. Actually the two different (off-shell) states must be gauge equivalent as

$$\begin{aligned} | k, \epsilon _{({\hat{E}})}\rangle = | k, \epsilon _{(E)}\rangle + \zeta \, {{\tilde{ {{\underline{A}}}}}}^-_{-1}(E) | {{\underline{k}}}_T\rangle . \end{aligned}$$

(6.25)

However, while the off-shell Brower state satisfies all Virasoro conditions but the $L_0$ one, it may have negative norm and only when it is on-shell it has zero norm.

As before the previous equality is valid for states with both ${{\underline{k}}}^+= E^{\underline{+}}_\mu k^\mu \ne 0$ and ${\widehat{ {{\underline{k}}}}}^+= {{\hat{E}}}^{\underline{+}}_\mu k^\mu \ne 0$ and therefore can be understood as a transition between two charts in the space of momenta. The interpretation is that when we consider on-shell states where the states generated by improved Brower operators are null, only one frame, e.g. E is sufficient to compute all the on-shell amplitudes by “analytic continuations”. This is no longer true for off-shell amplitudes.

Moreover this gauge symmetry is related to local frame rotations and replaces off-shell the gauge symmetry generated by Virasoro operators, i.e. BRST.

Again, it is a system of D equations in $D-1$ unknowns ${{\underline{\epsilon }}}_i$ and $\zeta$ when ${\hat{ {{\underline{\epsilon }}}}}_i$, ${\hat{E}}$, E and k are given. However, it is soluble because the action of $L_1$ kills both the right and left sides. Let us discuss this in more detail. Equating the states in global operator basis, we get (using (6.15))

$$\begin{aligned} {\hat{ {{\underline{\epsilon }}}}}_i\, \Pi ^{\underline{i}}_\mu ({\hat{E}})&= {{\underline{\epsilon }}}_i\, \Pi ^{\underline{i}}_\mu (E)\nonumber \\&\quad + \zeta \left( E_{\mu }^{\underline{-}} + \frac{ -2 {{\underline{k}}}^-_{T+1} {{\underline{k}}}^+ + 2 \vec {{\underline{k}}}^2}{ 2 ( {{\underline{k}}}^+)^2} E_{\mu }^{\underline{+}} - \frac{ {{\underline{k}}}^j}{ {{\underline{k}}}^+} E_{\mu }^{\underline{j}} \right) . \end{aligned}$$

(6.26)

Multiplying by $E^\mu _{\underline{-}}$ we can extract $\zeta$ as

$$\begin{aligned} \zeta&= {\hat{ {{\underline{\epsilon }}}}}_i\, \Pi ^{\underline{i}}_\mu ({\hat{E}})\, E^\mu _{\underline{-}} , \end{aligned}$$

(6.27)

and by $E^\mu _{\underline{i}}$ we can extract ${{\underline{\epsilon }}}_i$ as

$$\begin{aligned} {{\underline{\epsilon }}}_i&= {\hat{ {{\underline{\epsilon }}}}}_j\, \Pi ^{\underline{j}}_\mu ({\hat{E}})\, E^\mu _{\underline{i}} - \frac{ {{\underline{k}}}_i}{ {{\underline{k}}}^+}\, {\hat{ {{\underline{\epsilon }}}}}_j\, \Pi ^{\underline{j}}_\mu ({\hat{E}})\, E^\mu _{\underline{-}} . \end{aligned}$$

(6.28)

The equation which is obtained by multiplying with $E^\mu _{\underline{+}}$ is then identically satisfied.

6.2 Level 2 states

We now quickly examine the next level since computations are heavier and not very illuminating. In this case, we encounter for the first time two kinds of DDF state, even though they belong to the same $D=26$ irrep.

6.2.1 Level 2 state from ${{\underline{A}}}^i_{-2}$

Similar to the previous example using (2.10) we now compute

$$\begin{aligned} {{\underline{A}}}^i_{-2} |k_T\rangle&= \left[ {{\underline{\alpha }}}^i_{-2} - \frac{2}{ \sqrt{2\alpha '} {{\underline{k}}}_{ T}^+ } {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^i_{-1}\right. \nonumber \\&\quad \left. + \left( \frac{-1}{ \sqrt{2\alpha '} {{\underline{k}}}_{ T}^+} {{\underline{\alpha }}}^+_{-2} + \frac{1}{ ( \sqrt{2\alpha '} {{\underline{k}}}_{ T}^+)^2 } {{\underline{\alpha }}}^+_{-1} {{\underline{\alpha }}}^+_{-1} \right) \sqrt{2\alpha '} {{\underline{k}}}^i_T \right] \nonumber \\&\quad \times \bigg |{ {{\underline{k}}}_{T}^- + \frac{2 }{2\alpha ' {{\underline{k}}}_{ T}^+}, {{\underline{k}}}_{T}^+, {{\underline{k}}}_{T i}}\bigg \rangle \nonumber \\&= \left( T^{(i)}_\mu \, \alpha ^\mu _{-2} + S^{(i)}_{\mu \nu }\, \alpha ^\mu _{-1} \alpha ^\nu _{-1} \right) |k_\nu \rangle , \end{aligned}$$

(6.29)

so that the covariant polarizations are

$$\begin{aligned} T^{(i)}_\mu&= E^{\underline{i}}_{\mu } - \frac{ {{\underline{k}}}^i_T }{ {{\underline{k}}}_{ T}^+} E^{\underline{+}}_{\mu } = \Pi ^{\underline{i}}_\mu (E) , \nonumber \\ S^{(i)}_{\mu \nu }&= S^{(i)}_{\nu \mu } = -\frac{2}{ \sqrt{2\alpha '} {{\underline{k}}}_{T}^+} E^{\underline{+}}_{(\mu }~ E^{\underline{i}}_{\nu )} - 2 \frac{ {{\underline{k}}}^i_T }{ {{\underline{k}}}_{ T}^+} \frac{1}{ \sqrt{2\alpha '} {{\underline{k}}}_{T -}} E^{\underline{+}}_{\mu }~E^{\underline{+}}_{\nu } \nonumber \\&= \Pi ^{\underline{i}}_{( \mu }(E)\, E^{\underline{+}}_{\nu )} \frac{N}{ \sqrt{2\alpha '} {{\underline{k}}}^+} \frac{1}{1!} , \end{aligned}$$

(6.30)

and the momentum is

$$\begin{aligned} {{\underline{k}}}= {{\underline{k}}}_{T + 2}. \end{aligned}$$

(6.31)

The Virasoro conditions are given by,

$$\begin{aligned} L_2:~ S^{\mu }_\mu + 2 \sqrt{2\alpha '} k^\mu T_\mu = 0,\quad L_1:~ \sqrt{2\alpha '} k^\nu S_{\nu \mu } + T_\mu = 0,\nonumber \\ \end{aligned}$$

(6.32)

and can be verified using the expressions for the covariant polarizations above. In particular using the properties of the projector

$$\begin{aligned} k^\mu \, \Pi ^i_\mu (E) = \Pi ^i_\mu (E)\, E^+_\nu \eta ^{\mu \nu } = 0, \end{aligned}$$

(6.33)

we see that T is transverse, S is traceless but not transverse so $L_2$ condition is automatic while $L_1$ must be checked.

6.2.2 Level 2 from ${{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1}$

For two DDF operators successively acting on a pure momentum state we have some self-interactions.

The corresponding level 2 DDF state is given by

$$\begin{aligned} {{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1}{|{k_T}\rangle }&= \left\{ \left( {{\underline{\alpha }}}^i_{-1} + \frac{ {{\underline{k}}}^i_T}{ {{\underline{k}}}_{T-}} {{\underline{\alpha }}}^+_{-1} \right) \left( {{\underline{\alpha }}}^j_{-1} + \frac{ {{\underline{k}}}^j_T}{ {{\underline{k}}}_{T-}} {{\underline{\alpha }}}^+_{-1}\right) \right. \nonumber \\&\left. \quad + \delta ^{ij} \left[ \frac{1}{ \sqrt{2\alpha '} {{\underline{k}}}_{T-}}\frac{ {{\underline{\alpha }}}^+_{-2}}{2} + \frac{1}{2}\left( \frac{ {{\underline{\alpha }}}^+_{-1}}{ \sqrt{2\alpha '} {{\underline{k}}}_{T-}}\right) ^2 \right] \right\} \nonumber \\&\quad \times {|{ {{\underline{k}}}_{T}^- + \frac{2}{2\alpha ' {{\underline{k}}}_{T}^+}, {{\underline{k}}}_{T}^+, {{\underline{k}}}_{T}^i}\rangle }\nonumber \\&= \left( T^{(ij)}_\mu \, \alpha ^\mu _{-2} + S^{(ij)}_{\mu \nu }\, \alpha ^\mu _{-1} \alpha ^\nu _{-1} \right) |k_\nu \rangle , \end{aligned}$$

(6.34)

with curved polarizations

$$\begin{aligned} T^{(i j)}_\mu&= \frac{\delta ^{ij}}{2 \sqrt{2\alpha '} {{\underline{k}}}_{T-}} E^{\underline{+}}_\mu , \end{aligned}$$

(6.35)

$$\begin{aligned} S^{(i j)}_{\mu \nu }&= \Pi ^{\underline{i}}_{ (\mu }(E)\, \Pi ^{\underline{j}}_{ \nu )}(E) + \frac{\delta ^{ij}}{2( \sqrt{2\alpha '} {{\underline{k}}}_{T}^+)^2} E^{\underline{+}}_\mu \, E^{\underline{+}}_\nu . \end{aligned}$$

(6.36)

The polarizations for $i\ne j$ are transverse and traceless since $\Pi ^{\underline{i}}_{ (\mu }(E)\, \Pi ^{\underline{j}}_{ \nu )}(E)\, \eta ^{\mu \nu } = \delta ^{i j}$. On the contrary the ones for $i=j$ are neither transverse nor traceless.

As before the Virasoro conditions read,

$$\begin{aligned} L_2:~ S^{\mu }_\mu + 2 \sqrt{2\alpha '} k^\mu T_\mu = 0, \ L_1:~ \sqrt{2\alpha '} k^\nu S_{\nu \mu } + T_\mu = 0, \end{aligned}$$

(6.37)

and can be verified as follows,

$$\begin{aligned} S^\mu _\mu&= {{\underline{S}}}_{ii} = 1; ~ 2 \sqrt{2\alpha '} k^\mu T_\mu = - \delta ^{ii} = -1 \end{aligned}$$

(6.38)

$$\begin{aligned}&\implies S^{\mu }_\mu + 2 \sqrt{2\alpha '} k^\mu T_\mu = 0. \end{aligned}$$

(6.39)

and,

$$\begin{aligned} \sqrt{2\alpha '} k^\nu S_{\nu i}&= \sqrt{2\alpha '} \left( {{\underline{k}}}_T^+\frac{ {{\underline{k}}}_T^i}{ {{\underline{k}}}_{T-}} + {{\underline{k}}}^i_T\right) \nonumber \\&= (- {{\underline{k}}}^i_T + {{\underline{k}}}^i_T) = 0; \end{aligned}$$

(6.40)

$$\begin{aligned} \sqrt{2\alpha '} k^\nu S_{\nu +}&= \sqrt{2\alpha '} \left( {{\underline{k}}}_T^+\left\{ \frac{ {{\underline{k}}}^i_T {{\underline{k}}}^j_T}{ {{\underline{k}}}_{T-}^2} + \frac{\delta ^{ij}}{4\alpha ' {{\underline{k}}}_{T-}^2}\right\} \right) \nonumber \\&\quad + \frac{ {{\underline{k}}}^i_T {{\underline{k}}}^j_T}{ {{\underline{k}}}_{T-}} +\frac{\delta ^{ij}}{2 \sqrt{2\alpha '} {{\underline{k}}}_{T-}} = 0 \end{aligned}$$

(6.41)

$$\begin{aligned}&\implies \sqrt{2\alpha '} k^\nu S_{\nu \mu } + T_\mu = 0 . \end{aligned}$$

(6.42)

6.2.3 Level 2 Brower states in ${{\underline{k}}}^i=0$ frame

To make computations easier we can perform them in the ${{\underline{k}}}^i=0$ frame where the particle can still move in $x^1$ direction and therefore it is not the rest frame even if the rest frame belongs to this class.

Despite this “nice” kinematics, the results are quite opaque.

In this frame we have the following partial simplification

$$\begin{aligned} \Pi ^{\underline{i}}_\mu ~\rightarrow ~ E^{\underline{i}}_\mu ~~ \text{ but } ~~ \Pi ^{\underline{-}}_\mu (E, {{\underline{\alpha }}}_0^-) = E^{\underline{-}}_\mu - \frac{ {{\underline{\alpha }}}_0^-}{ {{\underline{k}}}_T^+} E^{\underline{+}}_\mu , \end{aligned}$$

(6.43)

since ${{\underline{\alpha }}}_0^-\sim {{\underline{k}}}^-\ne 0$ is not vanishing. Moreover, the actual value of ${{\underline{\alpha }}}_0^-$ in the state does depend on the order of ${{\underline{A}}}^-$ since ${{\underline{k}}}^-$ is altered by the action of the DDF and Brower operators.

We can then evaluate the following improved Brower states

$$\begin{aligned} {\tilde{ {{\underline{A}}}}}^-_{-2} | {{\underline{k}}}_T \rangle&= \left\{ \left[ {{\underline{\alpha }}}^-_{-2} - \frac{2\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^-}{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2}\, {{\underline{\alpha }}}^+_{-2} - \frac{2}{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)}\, {{\underline{\alpha }}}^-_{-1} {{\underline{\alpha }}}^+_{-1}\right. \right. \nonumber \\&\quad \left. +\frac{4\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^-}{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^3}\, \left( {{\underline{\alpha }}}^+_{-1} \right) ^2 \right] +\left[ - \frac{1}{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)}\right. \nonumber \\&\quad \times \left. \left( 3 \frac{ {{\underline{\alpha }}}_{-2}^+ }{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+) } - 5 \left( \frac{ {{\underline{\alpha }}}_{-1}^+ }{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+) }\right) ^2 \right) \right] \nonumber \\&\quad - \frac{1}{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)}\, \frac{1}{2}\left[ \vec {{\underline{\alpha }}}_{-1}^2 + \frac{D-2}{2} \left( - \frac{ {{\underline{\alpha }}}_{-2}^+ }{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+) }\right. \right. \nonumber \\&\quad \left. \left. +\frac{1}{2}\left( \frac{ {{\underline{\alpha }}}_{-1}^+ }{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+) }\right) ^2 \right) \right] | {{\underline{k}}}_{T+2}\rangle , \nonumber \\&= \Bigg \{ \left( E^{\underline{-}}_\mu + \frac{D-14 - 8\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^-}{4 ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2} E^{\underline{+}}_\mu \right) \, \alpha _{-2}^\mu \nonumber \\&\quad + \left( - \frac{2}{ \sqrt{2\alpha '} {{\underline{k}}}_T^+ }\, E^{\underline{-}}_\mu \, E^{\underline{+}}_\nu + \frac{42-D}{8 ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^3}\, E^{\underline{+}}_\mu \, E^{\underline{+}}_\nu \right. \nonumber \\&\quad \left. \left. - \frac{1}{2 ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)} E^{\underline{j}}_\mu \, E^{\underline{j}}_\nu \right) \, \alpha _{-1}^\mu \, \alpha _{-1}^\nu \right\} | {{\underline{k}}}_{T+2} \rangle , \end{aligned}$$

(6.44)

and

$$\begin{aligned}&({\tilde{ {{\underline{A}}}}}^-_{-1})^2 | {{\underline{k}}}_T \rangle \nonumber \\&\quad = \left\{ \left[ - \frac{1}{ \sqrt{2\alpha '} {{\underline{k}}}_T^+}\, {{\underline{\alpha }}}_{-2}^- + \left( {{\underline{\alpha }}}_{-1}^- \right) ^2 - \frac{4\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^- +1 }{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2}\, {{\underline{\alpha }}}_{-1}^- {{\underline{\alpha }}}_{-1}^+\right. \right. \nonumber \\&\qquad \left. + \frac{ (2\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^- +1)^2 }{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^4}\, \left( {{\underline{\alpha }}}_{-1}^+ \right) ^2 \right] \nonumber \\&\qquad + \left[ \frac{1}{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^3}\, {{\underline{\alpha }}}_{-2}^+ - \frac{1 }{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2}\, {{\underline{\alpha }}}_{-1}^- {{\underline{\alpha }}}_{-1}^+\right. \nonumber \\&\qquad \left. \left. + \frac{ 2\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^- }{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^4}\, \left( {{\underline{\alpha }}}_{-1}^+ \right) ^2 \right] \right\} | {{\underline{k}}}_{T+2}\rangle \nonumber \\&\quad = \left\{ \left( - \frac{1}{ \sqrt{2\alpha '} {{\underline{k}}}_T^+}\, E^{\underline{-}}_\mu + \frac{1}{( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^3}\, E^{\underline{+}}_\mu \right) \alpha _{-2}^\mu \right. \nonumber \\&\qquad + \Bigg ( E^{\underline{-}}_\mu \, E^{\underline{-}}_\nu + \frac{(2 \alpha ' {{\underline{k}}}_{T}^+ {{\underline{k}}}_{T}^-)^2 + 6\alpha ' {{\underline{k}}}_{T}^+ {{\underline{k}}}_{T}^- + 1 }{ ( \sqrt{2\alpha '} {{\underline{k}}}_{T}^+)^4 } E^{\underline{+}}_\mu \, E^{\underline{+}}_\nu \nonumber \\&\qquad \left. + \frac{-4\alpha ' {{\underline{k}}}_{T}^+ {{\underline{k}}}_{T}^- -2 }{( \sqrt{2\alpha '} {{\underline{k}}}_{T}^+)^2} E^{\underline{+}}_\mu \, E^{\underline{-}}_\nu \Bigg ) \alpha _{-1}^\mu \, \alpha _{-1}^\nu \right\} | {{\underline{k}}}_{T+2}\rangle , \end{aligned}$$

(6.45)

and

$$\begin{aligned}&{{\underline{A}}}^i_{-1}\,{\tilde{ {{\underline{A}}}}}^-_{-1} | {{\underline{k}}}_T \rangle \nonumber \\&\quad = \Bigg [ - \frac{ 1 }{ \sqrt{2\alpha '} {{\underline{k}}}_T^+ }\, {{\underline{\alpha }}}_{-2}^i + {{\underline{\alpha }}}_{-1}^i {{\underline{\alpha }}}_{-1}^- -\frac{2\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}^-_{T}}{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+ )^2}\, {{\underline{\alpha }}}_{-1}^i\, {{\underline{\alpha }}}_{-1}^+ \Bigg ] | {{\underline{k}}}_{T+2}\rangle \nonumber \\&\quad = \left\{ \left( - \frac{1}{ \sqrt{2\alpha '} {{\underline{k}}}_T^+} E^{\underline{+}}_\mu \right) \, \alpha ^\mu _{-2}\right. \nonumber \\&\qquad \left. + \left( E^{\underline{i}}_\mu \, E^{\underline{-}}_\mu - \frac{2\alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}^-_{T}}{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2 }\, E^{\underline{i}}_\mu \, E^{\underline{+}}_\mu \right) \alpha _{-1}^\mu \, \alpha _{-1}^\nu \right\} | {{\underline{k}}}_{T+2} \rangle , \end{aligned}$$

(6.46)

where in the state ${\tilde{ {{\underline{A}}}}}^-_{-2} | {{\underline{k}}}_T \rangle$ the first two $[\dots ]$ are ${{\underline{A}}}^-_{-2} | {{\underline{k}}}_T \rangle$ and the last one is the contribution from $\mathcal{L} _{-2}$. In particular, the first $[\dots ]$ is from $\oint \partial {{\underline{L}}}^- \dots$ and the second $[\dots ]$ is from $\oint \frac{\partial ^2 {{\underline{L}}}^+}{\partial {{\underline{L}}}^+} \dots$

In the state $({\tilde{ {{\underline{A}}}}}^-_{-1})^2 | {{\underline{k}}}_T \rangle$ the first $[\dots ]$ is from $\oint \partial {{\underline{L}}}^- \dots$ and the second $[\dots ]$ is from $\oint \frac{\partial ^2 {{\underline{L}}}^+}{\partial {{\underline{L}}}^+} \dots$. The contributions from $\mathcal{L} _{-1}$ are absent in the other states due to the choice of frame.

They have off-shell norms given by,

$$\begin{aligned}&\langle {{\underline{l}}}_T | {\tilde{ {{\underline{A}}}}}^-_{2}\, {\tilde{ {{\underline{A}}}}}^-_{-2} | {{\underline{k}}}_T \rangle = \frac{(26-D) +8 (1 + 2 \alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^-)}{2 ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2 } \delta ( {{\underline{k}}}- {{\underline{l}}}) \nonumber \\&\langle {{\underline{l}}}_T | \left( {\tilde{ {{\underline{A}}}}}^-_{1}\right) ^2\, \left( {\tilde{ {{\underline{A}}}}}^-_{-1}\right) ^2 | {{\underline{k}}}_T \rangle \nonumber \\&\quad = 8 \frac{ \left( 2 \alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^- +1 \right) \left( 4 \alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^- +3 \right) }{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^4 } \delta ( {{\underline{k}}}- {{\underline{l}}}) \nonumber \\&\langle {{\underline{l}}}_T | {\tilde{ {{\underline{A}}}}}^-_{1}\, {{\underline{A}}}^i_{1}\, {{\underline{A}}}^i_{-1}\,{\tilde{ {{\underline{A}}}}}^-_{-1} | {{\underline{k}}}_T \rangle = \delta ^{i j} \frac{2 \alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^- +1}{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2 } \delta ( {{\underline{k}}}- {{\underline{l}}}) , \end{aligned}$$

(6.47)

which vanish in the critical momentum and on-shell where $1 + 2 \alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^-=0$.

Notice that these states are not orthogonal off-shell, in fact

$$\begin{aligned} \langle {{\underline{l}}}_T | {\tilde{ {{\underline{A}}}}}^-_{2}\, \left( {\tilde{ {{\underline{A}}}}}^-_{-1}\right) ^2 | {{\underline{k}}}_T \rangle&= 3 \frac{2 \alpha ' {{\underline{k}}}_T^+ {{\underline{k}}}_T^- +1}{ ( \sqrt{2\alpha '} {{\underline{k}}}_T^+)^2 } \delta ( {{\underline{k}}}- {{\underline{l}}}) . \end{aligned}$$

(6.48)

6.2.4 Off-shell gauge equivalence of different frames

We are now in a position to observe that starting from an arbitrary frame, one can always choose the rest frame (or vice versa).

Consider for example the state,

$$\begin{aligned} {{\underline{A}}}^i_{-1}({\hat{E}})\, {{\underline{A}}}^j_{-1}({\hat{E}})\,|{\hat{ {{\underline{k}}}}}_T\rangle , \end{aligned}$$

(6.49)

with $i\ne j$. We want to show that the expansion

$$\begin{aligned}&{{\underline{A}}}^i_{-1}({\hat{E}})\, {{\underline{A}}}^j_{-1}({\hat{E}})\,|{\hat{ {{\underline{k}}}}}_T\rangle \nonumber \\&\quad = \Bigg [ R^{i j}_l\, {{\underline{A}}}^l_{-2}(E) + R^{i j}_{l m}\, {{\underline{A}}}^l_{-1}(E)\, {{\underline{A}}}^m_{-1}(E) + G^{i j}_2\,{\tilde{ {{\underline{A}}}}}^-_{-2}(E) \nonumber \\&\qquad + G^{i j}_1\,{\tilde{ {{\underline{A}}}}}^-_{-1}(E)\,{\tilde{ {{\underline{A}}}}}^-_{-1}(E) + G^{i j}_l\, {{\underline{A}}}^l_{-1}(E)\,{\tilde{ {{\underline{A}}}}}^-_{-1}(E) \Bigg ] \,| {{\underline{k}}}_T\rangle . \end{aligned}$$

(6.50)

is well-defined. To see this, note that the previous relation amounts to $D + \frac{1}{2}D(D+1)= \frac{1}{2}( D^2+ 3 D)$ equations for $\alpha ^\mu _{-2}$ and $\alpha ^\mu _{-1}\alpha ^\nu _{-1}$, in $2(D-2) + 2 + \frac{1}{2}(D-2)(D-1) =\frac{1}{2}( D^2 +D -2)$ unknown coefficients $R_l, G_l$, $G_1, G_2$ and $R_{l m}$. The excess of $D+1$ equations is only apparent since both states satisfy the off-shell $L_2$ and $L_1$ Virasoro conditions which yield precisely $D+1$ equations associated to $\alpha ^\mu _{-1}$ and 1.

As before the previous equality is valid for states with both ${{\underline{k}}}^+= E^{\underline{+}}_\mu k^\mu \ne 0$ and ${\widehat{ {{\underline{k}}}}}^+= {{\hat{E}}}^{\underline{+}}_\mu k^\mu \ne 0$ and as before the interpretation is that when we consider on-shell states where the states generated by improved Brower operators are null, only one frame, e.g. E is sufficient to compute all the on-shell amplitudes by “analytic continuations”.

6.3 A curiosity

With the formulation based on the local frame and the use of ${{\underline{p}}}_0^+$, the DDF operators are true zero-dimensional conformal operators. Therefore, it is possible to consider physical states built using different local frames. For example

$$\begin{aligned} {{\underline{A}}}^i({\tilde{E}})\, {{\underline{A}}}^j(E)\, | {{\underline{k}}}_T(E)\rangle , \end{aligned}$$

(6.51)

which is still a physical state.

7 Mean value of second order Casimir for Poincaré group for some physical states

In order to substantiate the map between DDF states and lightcone ones and in order to understand the average spin content of the DDF states, we compute the mean second order Casimir for some states.

In particular, at level $N=2$, we know that there is only one Poincaré irrep. despite the existence of two classes of DDF states, therefore, these must have the same Casimirs and we want to verify this.

The computation can be done in two ways. The first is to evaluate on the lightcone or equivalently using the Lorentz algebra of DDF operators and using Eq. (3.11). The second way is to express the DDF states in a covariant basis and then perform the polarization decomposition into Poincaré irreps.

We start with the lightcone computation and then move to the covariant one.

7.1 Poincaré algebra and Casimirs

The Poincaré algebra $iso(1,D-1)$ reads

$$\begin{aligned} {[} M_{\mu \nu },\, M_{\rho \sigma }]&= 2 i \eta _{\rho [\mu } M_{\nu ] \sigma } - 2 i \eta _{\sigma [\mu } M_{\nu ] \rho } , \nonumber \\ {[}M_{\mu \nu },\, P_\rho ]&= 2 i \eta _{\rho [\mu } P_{\nu ]} . \end{aligned}$$

(7.1)

We can then define the tensor

$$\begin{aligned} W_{[ \mu \nu \rho ] }= & {} M_{[\mu \nu }\, P_{\rho ]} = P_{[\rho }\, M_{\mu \nu ] } \equiv \frac{1}{3!} \sum _\sigma W_{ \sigma (\mu ) \sigma (\nu ) \sigma (\rho ) }\nonumber \\= & {} \frac{1}{3} \left[ M_{\mu \nu }\, P_{\rho } + 2 M_{\rho [\mu }\, P_{\nu ]} \right] , \end{aligned}$$

(7.2)

which is well defined independently of the order of M and P. It commutes with P, i.e.

$$\begin{aligned} {[} W_{[ \mu \nu \rho ] },\, P_\sigma ] = 0. \end{aligned}$$

(7.3)

Then the simplest Casimir invariant is the second order Casimir invariant, given by the scalar

$$\begin{aligned}&-\frac{3^2}{3!} C_2\left( iso(1,D-1)\right) \nonumber \\&\quad = W_{[ \mu \nu \rho ] } W^{[ \mu \nu \rho ] }= 3! \sum _{\mu<\nu <\rho } W_{[ \mu \nu \rho ] } W^{[ \mu \nu \rho ] } \nonumber \\&\quad = -\frac{1}{3} \left( P_\rho (M^2)_{\mu \mu } P_\rho + 2 P_\rho (M^2)_{\mu \rho } P_\mu \right) , \end{aligned}$$

(7.4)

where the normalization is chosen so that we get the simple relation (7.14) between the Poincare Casimir and the space rotation one.

Higher Casimir can be obtained by considering the square of

$$\begin{aligned} W_{[ \mu _1\nu _1 \dots \mu _n\nu _n \rho ] } = M_{[\mu _1\nu _1 }\, \cdots M_{\mu _n\nu _n }\, P_{\rho ]}, \end{aligned}$$

(7.5)

i.e.

$$\begin{aligned} C_{2 n} =&W_{[ \mu _1\nu _1 \cdots \rho ] } W^{[ \mu _1\nu _1 \cdots \rho ] } . \end{aligned}$$

(7.6)

Notice that the usual Minkowski algebra $so(1,D-1)$ Casimir is not a Casimir for the Poincaré algebra since

$$\begin{aligned} {[} M_{\mu \nu } M^{\mu \nu },\, P_\rho ] =&4 i M_{\rho \nu } P^\nu + 4 \frac{1-D}{2} P_\rho . \end{aligned}$$

(7.7)

7.1.1 $C_2(iso(1,D-1))$ in the rest frame

The expression for W can be simplified a lot if we notice that in string

$$\begin{aligned} M_{\mu \nu }&= 2 x_{[\mu } p_{0 \nu ]} + J_{\mu \nu } , \nonumber \\ P_\mu&= p_{0 \mu } , \end{aligned}$$

(7.8)

where

$$\begin{aligned} J^{\mu \nu } = i \sum _{n=1}^\infty \frac{2}{n} \alpha ^{[\mu }_{-n}\alpha ^{\nu ]}_{n}, \end{aligned}$$

(7.9)

is manifestly anti-symmetric and contains only the non-zero modes of the Lorentz generator $M_{\mu \nu }$. In fact we get

$$\begin{aligned} W_{[ \mu \nu \rho ] } = J_{[\mu \nu }\, p_{0 \rho ]}, \end{aligned}$$

(7.10)

where the only dependence on zero modes is through $p_0$.

The meaning of this expression is that the tachyonic string state $|k\rangle$ is a scalar, in fact

$$\begin{aligned} W_{[ \mu \nu \rho ] } |k \rangle =0 ~~\Rightarrow ~~ C_2\equiv 0. \end{aligned}$$

(7.11)

In the following, we consider massive states and in order to simplify the computations we choose the rest frame where

$$\begin{aligned} k_0 =M,~~~~ k_I=0,~~~~ I=1,2,\ldots D-1, \end{aligned}$$

(7.12)

then we get that the only non-vanishing W components are

$$\begin{aligned} W_{I J 0} = \frac{1}{3} J_{I J} p_{0\, 0} = \frac{1}{3} J_{I J}\, M. \end{aligned}$$

(7.13)

Then the Casimir $C_2( iso(1, D-1) )$ reads,

$$\begin{aligned} C_2( iso(1, D-1) )= & {} - M^2\, \sum _{I<J} J_{I J} J^{I J} \nonumber \\= & {} M^2\, C_2( so(D-1) ) . \end{aligned}$$

(7.14)

This means that the states are classified according to $SO(D-1)$, i.e. according to the “spin”.

7.2 Casimir of $\alpha ^i_{-N} |k\rangle$ physical lightcone states

We want now to compute the second Casimir $C_2$ for the level $N=2$ lightcone $\alpha ^i_{-N} |k\rangle$ state.

Since the computation has the same order of difficulty for generic N we will do this more general one and then restrict to $N=2$. Moreover it is also interesting ho have an estimate of the average “spin” contained in these states.

First we compute

$$\begin{aligned} i J^{i j} \alpha ^l_{-N ({lc} )} |k\rangle&= 2 \delta ^{l [i} \alpha ^{j ]}_{-N ({lc} )} |k\rangle , \end{aligned}$$

(7.15)

and its squared modulus

$$\begin{aligned} \sum _{i<j} \parallel i J^{i j} \alpha ^l_{-N ({lc} )} |k\rangle \parallel ^2&= N \cdot (D-3) . \end{aligned}$$

(7.16)

We then compute

$$\begin{aligned} i J^{- j} \alpha ^l_{-N ({lc} )} |k\rangle&= \frac{1}{ \sqrt{2\alpha '} k^+} \sum _{l=1}^{N-1} \sum _{r,s=2}^{D-1}\nonumber \\&\quad \times \left[ \frac{1}{2}\delta ^{j l} \delta ^{r s} - \frac{N}{l} \delta ^{j (r} \delta ^{s) l} \right] \alpha ^r_{-1 ({lc} )} \alpha ^s_{-1 ({lc} )} |k\rangle , \end{aligned}$$

(7.17)

and its squared modulus

$$\begin{aligned}&\sum _{j} \parallel i J^{- j} \alpha ^l_{-N ({lc} )} |k\rangle \parallel ^2\nonumber \\&\quad = \frac{1}{2\alpha '(k^+)^2} \sum _{l=1}^{N-1} \sum _{j,r,s=2}^{D-1} l (N -l)\nonumber \\&\qquad \times \Biggl [ \left( \frac{1}{2}\delta ^{j l} \delta ^{r s} - \frac{N}{N-l} \delta ^{j (r} \delta ^{s) l} \right) \left( \frac{1}{2}\delta ^{j l} \delta ^{s r}- \frac{N}{l} \delta ^{j (s} \delta ^{r) l} \right) \nonumber \\&\qquad + \left( \frac{1}{2}\delta ^{j l} \delta ^{r s} - \frac{N}{l} \delta ^{j (r} \delta ^{s) l} \right) \left( \frac{1}{2}\delta ^{j l} \delta ^{r s}-\frac{N}{l} \delta ^{j (r} \delta ^{s) l} \right) \Biggr ] \nonumber \\&= \frac{1}{2\alpha '(k^+)^2} \left[ \sum _{l=1}^{N-1} l (N -l) \frac{D-2}{4} + N^2 (N-1) \frac{D-2}{2}\right. \nonumber \\&\qquad \left. + \sum _{l=1}^{N-1} l (N -l) \frac{D-2}{4} - N \sum _{l=1}^{N-1} (N -l)\right. \nonumber \\&\qquad \left. + N^2 \sum _{l=1}^{N-1} \frac{N -l}{l} \frac{D-1}{2} \right] . \end{aligned}$$

(7.18)

Using the relations

$$\begin{aligned} \sum _{l=1}^{m-1} l (m -l)&= \frac{ m (m^2 -1)}{6} , \nonumber \\ \sum _{l=1}^{m-1} \frac{m -l}{l}&= m \sum _{l=1}^{m-1} \frac{1}{l} - (m-1) = m H_{m-1} - (m-1) , \nonumber \\ \alpha 'M_N^2&= N-1 , \end{aligned}$$

(7.19)

where $H_m$ is the m-th harmonic number we can evaluate the average second Casimir to be

$$\begin{aligned}&\frac{1}{M_N^2} \langle {\hat{C}}_2( iso(1, D-1) ) \rangle \nonumber \\&\quad = \langle {\hat{C}}_2(so(D-1)) \rangle = \frac{D-3}{N} + \frac{1}{24 (N-1)} \nonumber \\&\qquad \times \left[ \left( (6 H_{N-1} +1 ) N^2 -1 \right) (D-1)- 18 N^2 + 12 N + 1 \right] \nonumber \\&\quad \sim \frac{1}{4} N \ln (N) (D-1) +O(N) , \end{aligned}$$

(7.20)

where ${\hat{C}}_2$ is the operator associated with the Casimir. This value is rather suppressed w.r.t. the Casimir we can expect from the symmetric traceless irrep. which should be of order $N^2$ as suggested by eq. (7.29).

In the case of interest $N=2$ we get

$$\begin{aligned} \alpha '\langle {\hat{C}}_2( iso(1, D-1) ) \rangle&= \langle {\hat{C}}_2(so(D-1)) \rangle \nonumber \\&= (D-3) + \frac{9}{8} (D-2) = 50 , . \end{aligned}$$

(7.21)

which matches exactly the covariant expressing eq. (7.42) only for $D=26$ and the lightcone Casimir for the $(\alpha ^i_{-1})^N |k\rangle$ state in (7.30) for all D.

7.3 Casimir of $(\alpha ^i_{-1})^N |k\rangle$ physical lightcone states

As in the previous case since the computation has the same order of difficulty for generic N we will do this more general one and then restrict to $N=2$.

We start from the easiest part

$$\begin{aligned} -i J^{i j} ( \alpha ^l_{-1 ({lc} )} )^N |k \rangle&= -N\, \delta ^{i l} (\alpha ^l_{-1 ({lc} )} )^{N-1} \alpha ^j_{-1 ({lc} )} |k \rangle , \end{aligned}$$

(7.22)

then we can compute the square norm

$$\begin{aligned} \sum _{i<j} \parallel -i J^{i j} ( \alpha ^l_{-1 ({lc} )} )^N |k \rangle \parallel ^2&= N^2\, (D-3)\, N! . \end{aligned}$$

(7.23)

For computing the remaining contribution we start from

$$\begin{aligned} {[}i J^{- j},\, \alpha ^l_{-1 ({lc} )}]&= \delta ^{j l} \hat{\alpha }^-_{-1 ({lc} )} + \frac{1}{\alpha ^+_{0 ({lc} )} } \sum _{n=1}^\infty \frac{1}{n} \alpha ^l_{-n-1 ({lc} )} \alpha ^j_{n ({lc} )}\nonumber \\&\quad - \frac{1}{\alpha ^+_{0 ({lc} )}} \sum _{n=1}^\infty \frac{1}{n} \alpha ^j_{-n ({lc} )} \alpha ^l_{n-1 ({lc} )} \nonumber \\ \implies&\delta ^{j l} \frac{ \vec \alpha _{-2 ({lc} )} \cdot \vec \alpha _1({lc} )}{\alpha ^+_{0 ({lc} )}} + \frac{1}{\alpha ^+_{0 ({lc} )}} \alpha ^l_{-2 ({lc} )} \alpha ^j_{1 ({lc} )}\nonumber \\&\quad -\frac{1}{\alpha ^+_{0 ({lc} )}} \frac{1}{2} \alpha ^j_{-2 ({lc} )} \alpha ^l_{1 ({lc} )} , \end{aligned}$$

(7.24)

where we have kept the terms which give a non-vanishing contribution in

$$\begin{aligned}&i J^{- j} ( \alpha ^l_{-1 ({lc} )})^N |k \rangle \nonumber \\&\quad = \sum _{k=0}^{N-1} ( \alpha ^l_{-1 ({lc} )})^k [i J^{- j},\, \alpha ^l_{-1 ({lc} )}] ( \alpha ^l_{-1 ({lc} )})^{N-k-1} |k \rangle ,\quad j\ne l , \end{aligned}$$

(7.25)

taking also into account that we have chosen the rest frame for which $\alpha ^i_{0 ({lc} )} \rightarrow k^i=0$. It is then easy to show that

$$\begin{aligned} i J^{- j} ( \alpha ^l_{-1 ({lc} )} )^N |k \rangle&= - \frac{1}{ \sqrt{2\alpha '} k^+} \frac{N (N-1)}{2} ( \alpha ^l_{-1 ({lc} )} )^{N-1}\nonumber \\&\quad \times \alpha ^j_{-2 ({lc} )} |k \rangle ,\quad j\ne l . \end{aligned}$$

(7.26)

The case $j=l$ is the same but with a different overall coefficient which arises because of $\delta ^{j l}$ in (7.24). We get therefore

$$\begin{aligned} i J^{- l} ( \alpha ^l_{-1 ({lc} )} )^N |k \rangle&= + \frac{3}{2} \frac{1}{ \sqrt{2\alpha '} k^+} \frac{N (N-1)}{2} ( \alpha ^l_{-1 ({lc} )} )^{N-1}\nonumber \\&\quad \times \alpha ^l_{-2 ({lc} )} |k \rangle . \end{aligned}$$

(7.27)

Now we can compute the square norm

$$\begin{aligned}&\sum _{j\ne l} \parallel i J^{- j} ( \alpha ^l_{-1 ({lc} )} )^N |k \rangle \parallel ^2 + \parallel i J^{- l} ( \alpha ^l_{-1} )^N |k \rangle \parallel ^2 \nonumber \\&\quad = [ 9 + D-3 ]\, \frac{1}{ 8 \alpha '(k^+)^2 }\, \frac{N (N-1)}{2}\, N! . \end{aligned}$$

(7.28)

Now we can evaluate the average second Casimir to be

$$\begin{aligned}&\frac{1}{M_N^2} \langle {\hat{C}}_2( iso(1, D-1) ) \rangle \nonumber \\&\quad = \langle {\hat{C}}_2(so(D-1)) \rangle = N\, (D-3) + \frac{N (N-1)}{16} (D+6) \nonumber \\&\quad \sim \frac{N^2}{16} (D+6) +O(N) . \end{aligned}$$

(7.29)

This value of order $N^2$ is what we can expect from a symmetric traceless irrep. The coefficient is very likely, less than the pure irrep because of the contamination with other irreps.

In the case of interest $N=2$ we get

$$\begin{aligned} . \alpha '\langle {\hat{C}}_2( iso(1, D-1) ) \rangle&= \langle {\hat{C}}_2(so(D-1)) \rangle \nonumber \\&= 2 (D-3) + \frac{1}{8} (D+6) = 50 , \end{aligned}$$

(7.30)

which matches exactly the covariant expression in eq. (7.42) only for $D=26$ and the lightcone state of the previous section (7.21) for all D.

7.4 Casimir of $N=2$ physical DDF states: decomposing tensors and computing the Casimirs

We know that ${{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle$ and ${{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1}| {{\underline{k}}}_T\rangle$ are two different pieces of the same irrep then we expect that their Casimirs be equal, i.e

$$\begin{aligned} C_2\left( {{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle \right)&\equiv \frac{ \langle {{\underline{l}}}_T | {{\underline{A}}}^i_2\, {\hat{C}}_2\, {{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle }{ \langle {{\underline{l}}}_T | {{\underline{A}}}^i_2\, {{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle } = C_2\left( {{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1}| {{\underline{k}}}_T\rangle \right) \nonumber \\&\equiv \frac{ \langle {{\underline{l}}}_T | {{\underline{A}}}^i_{1} {{\underline{A}}}^j_{1}\, {\hat{C}}_2\, {{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1} | {{\underline{k}}}_T\rangle }{ \langle {{\underline{l}}}_T | {{\underline{A}}}^i_{1} {{\underline{A}}}^j_{1}\, {{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1} | {{\underline{k}}}_T\rangle }, \end{aligned}$$

(7.31)

where ${\hat{C}}_2$ is the Casimir operator which acts for example as

(7.32)

Let us start writing the states for $i\ne j$ and $N=2$

$$\begin{aligned}&{{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle = \left[ \left( \Pi ^i {{\underline{\alpha }}}_{-2} \right) + \left( \Pi ^i {{\underline{\alpha }}}_{-1} \right) {{\underline{\alpha }}}^+_{-1} \frac{N}{ \sqrt{2\alpha '} {{\underline{k}}}^+} \frac{1}{1!} \right] | {{\underline{k}}}\rangle , \nonumber \\&{{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1}| {{\underline{k}}}_T\rangle = \left( \Pi ^i {{\underline{\alpha }}}_{-1} \right) \left( \Pi ^j {{\underline{\alpha }}}_{-1} \right) | {{\underline{k}}}\rangle . \end{aligned}$$

(7.33)

Then we introduce the following transverse tensors

$$\begin{aligned} \Pi ^i_\mu&= \delta ^i_\mu - \frac{k^i}{k^+} \delta ^+_\mu , \nonumber \\ P^\rho _\mu&= \eta ^{\perp \, \rho }_{\,\,\,\, \mu } = \delta ^\rho _\mu + \frac{k^\rho k_\nu }{ M^2 } = \delta ^\rho _\mu - \frac{k^\rho k_\nu }{ k^2 } , \end{aligned}$$

(7.34)

which have the following properties

$$\begin{aligned} \Pi ^i_\mu k^\mu =&P^\rho _\mu k^\mu = 0 , \nonumber \\ \Pi ^i \cdot \Pi ^j&= \delta ^{ i j} , \nonumber \\ P^\rho _\mu \, \eta ^{\mu \nu }\, P^\sigma _\nu&= P^{\rho \sigma } , \nonumber \\ \Pi ^i_\mu \, \eta ^{\mu \nu }\, P^\sigma _\nu&= \eta ^{i \sigma } . \end{aligned}$$

(7.35)

Using these objects we can rewrite the covariant states associated with the DDF states using the irreps as

(7.36)

where the last equation is true only because we have chosen $i\ne j$ and the antisymmetric is obviously zero.

Now we can check the splitting into irreps by computing the states norm. We expect the norm to be the sum of the squares of the different irreps up to coefficients associated with the norms of the states.

If we compute the norm of the generic sum of irreps

(7.37)

we get

(7.38)

where in our case $\alpha 'M_N^2 = N-1 = 1$ since $N=2$.

Then we can easily compare with the direct computation of the norms as

$$\begin{aligned}&\langle {{\underline{k}}}_T | {{\underline{A}}}^i_2\, {{\underline{A}}}^i_{-2} | {{\underline{k}}}_T \rangle \nonumber \\&\quad = 2\, \Pi ^i \cdot \Pi ^i + 2 \left( \frac{N}{ \sqrt{2\alpha '} {{\underline{k}}}^+} \right) ^2 \frac{1}{2}\Pi ^i \cdot \Pi ^i\, P^+ \cdot P^+\nonumber \\&\qquad - \frac{1}{M^2} \left( \frac{N}{ \sqrt{2\alpha '} {{\underline{k}}}^+} \right) ^2 \frac{1}{2}\Pi ^i \cdot \Pi ^i \nonumber \\&\quad = 2 + \frac{N^2}{2 \alpha 'M^2} - \frac{N^2}{2 \alpha 'M^2} =2 , \nonumber \\&\langle {{\underline{l}}}_T | {{\underline{A}}}^i_{1} {{\underline{A}}}^j_{1}\, {{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1} | {{\underline{k}}}_T = 2\, \frac{1}{2}\Pi ^i \cdot \Pi ^i \Pi ^j \cdot \Pi ^j = 1 . \end{aligned}$$

(7.39)

Finally we can compute the Casimirs

(7.40)

where the first factor is the normalization. When we use

(7.41)

where $D-1$ is the space dimension, i.e. $D-1=25$ and $C_2(*,d)$ is the Casimir for the $so(D-1)$ irrep $*$ we get

$$\begin{aligned}{} & {} C_2\left( {{\underline{A}}}^i_{-2} | {{\underline{k}}}_T\rangle \right) = (D-2) + 2(D-1) -(D-2)\nonumber \\{} & {} \quad = C_2\left( {{\underline{A}}}^i_{-1} {{\underline{A}}}^j_{-1}| {{\underline{k}}}_T\rangle \right) = 2(D-1), \end{aligned}$$

(7.42)

which is valid in all dimension as long as the mass shell condition is true.

8 Conclusions

In this paper, we have presented a reformulation of DDF operators which makes the underlying structure more clear and repackages all data in a nice simple structure, the local frame E.

These operators are then decoupled from the usual associated tachyon.

Moreover, in the proposed formulation the framed DDFs are true conformal operators and do not have a cut, which is present in the usual formulation as shown in Sect. 3.4.

This allows us to go off the mass shell easily, while maintaining all the Virasoro constraints but the $L_0$ one as discussed in Sect. 4.2.

Another important point worth stressing in this off-shell formulation is that two off shell DDF states related by an infinitesimal lightcone boost differ by an off shell Brower state and not by a null state. This is discussed in Sect. 6 using examples. Null states are generated by using Virasoro algebra or BRST. These states can then be associated with infinitesimal gauge transformations. Then the fact that Brower states are needed off shell can be interpreted as they are the proper states to describe the off shell gauge transformations in the restricted Fock space generated by DDF and Brower operators. Hence these gauge symmetries are not the usual ones obtained using $L_n$ as in the string field theory, but the states generated by using the improved Brower operators. This can be interpreted by saying that off-shell, Brower states replace the gauge symmetry generated by Virasoro operators, i.e. BRST invariance. Therefore, using the Brower states off-shell, one can cover the whole momentum space except the $k_\mu =0$ point.

Using the framed DDFs in Sect. 4 we have written the general covariant solution of the Virasoro constraints in terms of tangent space lightcone polarizations albeit in an unconventional gauge. In Sect. 5 we have also described how they can be used to extend the natural idea that DDFs are the embedding of lightcone states into the covariant formulation in such a way that we need only one set of tangent space lightcone polarizations associated with only one frame. It is worth stressing again that the improved Brower operators are the proper operators to consider.

Finally, in Sect. 7 we check that the identification is correct by comparing the Lorentz algebras and the expectation values of the second Casimir for some lightcone states and the corresponding covariant states obtained by the stated embedding.

Using this formalism, we can easily get the N Reggeon vertex, i.e. the generating function of N point open string framed DDF correlators [55]. In this paper we also present some higher-level scalar amplitudes which require the precise knowledge of the states besides its existence [56,57,58]. This is obtained from lightcone formalism [59].

It would be interesting to investigate how and whether it is possible to extend these ideas and findings to the supersymmetric case and to other open string backgrounds, for example with a constant magnetic field.

Appendices

Appendix A: ${{\underline{A}}}^-$ and ${\tilde{ {{\underline{A}}}}}^-$ operators

If we proceeded to define the ${{\underline{A}}}^-_m$ operators in direct analogy with the framed DDF operators ${{\underline{A}}}^i_{m}$, we run into problems with cubic poles. To differentiate with the proper definition derived at the end of this section, we refer to this naive version as,

$$\begin{aligned} {{\widehat{ {{\underline{A}}}}}}^-_m(E) = i \sqrt{\frac{2}{\alpha '}} \oint _{z=0} \frac{d z}{ 2\pi i}: \partial _z {{\underline{L}}}^-(z) e^{i m \frac{ {{\underline{L}}}^+(z) }{\alpha ' {{\underline{p}}}^+_0} }:. \end{aligned}$$

(A.1)

To see the non-vanishing cubic pole, we use $L_n = \oint \frac{dz}{2\pi i} z^{n+1}\left( -\frac{1}{\alpha '}:\partial {{\underline{L}}}(z)\partial {{\underline{L}}}(z):\right)$ to obtain,

$$\begin{aligned}&{[}L_n,{\widehat{ {{\underline{A}}}}}^-_m] \nonumber \\&\quad = \left[ \oint _z z^{n+1}\left( -\frac{2}{\alpha '}\right) e^{\delta \cdot \partial {{\underline{L}}}(z)},\right. \nonumber \\&\qquad \times \left. i \sqrt{2\alpha '} \oint _w e^{{\epsilon }\cdot \partial {{\underline{L}}}(w) + i {{\underline{k}}}\cdot {{\underline{L}}}(w)}\right] \Biggm |_{\delta ^2,{\epsilon }_-,k_+} \nonumber \\&\quad = -i \frac{2}{\alpha '} \sqrt{2\alpha '} \oint _w \oint _{z=w} z^{n+1} :e^{\delta \cdot \partial {{\underline{L}}}(z)} e^{{\epsilon }\cdot \partial {{\underline{L}}}(w) + i {{\underline{k}}}\cdot {{\underline{L}}}(w)}:\nonumber \\&\qquad \times e^{-\frac{\alpha '}{2}\frac{\delta \cdot {\epsilon }}{(z-w)^2}}e^{-i\frac{\alpha '}{2}\frac{\delta \cdot k}{(z-w)}}\Biggm |_{\delta ^2,{\epsilon }_-,k_+} \nonumber \\&\quad = i \sqrt{2\alpha '} \oint _w \oint _{z=w} z^{n+1} \left[ -i\frac{\alpha '}{2} \frac{(\delta \cdot {\epsilon })(\delta \cdot k)}{(z-w)^3} \right. \nonumber \\&\qquad + \frac{\delta \cdot {\epsilon }}{(z-w)^2}\delta \cdot \partial {{\underline{L}}}(z)\nonumber \\&\qquad \left. + \delta \cdot i\partial {{\underline{L}}}(z) {\epsilon }\cdot \partial {{\underline{L}}}(w) \frac{\delta \cdot k}{(z-w)}\right] e^{i {{\underline{k}}}\cdot {{\underline{L}}}(w)} \end{aligned}$$

(A.2)

Using $\delta _\mu \delta _\nu = \eta _{\mu \nu }$ and ${\epsilon }\cdot k = -\frac{m}{\alpha ' p_0^+}$ we get,

$$\begin{aligned}&{[}L_n,{\widehat{ {{\underline{A}}}}}^-_m] \nonumber \\&\quad = i \sqrt{2\alpha '} \oint _{w=0}:\left[ {\epsilon }\cdot \partial (w^{n+1}\partial {{\underline{L}}}(w))\right. \nonumber \\&\qquad \left. + i w^{n+1}{\epsilon }\cdot \partial {{\underline{L}}}(w) {{\underline{k}}}\cdot \partial {{\underline{L}}}(w)\right] e^{i {{\underline{k}}}\cdot {{\underline{L}}}(w)}: \nonumber \\&\qquad + i \sqrt{2\alpha '} \oint _{w}\oint _{z=w}z^{n+1}\left( -i \frac{\alpha '}{2}\frac{{\epsilon }\cdot {{\underline{k}}}}{(z-w)^3}\right) e^{i {{\underline{k}}}\cdot {{\underline{L}}}(w)} \nonumber \\&\quad = i \sqrt{2\alpha '} \oint _{z=0} \frac{im}{4p_0^+}\partial ^2(z^{n+1})e^{\frac{im {{\underline{L}}}^+(z)}{\alpha 'p_0^+}} + \cdots , \end{aligned}$$

(A.3)

where, “...” denotes the usual total derivative terms which vanish. To remove this cubic pole contribution, we first compute,

$$\begin{aligned}&\left[ \partial _w {{\underline{L}}}^-, \partial ^2_z {{\underline{L}}}^+/\partial _z {{\underline{L}}}^+\right] \nonumber \\&\quad = -\partial _z \int _0^\infty \frac{d\xi }{\xi } [\partial _w {{\underline{L}}}^-, e^{- \xi \partial _z {{\underline{L}}}^+}]\nonumber \\&\quad = \frac{\alpha '}{2} \partial _z \int _0^\infty d\xi \frac{1}{(w-z)^2}e^{-\xi \partial _z {{\underline{L}}}^+} + :..:\nonumber \\&\quad = \frac{\alpha '}{2} \partial _z \left[ \frac{1}{(w-z)^2} \frac{1}{\partial _z {{\underline{L}}}^+} \right] + :\partial _w {{\underline{L}}}^-\, \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+} :, \end{aligned}$$

(A.4)

where we used the OPE for $\partial _w {{\underline{L}}}^- \partial _z {{\underline{L}}}^+$ directly in the second step. Using (A.4), we get

$$\begin{aligned}&{[}L_n,\partial ^2_z {{\underline{L}}}^+/\partial _z {{\underline{L}}}^+] \nonumber \\&\quad = -\frac{2}{\alpha '}\oint _w\left[ \partial _w {{\underline{L}}}^- , \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+} \right] w^{n+1}\partial _w {{\underline{L}}}^+ \nonumber \\&\quad = \partial _z \left[ \partial _w\left( w^{n+1}\partial _w {{\underline{L}}}^+\right) \Biggm |_{w=z} \frac{1}{\partial _z {{\underline{L}}}^+}\right] \nonumber \\&\quad = \partial ^2_z(w^{n+1}) + \partial _z\left( z^{n+1} \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}\right) \end{aligned}$$

(A.5)

Finally we observe that

$$\begin{aligned}&\Bigg [L_n, i \sqrt{ \frac{ 2 }{ \alpha '} } \oint _{z=0} \frac{d z}{ 2\pi i} \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+} e^{i m \frac{ {{\underline{L}}}^+(z) }{\alpha ' {{\underline{p}}}^+_0} }\Bigg ]\nonumber \\&\quad = i \sqrt{ \frac{ 2 }{ \alpha '} } \oint _z \left( \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}\Big [L_n,e^{i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\Big ] + \left[ L_n,\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}\right] e^{i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\right) \nonumber \\&\quad = i \sqrt{ \frac{ 2 }{ \alpha '} } \oint _z \left( \partial ^2_z(z^{n+1})e^{i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}} + \partial _z\left( z^{n+1}\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}\right) \right. \nonumber \\&\qquad \left. + \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+} z^{n+1}\partial _z\left( e^{i \frac{m {{\underline{L}}}^+}{\alpha 'p_0^+}}\right) \right) \nonumber \\&\quad = i \sqrt{ \frac{ 2 }{ \alpha '} } \oint _z \partial ^2_z(z^{n+1})e^{i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\nonumber \\&\qquad + i \sqrt{ \frac{ 2 }{ \alpha '} } \oint _z \partial _z\left( z^{n+1}\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}e^{i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\right) , \end{aligned}$$

(A.6)

where the second term is zero since it does not contain any branch cuts and is a total derivative. Therefore, we can define the ‘good’ conformal operator (free of cubic pole contributions) as,

$$\begin{aligned} {{\underline{A}}}^-_m(E)= & {} i \sqrt{\frac{2}{\alpha '}} \oint _{z=0} \frac{d z}{ 2\pi i}: \nonumber \\{} & {} \times \left[ \partial _z {{\underline{L}}}^-(z) - i \frac{m}{4 {{\underline{p}}}_0^+} \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+} \right] e^{i m \frac{ {{\underline{L}}}^+(z) }{\alpha ' {{\underline{p}}}^+_0} }:, \end{aligned}$$

(A.7)

which satisfies $[L_n, {{\underline{A}}}^-_m] = 0$ by construction.

Appendix B: Derivation of the algebra

The ${{\underline{A}}}^i_m$ algebra can be obtained via the usual means. Let us write

$$\begin{aligned} {{\underline{A}}}^i_m = \mathcal{N} \oint _{z=0} \frac{d z}{ 2\pi i} : \partial _z {{\underline{L}}}^i(z) e^{i m \delta _+ {{\underline{L}}}^+(z)} : , \end{aligned}$$

(B1)

with

$$\begin{aligned} \mathcal{N} = i \sqrt{\frac{2}{\alpha '}},~~~~ \delta _+ = \frac{1}{\alpha ' {{\underline{p}}}_0^+}, ~~~~ (\text{ when } {{\underline{p}}}^+_0\ne 0) , \end{aligned}$$

(B2)

then we get,

$$\begin{aligned}&{[} {{\underline{A}}}^i_m, {{\underline{A}}}^j_n] \nonumber \\&\quad = \mathcal{N} ^2 \left[ \oint _{z=0, |z|>|w|} \frac{d z}{ 2\pi i} \oint _{w=0} \frac{d w}{ 2\pi i}\right. \nonumber \\&\qquad \left. - \oint _{w=0} \frac{d w}{ 2\pi i} \oint _{z=0, |z|<|w|} \frac{d z}{ 2\pi i} \right] \nonumber \\&\qquad \times R\left[ : \left( \partial _z {{\underline{L}}}^i e^{i m \delta _+ {{\underline{L}}}^+} \right) (z) : : \left( \partial _w {{\underline{L}}}^j e^{i n \delta _+ {{\underline{L}}}^+} \right) (w) : \right] \nonumber \\&\quad = \mathcal{N} ^2 \oint _{w=0} \frac{d w}{ 2\pi i} \oint _{z=w} \frac{d z}{ 2\pi i} \Biggl [ - \frac{\alpha '}{2} \frac{ \delta ^{i j} }{(z-w)^2} e^{i (m+n) \delta _+ {{\underline{L}}}^+(w) } \nonumber \\&\qquad - \frac{\alpha '}{2} \frac{ \delta ^{i j} }{(z-w)} i \delta _+ m \partial _w {{\underline{L}}}^+ e^{i (m+n) \delta _+ {{\underline{L}}}^+(w) } +\dots \Biggr ] \nonumber \\&\quad = \left( -\frac{1}{2}(\alpha '\mathcal{N} )^2 \delta _+ {{\underline{p}}}^+_0 \right) m \delta _{m+n,0} \delta ^{i j} . \end{aligned}$$

(B3)

Using the definitions above we can compute,

$$\begin{aligned}&{[} {{\underline{A}}}^i_m,\alpha _0^+ {{\underline{A}}}^-_n] \nonumber \\&\quad = -\frac{2}{\alpha '} \alpha _0^+\oint _{w=0}\oint _{z=w} :e^{\delta \cdot \partial _z {{\underline{L}}}+ iq\cdot {{\underline{L}}}(z)}::e^{\gamma \cdot \partial _w {{\underline{L}}}+ ik\cdot {{\underline{L}}}(w)}:\Bigm |_{\delta _i,\gamma _-} \nonumber \\&\quad = -\frac{2}{\alpha '} \alpha _0^+\oint _{w=0}\oint _{z=w} :e^{\delta \cdot \partial _z {{\underline{L}}}+ iq\cdot {{\underline{L}}}(z)} e^{\gamma \cdot \partial _w {{\underline{L}}}+ ik\cdot {{\underline{L}}}(w)}:\nonumber \\&\qquad \times e^{i \frac{\alpha '}{2} \frac{q\cdot \gamma }{(z-w)}}\Bigm |_{\delta _i,\gamma _-}\nonumber \\&\quad =-\frac{2}{\alpha '}\alpha _0^+\oint _{w=0}\oint _{z=w}:\partial _z {{\underline{L}}}^i e^{i\frac{(m {{\underline{L}}}(z) + n {{\underline{L}}}(w))}{\alpha 'p_0^+}}:\nonumber \\&\qquad \times \left( -\frac{im}{\alpha 'p_0^+}\right) \frac{\alpha '}{2} \frac{1}{(z-w)}\implies [ {{\underline{A}}}^i_m,\alpha _0^+ {{\underline{A}}}^-_n] = m {{\underline{A}}}^i_{m+n}, \end{aligned}$$

(B4)

where, $\delta _i = 1, q_+ = m/(\alpha 'p_0^+), \gamma _- = 1, k_+ = n/(\alpha 'p_0^+)$ are the only non-zero components.

We shifted to the exponential trick for evaluating commutators since the second term in the definition of ${{\underline{A}}}^-_n$ obviously commutes with ${{\underline{A}}}^i_m$.

We now calculate the commutator,

$$\begin{aligned} {[}{ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_m. { {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_n]&= ({ {{\underline{\alpha }}}_0^+})^2([{\widehat{ {{\underline{A}}}}}^-_m,\widehat{{\underline{A}}}^-_n] - [{\widehat{ {{\underline{A}}}}}^-_m,C_n]\nonumber \\&\quad - [C_m,{\widehat{ {{\underline{A}}}}}^-_n] +[C_m,C_n]), \end{aligned}$$

(B5)

where, $C_m =i \sqrt{2\alpha '} \left( \frac{im}{4p_0^+}\right) \oint _{z=0}:\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+} e^{\frac{im {{\underline{L}}}^+}{\alpha 'p_0^+}}:$. We see that,

$$\begin{aligned}&{[}{\widehat{ {{\underline{A}}}}}^-_m,{\widehat{ {{\underline{A}}}}}^-_n] \nonumber \\&\quad = -\frac{2}{\alpha '}\oint _{w=0}\oint _{z=w}:e^{\delta \cdot \partial _z {{\underline{L}}}+ iq\cdot {{\underline{L}}}(z)}e^{{\epsilon }\cdot \partial _w {{\underline{L}}}+ik\cdot {{\underline{L}}}(w)}:\nonumber \\&\qquad \times e^{-i\frac{\delta \cdot k}{(z-w)} \frac{\alpha '}{2} }e^{i \frac{{\epsilon }\cdot q}{(z-w)} \frac{\alpha '}{2} }\Bigm |_{\delta _-,{\epsilon }_-}\nonumber \\&\quad =-\frac{2}{\alpha '}\oint _{w=0}\oint _{z=w}\left[ -\frac{im}{\alpha 'p_0^+}\frac{\partial _z {{\underline{L}}}^-}{(z-w)} \frac{\alpha '}{2} + \frac{in}{\alpha 'p_0^+}\frac{\partial _w {{\underline{L}}}^-}{(z-w)}\right. \nonumber \\&\qquad \left. +\left( \frac{\alpha '}{2} \right) ^2\frac{mn}{(z-w)^2(\alpha 'p_0^+)^2}\right] \nonumber \\&\qquad \times e^{i\frac{(m {{\underline{L}}}(z) + n {{\underline{L}}}(w))}{\alpha 'p_0^+}} \nonumber \\&\quad =~i\frac{(m-n)}{\alpha 'p_0^+} \oint _z :\partial _z {{\underline{L}}}^- e^{i \frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:\nonumber \\&\qquad - \frac{\alpha '}{2} \frac{inm^2}{(\alpha 'p_0^+)^3}:\partial _z {{\underline{L}}}^+ e^{i \frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:, \end{aligned}$$

(B6)

and,

$$\begin{aligned} {[}{\widehat{ {{\underline{A}}}}}^-_m,C_n]&= \left( \frac{-in}{2\alpha 'p_0^+}\right) \left( \left[ \oint _w \partial _w {{\underline{L}}}^-,\oint _z \frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}\right] e^{i\frac{(m {{\underline{L}}}^+(w) + n {{\underline{L}}}^+(z))}{\alpha 'p_0^+}} \right. \nonumber \\&\quad \left. + \left[ \oint _w \partial _w L^-,\oint _z e^{i\frac{n {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\right] e^{i\frac{m {{\underline{L}}}^+(w)}{\alpha 'p_0^+}}\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}\right) \nonumber \\&= \frac{inm^2}{4(\alpha ')^2 (p_0^+)^3}\oint _z :\partial _z {{\underline{L}}}^+ e^{i\frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:\nonumber \\&\quad + \frac{n^2}{4\alpha '(p_0^+)^2}\oint _z:\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}e^{i\frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:. \end{aligned}$$

(B7)

Using (B6) and (B7) in (B5), we get,

$$\begin{aligned} {[}{ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_m, { {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_n]&= \oint _z \Bigg [i\frac{(m-n)}{\alpha 'p_0^+} \oint _z :\partial _z {{\underline{L}}}^- e^{i \frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:\nonumber \\&\quad - \frac{\alpha '}{2} \frac{inm^2}{(\alpha 'p_0^+)^3}:\partial _z {{\underline{L}}}^+ e^{i \frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}: \nonumber \\&\quad - \frac{inm^2}{4(\alpha ')^2 (p_0^+)^3}\oint _z :\partial _z {{\underline{L}}}^+ e^{i\frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:\nonumber \\&\quad - \frac{n^2}{4\alpha '(p_0^+)^2}\oint _z:\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}e^{i\frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}: \nonumber \\&\quad + \frac{imn^2}{4(\alpha ')^2 (p_0^+)^3}\oint _z :\partial _z {{\underline{L}}}^+ e^{i\frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:\nonumber \\&\quad + \frac{m^2}{4\alpha '(p_0^+)^2}\oint _z:\frac{\partial ^2_z {{\underline{L}}}^+}{\partial _z {{\underline{L}}}^+}e^{i\frac{(m+n) {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}:\Bigg ] \nonumber \\&\implies [{ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_m, { {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_n] = (m-n){ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_{m+n}\nonumber \\&\quad + 2m^3 \delta _{m+n,0} \end{aligned}$$

(B8)

We now define the following operator,

$$\begin{aligned} {{\widetilde{ {{\underline{A}}}}}}^-_m(E) = { {{\underline{A}}}}^-_m(E) - \frac{1}{ {{\underline{\alpha }}}_0^+} \mathcal{L} _m(E) - \frac{D-2}{24} \frac{1}{ {{\underline{\alpha }}}_0^+}\, \delta _{m,0}, \end{aligned}$$

(B9)

where we have defined the Virasoro generators as,

$$\begin{aligned} \mathcal{L} _m(E)&= \frac{1}{2} \sum _{j=2}^{D-1} \sum _{l\in {\mathbb {Z}} }: {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^j_{m-l}(E) : , \end{aligned}$$

(B10)

and they satisfy the standard Virasoro algebra for a theory with $D-2$ bosons, namely,

$$\begin{aligned} {[}\mathcal{L} _m,\, \mathcal{L} _n] = (m-n) \mathcal{L} _{m+n} + (D-2) \frac{1}{12} m(m^2-1) \delta _{m+n,0}.\nonumber \\ \end{aligned}$$

(B11)

This is easy to check since the ${{\underline{A}}}^i_m$ and ${{\underline{\alpha }}}^i_m$ satisfy the exact same algebra. We then observe that,

$$\begin{aligned} {[} {{\underline{A}}}^i_n,{\widetilde{ {{\underline{A}}}}}^-_m]&= [ {{\underline{A}}}^i_n, {{\underline{A}}}^-_m] - \frac{1}{{ {{\underline{\alpha }}}_0^+}}[ {{\underline{A}}}^i_n, \mathcal{L} _m] \nonumber \\&= \frac{n}{{ {{\underline{\alpha }}}_0^+}}A^i_{m+n} - \frac{1}{{ {{\underline{\alpha }}}_0^+}}\frac{1}{2}\sum _{j=2}^{D-1}\sum _{l \in {\mathbb {Z}}}[ {{\underline{A}}}^i_n, : {{\underline{A}}}^j_{l} {{\underline{A}}}^j_{m-l}:] \nonumber \\&= \frac{n}{{ {{\underline{\alpha }}}_0^+}}A^i_{m+n} - \frac{1}{{ {{\underline{\alpha }}}_0^+}}\frac{1}{2}(n {{\underline{A}}}^i_{m+n} + n {{\underline{A}}}^i_{m+n}) = 0, \end{aligned}$$

(B12)

where in reaching the last line, we have used the commutator $[A^i_m,A^j_n] = m \delta _{m+n,0}\delta ^{ij}$.

We also have,

$$\begin{aligned}&{[}{ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_m,\mathcal{L} _n] = \frac{1}{2}\sum _{j=1}^{D-2} \left( \sum _{p=-\infty }^0[{ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_m, {{\underline{A}}}^j_p {{\underline{A}}}^j_{n-p}]\right. \nonumber \\&\qquad \left. + \sum _{p=1}^\infty [{ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_m, {{\underline{A}}}^j_{n-p} {{\underline{A}}}^j_p]\right) \nonumber \\&\quad = -\frac{1}{2} \sum _j \left[ \sum _{p=-\infty }^0\left( p {{\underline{A}}}^j_{m+p} {{\underline{A}}}^j_{n-p} + (n-p) {{\underline{A}}}^j_p {{\underline{A}}}^j_{m+n-p}\right) \right. \nonumber \\&\qquad \left. + \sum _{p=1}^\infty \left( (n-p) {{\underline{A}}}^j_{m+n-p} {{\underline{A}}}^j_p + p {{\underline{A}}}^j_{n-p} {{\underline{A}}}^j_{m+p}\right) \right] \nonumber \\&\quad = -\frac{1}{2}\sum _j\left[ \sum _{p=-\infty }^0(n-p) {{\underline{A}}}^j_{p} {{\underline{A}}}^j_{m+n-p}\right. \nonumber \\&\qquad \left. + \sum _{q=-\infty }^m (q-m) {{\underline{A}}}^j_{q} {{\underline{A}}}^j_{m+n-q} \right. \nonumber \\&\qquad \left. + \sum _{p=1}^\infty (n-p) {{\underline{A}}}^j_{m+n+p} {{\underline{A}}}^j_{p} + \sum _{q=m+1}^\infty (q-m) {{\underline{A}}}^j_{m+n-q} {{\underline{A}}}^j_{q}\right] \nonumber \\&\quad = -\frac{1}{2}\sum _j\left[ \sum _{p=-\infty }^0(n-m) {{\underline{A}}}^j_{p} {{\underline{A}}}^j_{m+n-p}\right. \nonumber \\&\qquad \left. + \sum _{q=1}^m (q-m) {{\underline{A}}}^j_{q} {{\underline{A}}}^j_{m+n-q} \right. \nonumber \\&\qquad \left. + \sum _{p=m+1}^\infty (n-m) {{\underline{A}}}^j_{m+n+p} {{\underline{A}}}^j_{p} + \sum _{q=1}^m(q-m) {{\underline{A}}}^j_{m+n-q} {{\underline{A}}}^j_{q}\right] \nonumber \\&\quad = (m-n)\frac{1}{2}\sum _j\sum _{l\in {\mathbb {Z}}}: {{\underline{A}}}^j_{p} {{\underline{A}}}^j_{m+n-p}:\nonumber \\&\qquad - \frac{1}{2}\sum _j\left( \sum _{q=1}^m q(q-m) \delta _{m+n,0}\right) , \end{aligned}$$

(B13)

where, the last summation of the last line in obtained from normal ordering the second term in the penultimate line (all others are already normal ordered!). Finally, we can evaluate the summation using,

$$\begin{aligned} \sum _{q=1}^m q^2 = \frac{1}{6}m(m+1)(2m+1), ~~ \sum _{q=1}^m q = \frac{1}{2}m(m+1), \end{aligned}$$

(B14)

to obtain,

$$\begin{aligned} {[}{ {{\underline{\alpha }}}_0^+} {{\underline{A}}}^-_m,\mathcal{L} _n]=(m-n) \mathcal{L} _{m+n} + \frac{D-2}{12}m(m^2-1)\delta _{m+n,0}. \end{aligned}$$

(B15)

Using (B12) and (B15), we can finally calculate,

$$\begin{aligned}&{[}{\widetilde{ {{\underline{A}}}}}^-_m, {\widetilde{ {{\underline{A}}}}}^-_n] \nonumber \\&\quad = [ {{\underline{A}}}^-_m, {{\underline{A}}}^-_n] - \frac{1}{{ {{\underline{\alpha }}}_0^+}}[\mathcal{L} _m, {{\underline{A}}}^-_n] - \frac{1}{{ {{\underline{\alpha }}}_0^+}}[ {{\underline{A}}}^-_m,\mathcal{L} _n]\nonumber \\&\qquad +\frac{1}{({ {{\underline{\alpha }}}_0^+})^2}[\mathcal{L} _m,\mathcal{L} _n]\nonumber \\&\quad = \frac{(m-n)}{{ {{\underline{\alpha }}}_0^+}} {{\underline{A}}}^-_{m+n} + \frac{2m^3}{({ {{\underline{\alpha }}}_0^+})^2}\delta _{m+n,0} \nonumber \\&\qquad + \frac{1}{({ {{\underline{\alpha }}}_0^+})^2} \left[ (n-m)\mathcal{L} _{m+n} + \frac{D-2}{12}n(n^2-1)\delta _{m+n,0}\right] \nonumber \\&\qquad - \frac{1}{({ {{\underline{\alpha }}}_0^+})^2} \Bigg [(m-n)\mathcal{L} _{m+n} + \frac{D-2}{12}m(m^2-1)\delta _{m+n,0} \Bigg ] \nonumber \\&\qquad + \frac{1}{({ {{\underline{\alpha }}}_0^+})^2} \left[ (m-n)\mathcal{L} _{m+n} + \frac{D-2}{12}m(m^2-1)\delta _{m+n,0} \right] \nonumber \\&\quad = \frac{(m-n)}{{ {{\underline{\alpha }}}_0^+}} \left[ {{\underline{A}}}^-_{m+n} - \frac{1}{{ {{\underline{\alpha }}}_0^+}}\mathcal{L} _{m+n} \right] - \frac{D-2}{12({ {{\underline{\alpha }}}_0^+})^2}n\delta _{m+n,0}\nonumber \\&\qquad +\frac{D-26}{12}\frac{n^3}{({ {{\underline{\alpha }}}_0^+})^2}\delta _{m+n,0}\nonumber \\&\quad =\frac{(m-n)}{{ {{\underline{\alpha }}}_0^+}} \left[ {{\underline{A}}}^-_{m+n} - \frac{1}{{ {{\underline{\alpha }}}_0^+}}\mathcal{L} _{m+n} +\frac{D-2}{24({ {{\underline{\alpha }}}_0^+})^2}\delta _{m+n,0}\right] \nonumber \\&\qquad +\frac{26-D}{12}\frac{m^3}{({ {{\underline{\alpha }}}_0^+})^2}\delta _{m+n,0}\nonumber \\&\quad \implies \left[ { {{\underline{\alpha }}}_0^+}{\widetilde{ {{\underline{A}}}}}^-_m, { {{\underline{\alpha }}}_0^+}\widetilde{{\underline{A}}}^-_n\right] = (m-n){ {{\underline{\alpha }}}_0^+}{\widetilde{ {{\underline{A}}}}}^-_{m+n}\nonumber \\&\qquad +\frac{26-D}{12}m^3\delta _{m+n,0}. \end{aligned}$$

(B16)

Appendix C: Derivation of hermiticity properties

We calculate the Hermitian conjugation of the $\widetilde{A}^-_m$ operators using explicitly, the mode expansion of the string solution. Thereby, we note that,

$$\begin{aligned} \left[ e^{i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\right] ^\dag&= e^{\left[ i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}\right] ^\dag } \nonumber \\&= \exp \left[ \frac{im}{\alpha 'p_0^+}\left( \frac{1}{2}x^+_0 - i \alpha 'p^+_0 \ln (z)\right. \right. \nonumber \\&\quad \left. \left. +i \sqrt{\frac{\alpha '}{2}} \sum _{n\ne 0} \frac{\alpha ^+_n}{n} z^{-n}\right) \right] ^\dag \nonumber \\&=\exp \left[ -\frac{im}{\alpha 'p_0^+}\left( \frac{1}{2}x^+_0 + i \alpha 'p^+_0 \ln ({{\bar{z}}})\right. \right. \nonumber \\&\quad \left. \left. - i \sqrt{\frac{\alpha '}{2}} \sum _{n\ne 0} \frac{\alpha ^+_{-n}}{n} {{\bar{z}}}^{-n}\right) \right] \nonumber \\&= \exp \left[ -\frac{im}{\alpha 'p_0^+}\left( \frac{1}{2}x^+_0 - i \alpha 'p^+_0 \ln (\frac{1}{{{\bar{z}}}})\right. \right. \nonumber \\&\quad \left. \left. + i \sqrt{\frac{\alpha '}{2}} \sum _{m\ne 0} \frac{\alpha ^+_{m}}{m} \left( \frac{1}{{{\bar{z}}}}\right) ^{-m}\right) \right] \nonumber \\&\implies \left[ e^{i\frac{m {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\right] ^\dag = e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}}, \end{aligned}$$

(C.1)

where we have used that $x_0^\mu$ and $p_0^\mu$ are Hermitian and $(\alpha ^+_n)^\dag =\alpha ^+_{-n}$. We also compute,

$$\begin{aligned}&[\partial {{\underline{L}}}^\mu (z)]^\dag \nonumber \\&\quad = \left[ -i \sqrt{\frac{\alpha '}{2}} \sum _{n\in {\mathbb {Z}}}\alpha ^\mu _n z^{-n-1}\right] ^\dag = i \sqrt{\frac{\alpha '}{2}} \sum _{n\in {\mathbb {Z}}} \alpha ^\mu _{-n}{{\bar{z}}}^{-n-1}\nonumber \\&\quad = i \sqrt{\frac{\alpha '}{2}} \frac{1}{{{\bar{z}}}^2}\sum _{m\in {\mathbb {Z}}}\alpha ^\mu _m \left( \frac{1}{{{\bar{z}}}}\right) ^{-m-1} \nonumber \\&\quad = -\frac{1}{{{\bar{z}}}^2}\left[ -i \sqrt{\frac{\alpha '}{2}} \sum _{m\in {\mathbb {Z}}}\alpha ^\mu _m \left( \frac{1}{{{\bar{z}}}}\right) ^{-m-1}\right] \nonumber \\&\quad \implies [\partial {{\underline{L}}}^\mu (z)]^\dag = -\frac{1}{{{\bar{z}}}^2} \partial {{\underline{L}}}^\mu \left( \frac{1}{{{\bar{z}}}}\right) , \end{aligned}$$

(C.2)

and the Hermitian conjugate of the second derivative,

$$\begin{aligned} {[}\partial ^2 {{\underline{L}}}^\mu (z)]^\dag&= -i \sqrt{\frac{\alpha '}{2}} \sum _{n}(n+1)\alpha ^\mu _{-n}{{\bar{z}}}^{-n-2} \nonumber \\&= -i \sqrt{\frac{\alpha '}{2}} \sum _m (1-m)\alpha ^\mu _m \left( \frac{1}{{{\bar{z}}}}\right) ^{-m-2} \frac{1}{{{\bar{z}}}^4} \nonumber \\&= \frac{1}{{{\bar{z}}}^4}i \sqrt{\frac{\alpha '}{2}} \sum _m (m+1) \alpha ^\mu _m \left( \frac{1}{{{\bar{z}}}}\right) ^{-m-2} \nonumber \\&\quad - 2 i \sqrt{\frac{\alpha '}{2}} \sum _m \alpha ^\mu _m \left( \frac{1}{{{\bar{z}}}}\right) ^{-m-1}\frac{1}{{{\bar{z}}}^{3}} \nonumber \\&\implies [\partial ^2 {{\underline{L}}}^\mu (z)]^\dag = \frac{1}{{{\bar{z}}}^4}\partial ^2 {{\underline{L}}}^\mu \left( \frac{1}{{{\bar{z}}}}\right) \nonumber \\&\quad + 2 \frac{1}{{{\bar{z}}}^{3}} \partial {{\underline{L}}}^\mu \left( \frac{1}{{{\bar{z}}}}\right) . \end{aligned}$$

(C.3)

Finally we compute the last piece,

$$\begin{aligned} \left[ {\hat{Q}}_{l; m}(E) \right] ^\dagger = -\oint \frac{d{{\bar{z}}}}{2\pi i}\frac{1}{{{\bar{z}}}^{l+1}}e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}}, \end{aligned}$$

(C.4)

where the $'-'$ outside fixes the direction of the loop to anti-clockwise after taking the complex conjugate. Then,

$$\begin{aligned} \left[ {\hat{Q}}_{l; m}(E) \right] ^\dagger&= ({{\bar{z}}})^2 \oint \frac{d\left( \frac{1}{{{\bar{z}}}}\right) }{2\pi i}\frac{1}{\left( \frac{1}{{{\bar{z}}}}\right) ^{-l+1} \left( \frac{1}{{{\bar{z}}}}\right) ^{-2}}e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}} \nonumber \\&= \oint \frac{d\left( \frac{1}{{{\bar{z}}}}\right) }{2\pi i}\frac{1}{\left( \frac{1}{{{\bar{z}}}}\right) ^{-l+1}}e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}} \nonumber \\&\implies \left[ {\hat{Q}}_{l; m}(E) \right] ^\dag = {\hat{Q}}_{-l;-m}(E). \end{aligned}$$

(C.5)

To compute the Hermitian conjugate of ${\widehat{ {{\underline{A}}}}}^-_m$, we first note that the action of H.C on a normal ordered product of two operators $A:= \partial {{\underline{L}}}^-$ and $B:= e^{\frac{im {{\underline{L}}}^+}{\alpha 'p_0^+}}$ is given by,

$$\begin{aligned} (:AB:)^\dag = B^\dag A^\dag - \langle {AB}\rangle ^\dag \nonumber \\ =~:A^\dag B^\dag : + [B^\dag ,A^\dag ]. \end{aligned}$$

(C.6)

Using the above results, we then observe that,

$$\begin{aligned} :\left( \partial {{\underline{L}}}^-(z) e^{\frac{im {{\underline{L}}}^+(z)}{\alpha 'p_0^+}}\right) :&=~:\left( -\frac{1}{{{\bar{z}}}^2}\right) \partial {{\underline{L}}}^-\left( \frac{1}{{{\bar{z}}}}\right) e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}}:\nonumber \\&\quad + \frac{im}{\alpha 'p_0^+}\left( -\frac{1}{{{\bar{z}}}^2}\right) (-i\alpha ')\frac{{{\bar{z}}}}{2}i e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}} \nonumber \\&\implies [{\widehat{ {{\underline{A}}}}}^-_m]^\dag = {\widehat{ {{\underline{A}}}}}^-_{-m} + \frac{m}{\alpha ^+_0}{\widehat{Q}}_{0;-m} \end{aligned}$$

(C.7)

Finally, we can see that,

$$\begin{aligned} {[} {{\underline{A}}}^-_m(E)]^\dag&= [{\widehat{ {{\underline{A}}}}}^-_m(E)]^\dag + i \sqrt{ \frac{ 2 }{ \alpha '} } \oint \frac{d{{\bar{z}}}}{2\pi i}:\nonumber \\&\quad \times \left[ \frac{im}{4p_0^+}\left( -\frac{1}{{{\bar{z}}}^2}\frac{\partial ^2 {{\underline{L}}}^+}{\partial {{\underline{L}}}^+} - \frac{2}{{{\bar{z}}}}\right) \right] e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}}: \nonumber \\&= {\widehat{ {{\underline{A}}}}}^-_{-m} + \frac{m}{\alpha ^+_0}{\widehat{Q}}_{0;-m} + i \sqrt{ \frac{ 2 }{ \alpha '} } \oint \frac{d\left( \frac{1}{{{\bar{z}}}}\right) }{2\pi i}:\nonumber \\&\quad \times \left( - \frac{i(-m)}{4 p_0^+}\frac{\partial ^2 {{\underline{L}}}^+}{\partial {{\underline{L}}}^+}\right) e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}}: \nonumber \\&\quad + i \sqrt{ \frac{ 2 }{ \alpha '} } \oint \frac{d\left( \frac{1}{{{\bar{z}}}}\right) }{2\pi i}:\frac{2im}{4 p_0^+}\frac{1}{\left( \frac{1}{{{\bar{z}}}}\right) ^1} e^{-\frac{im {{\underline{L}}}^+\left( \frac{1}{{{\bar{z}}}}\right) }{\alpha 'p_0^+}}: \nonumber \\&= {{\underline{A}}}^-_{-m}(E) + \frac{m}{\alpha ^+_0}{\widehat{Q}}_{0;-m} - \frac{m}{\alpha ^+_0}{\widehat{Q}}_{0;-m} \end{aligned}$$

(C.8)

$$\begin{aligned}&\implies [ {{\underline{A}}}^-_m(E)]^\dag = {{\underline{A}}}^-_{-m}(E) \end{aligned}$$

(C.9)

$$\begin{aligned}&\implies [ {{\underline{A}}}^-_{m}(E)]^{\dag \dag } = {{\underline{A}}}^-_m(E). \end{aligned}$$

(C.10)

We also note that the ${\widetilde{ {{\underline{A}}}}}^-_m$ follows the same algebra under Hermitian conjugation as ${{\underline{A}}}^-_m$. This is evident from its definition and the properties of the involved quantities derived in the preceding sections of the Appendix.

Appendix D: Details on Lorentz algebra

1.1 D.1 $M^{+ 2}$ Infinitesimal rotation

To better understand the discussion above, we explicitly compute the variation of the flat space DDF operators using an infinitesimal rotation such that,

$$\begin{aligned} E^{1'} = E^1 + \sqrt{2} \epsilon E^2,\quad E^{2'} = E^2 - \sqrt{2} \epsilon E^1, \end{aligned}$$

(D.1)

where the factor $\sqrt{2}$ has been chosen so that,

$$\begin{aligned} E^{\pm '} = E^\pm \pm \epsilon E^2,\quad E^{2'} = E^2 - \epsilon E^+ + \epsilon E^-. \end{aligned}$$

(D.2)

Under this condition we can write,

$$\begin{aligned}&e^{i m \frac{ E^{+'} \cdot L(z) }{\alpha 'E^{+'} \cdot p_0} }\nonumber \\&\quad = e^{i m \frac{ E^{+} \cdot L(z) }{\alpha 'E^{+} \cdot p_0} } \left[ 1 + \epsilon \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} \left( {{\underline{p}}}_0^+ {{\underline{L}}}^2 - {{\underline{p}}}_0^2 {{\underline{L}}}^+ \right) + O(\epsilon ^2) \right] \nonumber \\&= e^{i m \frac{ E^{+} \cdot L(z) }{\alpha 'E^{+} \cdot p_0} } \left[ 1 + \epsilon \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} \left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2 - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+ \right) + O(\epsilon ^2) \right] , \end{aligned}$$

(D.3)

where we have defined,

$$\begin{aligned} {{\underline{L}}}_{(\ne )}(z) = {{\underline{L}}}(z) - i \alpha ' {{\underline{p}}}_0 \ln (z), \end{aligned}$$

(D.4)

i.e. the part of ${{\underline{L}}}$ without the logarithmic cut. Note the normal ordering relations,

$$\begin{aligned} {{\underline{L}}}_{(\ne )}(z)\, {{\underline{L}}}_{(\ne )}(w) = :&{{\underline{L}}}_{(\ne )}(z)\, {{\underline{L}}}_{(\ne )}(w) : \nonumber \\&-\frac{\alpha '}{2} \ln \left( 1 - \frac{z}{w} \right) ,\ |z|>|w| \nonumber \\ {{\underline{L}}}(z)\, {{\underline{L}}}_{(\ne )}(w) = :&{{\underline{L}}}(z)\, {{\underline{L}}}_{(\ne )}(w) : -\frac{\alpha '}{2} \ln \left( z -w \right) , \end{aligned}$$

(D.5)

where the first one has no cuts.

It is important to use ${{\underline{L}}}_{(\ne )}$ if we want to use the usual formula

$$\begin{aligned} \left[ \oint _{z=0} \mathcal{A} (z), \oint _{w=0} \mathcal{E} (w)\right] = \oint _{w=0} \oint _{z=w} R[\mathcal{A} (z) \mathcal{E} (w)],\nonumber \\ \end{aligned}$$

(D.6)

since its derivation requires the absence of cuts.

From the previous expression we get,

$$\begin{aligned} \delta _\epsilon A^i_m(E)&= A^i_m(E') - A^i_m(E) \nonumber \\&= \epsilon \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} i \sqrt{\frac{2}{\alpha '}} \oint _{z=0} \frac{d z}{ 2\pi i} : \partial _z {{\underline{L}}}^i(z)\nonumber \\&\quad \times \left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2(z) - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+(z) \right) e^{i m \frac{ {{\underline{L}}}^+(z) }{\alpha ' {{\underline{p}}}^+_0} } : \nonumber \\&\quad + \delta ^{i 2} (-\epsilon ) i \sqrt{\frac{2}{\alpha '}} \oint _{z=0} \frac{d z}{ 2\pi i} :\nonumber \\&\quad \times \left( \partial _z {{\underline{L}}}^+(z) - \partial _z {{\underline{L}}}^-(z) \right) e^{i m \frac{ {{\underline{L}}}^+(z) }{\alpha ' {{\underline{p}}}^+_0} } : , \end{aligned}$$

(D.7)

where the contribution in the last line is only for $i=2$ and does not use ${{\underline{L}}}_{(\ne )}$, but ${{\underline{L}}}$ since it comes from the rotation of $\partial {{\underline{L}}}^2$. We shall study the variation above separately for two cases ($i\ne 2$ and $i=2$).

1.1.1 D.1.1 Rewriting the variation $\delta _\epsilon A^i_m(E)$ for $i\ne 2$

If the map** $A^i_n \leftrightarrow \alpha ^i_{n (lightcone)}$ is indeed true, we should have that the previous variation $\delta _\epsilon A^i_m(E)= \delta _\epsilon \alpha ^i_{m (lightcone)}$, up to terms which vanish in the lightcone gauge.

To check this, we evaluate the commutator,

$$\begin{aligned} {[}A^i_m(E), \delta _\epsilon A^j_n(E)], \end{aligned}$$

(D.8)

and then determine the coefficients $c_l$ and $\mathcal{N}$ in

$$\begin{aligned} \delta _\epsilon {{\underline{A}}}^j_n(E) =\sum _l c_l\, {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^2_{n-l}(E) + \mathcal{N} {{\tilde{ {{\underline{A}}}}}}^-_n(E), \end{aligned}$$

(D.9)

in order to reproduce the previous commutators. The expression in (D.9) is an ansatz based on the operator’s level and $SO(D-2)$ rotations.

We evaluate (D.7) for $j\ne 2$ as follows,

$$\begin{aligned}&{[} {{\underline{A}}}^i_m(E), ~ \delta _\epsilon {{\underline{A}}}^j_n(E)] \nonumber \\&\quad = \epsilon \frac{i n}{\alpha '( {{\underline{p}}}^+_0)^2}\cdot \oint _{w=0} \frac{d w}{2\pi i} \oint _{z=w} \frac{d z}{2\pi i} \Biggl [ \frac{ \delta ^{i j} }{ (z-w)^2 } \nonumber \\&\qquad \times \left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2(w) - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+(w) \right) + \frac{ \delta ^{i 2} }{ z-w } {{\underline{p}}}_0^+ \partial {{\underline{L}}}^j(w)\nonumber \\&\qquad + \frac{-2}{\alpha '} : \partial {{\underline{L}}}^i(z) \partial {{\underline{L}}}^j(w) \left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2(w)\right. \nonumber \\&\qquad \left. - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+(w) \right) : \Biggr ] \, e^{ i m \frac{ {{\underline{L}}}^+(z)}{\alpha 'p^+_0} } e^{i n \frac{ {{\underline{L}}}^+(w) }{\alpha ' {{\underline{p}}}^+_0} } \nonumber \\&\quad = \epsilon \frac{i n}{\alpha '( {{\underline{p}}}^+_0)^2} \oint _{w=0} \frac{d w}{2\pi i} \Biggl [ \delta ^{i j} \, \left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2(w)\right. \nonumber \\&\qquad \left. - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+(w) \right) \, \frac{i m}{\alpha ' {{\underline{p}}}^+_0} \, \partial {{\underline{L}}}^+(w) \, e^{i (m+n) \frac{ {{\underline{L}}}^+(w) }{\alpha ' {{\underline{p}}}^+_0} } \nonumber \\&\qquad + \delta ^{i 2} \, {{\underline{p}}}_0^+ \, \partial {{\underline{L}}}^j(w) \, e^{i (m+n) \frac{ {{\underline{L}}}^+(w) }{\alpha ' {{\underline{p}}}^+_0} } \Biggr ] . \end{aligned}$$

(D.10)

There are now two cases: $m+n\ne 0$ and $m+n=0$. For the first case, we can integrate by parts since, $\left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2(w) - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+(w) \right)$ has no cut and we can write,

$$\begin{aligned}&{[} {{\underline{A}}}^i_m, ~ \delta _\epsilon {{\underline{A}}}^j_n]\nonumber \\&\quad = \epsilon \frac{i n}{ \sqrt{2\alpha '} ( {{\underline{p}}}^+_0)^2 } \Biggl [ \left( - {{\underline{p}}}_0^+ {{\underline{A}}}^2_{m+n} \frac{m}{m+n} \delta ^{i j} + {{\underline{p}}}_0^+ {{\underline{A}}}^j_{m+n} \delta ^{i 2} \right) \nonumber \\&\qquad + {{\underline{p}}}_0^2 {{\underline{A}}}^+_{m+n} \frac{m}{m+n} \delta ^{i j} \Biggr ] , \end{aligned}$$

(D.11)

where we have defined,

$$\begin{aligned} {{\underline{A}}}^+_m(E) = i \sqrt{\frac{2}{\alpha '}} \oint \frac{d z}{2 \pi i} \partial {{\underline{L}}}^+(z) e^{ i m \frac{ {{\underline{L}}}^+(z)}{\alpha ' {{\underline{p}}}^+_0} } = \delta _{m,0} {{\underline{\alpha }}}^+_0,\nonumber \\ \end{aligned}$$

(D.12)

which is trivial for $m\ne 0$.

In order to determine the $c_l$ coefficients in Eq. (D.9), we insert the ansatz into the commutator and obtain,

$$\begin{aligned}&{[}A^i_m(E), \delta _\epsilon A^j_n(E)] = m \delta ^{i j}\, c_{-m}\, {{\underline{A}}}^2_{m+n}(E)\nonumber \\&\qquad + m \delta ^{i 2}\, c_{n+m}\, {{\underline{A}}}^j_{m+n}(E) , \nonumber \\&\quad \implies m\, c_{-m} = -\epsilon \frac{i}{\alpha ' {{\underline{p}}}^+_0} \frac{m n}{m+n} \left( -i \sqrt{ \frac{\alpha '}{2} } \right) ;~\nonumber \\&m\, c_{n+m} = +\epsilon \frac{i}{\alpha ' {{\underline{p}}}^+_0} n \left( -i \sqrt{ \frac{\alpha '}{2} } \right) , \end{aligned}$$

(D.13)

when compared with the actual result. Taking the ratio we get

$$\begin{aligned} \frac{ c_{-m} }{ c_{n+m} } = \frac{-m}{ n+m}, \end{aligned}$$

(D.14)

which suggests two possible classes of solutions and a combination of the two. The two basic possible solutions are,

$$\begin{aligned} c_l = {\tilde{c}}_1\, l,\quad c_l = {\hat{c}}_1 \frac{1}{n - l}. \end{aligned}$$

(D.15)

Inserting them into the original equations we get,

$$\begin{aligned} c_l&= {\tilde{c}}_1\, l \Rightarrow {\tilde{c}}_1 = \epsilon \frac{1}{ \sqrt{2\alpha '} {{\underline{p}}}_0^+} \frac{n}{m} \frac{1}{m+n} , \nonumber \\ c_l&= {\hat{c}}_1 \frac{1}{n - l} \Rightarrow {\hat{c}}_1 = - \epsilon \frac{1}{ \sqrt{2\alpha '} {{\underline{p}}}_0^+} n . \end{aligned}$$

(D.16)

Since $c_l$ may depend on n, but must not depend on m, only the second solution is viable.

Now, for the second case ($n+m=0$), we see that we get two contributions. The first one corresponds to the term $l=0$ in (D.9). The second one is related to the $\underline{M}^{+ 2}$ generator. Explicitly we get

$$\begin{aligned} {[} {{\underline{A}}}^i_{-n}(E), \delta _\epsilon {{\underline{A}}}^j_n(E) ]&= + \epsilon \frac{n}{ \sqrt{2\alpha '} {{\underline{p}}}_0^+} \delta ^{i 2} {{\underline{A}}}^j_0(E) \nonumber \\&\quad +~ \epsilon \frac{n^2}{(\alpha ' {{\underline{p}}}_0^+)^2} \delta ^{i j} \oint \frac{d z}{2 \pi i} ( {{\underline{p}}}_0^+ {{\underline{L}}}^2_{(\ne )}(z)\nonumber \\&\quad - {{\underline{p}}}_0^2 {{\underline{L}}}^+_{(\ne )}(z) )\partial {{\underline{L}}}^+(z). \end{aligned}$$

(D.17)

From the first term above, we see that, $c_0=- \epsilon \frac{1}{ \sqrt{2\alpha '} {{\underline{p}}}^+_0}$ as expected for the $l=0$ term.

We are left with the term $\epsilon \frac{n^2}{ \sqrt{2\alpha '} {{\underline{p}}}_0^+}\delta ^{ij}I^{(2)}$, where,

$$\begin{aligned} I^{(2)}= & {} \oint \frac{d z}{2 \pi i} ( {{\underline{p}}}_0^+ {{\underline{L}}}^2_{(\ne )}(z) - {{\underline{p}}}_0^2 {{\underline{L}}}^+_{(\ne )}(z) ) \partial {{\underline{L}}}^+(z)\nonumber \\\equiv & {} \frac{1}{2} {{\underline{p}}}_0^+ \underline{M}^{2 +}(E), \end{aligned}$$

(D.18)

and $\underline{M}^{+ 2}(E)$ is the Lorentz generator in the frame E. To see why this is so, we observe that,

$$\begin{aligned}&\oint \frac{d z}{2 \pi i} {{\underline{L}}}^+_{(\ne )}(z) \partial {{\underline{L}}}^+(z) \nonumber \\&\quad = \oint \frac{d z}{2 \pi i} {{\underline{L}}}^+_{(\ne )}(z) \partial \left( {{\underline{L}}}_{(\ne )}^+(z) -i \alpha ' {{\underline{p}}}_0^+ \ln (z) \right) \nonumber \\&\quad =~ -i \alpha ' {{\underline{p}}}_0^+ \oint \frac{d z}{2 \pi i z} {{\underline{L}}}^+_{(\ne )}(z) = -\frac{1}{2}i \alpha ' {{\underline{p}}}_0^+ {{\underline{x}}}_0^+ . \end{aligned}$$

(D.19)

Then the first term can be rewritten as,

$$\begin{aligned}&\oint \frac{d z}{2 \pi i} {{\underline{L}}}^2_{(\ne )}(z) \partial {{\underline{L}}}^+(z) \nonumber \\&\quad = + \oint \frac{d z}{2 \pi i} {{\underline{L}}}^2_{(\ne )}(z) \partial {{\underline{L}}}_{(\ne )}^+(z) -\frac{1}{2}i \alpha ' {{\underline{p}}}_0^+ {{\underline{x}}}_0^2 \nonumber \\&\quad = - \oint \frac{d z}{2 \pi i} {{\underline{L}}}^+_{(\ne )}(z) \partial {{\underline{L}}}_{(\ne )}^2(z) - \frac{1}{2}i \alpha ' {{\underline{p}}}_0^+ {{\underline{x}}}_0^2 , \end{aligned}$$

(D.20)

or taking the semi-sum as,

$$\begin{aligned} \displaystyle \oint \frac{d z}{2 \pi i} {{\underline{L}}}^2_{(\ne )}(z) \partial {{\underline{L}}}^+(z)= & {} \frac{1}{2}\displaystyle \oint \frac{d z}{2 \pi i} ( {{\underline{L}}}^2_{(\ne )}(z) \partial {{\underline{L}}}_{(\ne )}^+(z)\nonumber \\{} & {} - {{\underline{L}}}^+_{(\ne )}(z) \partial {{\underline{L}}}_{(\ne )}^2(z) ) - \frac{1}{2}i \alpha ' {{\underline{p}}}_0^+ {{\underline{x}}}_0^2.\nonumber \\ \end{aligned}$$

(D.21)

Assembling all the contributions and using $\partial {{\underline{L}}}$ instead of $\partial {{\underline{L}}}_{(\ne )}$ we get,

$$\begin{aligned} I^{(2)}&= \frac{1}{2} {{\underline{p}}}_0^+ \left[ \oint \frac{d z}{2 \pi i} ( {{\underline{L}}}^2_{(\ne )}(z) \partial {{\underline{L}}}^+(z) - {{\underline{L}}}^+_{(\ne )}(z) \partial {{\underline{L}}}^2(z) )\right. \nonumber \\&\quad \left. + \frac{1}{2}i \alpha '( {{\underline{p}}}_0^2 {{\underline{x}}}_0^+ - {{\underline{p}}}_0^+ {{\underline{x}}}_0^2 ) \right] \nonumber \\&= \frac{1}{2} {{\underline{p}}}_0^+ \underline{M}^{2 +}(E) , \end{aligned}$$

(D.22)

which matches (D.18) up to an overall factor in the frame E.

Noting that $M^{+2}$ rotates $x^2 \rightarrow x^+$ and $x^- \rightarrow x^2$ but keeps $x^+$ fixed, the term $I^{(2)}$ commutes with all DDFs $A^i_m$ with $i\ne 2$. Explicitly we get,

$$\begin{aligned} {[} {{\underline{A}}}^i_{-m}(E), I^{(2)}] = -\frac{1}{2}\sqrt{ \frac{\alpha '}{2} } \delta ^{i 2} ( {{\underline{\alpha }}}^+_0)^2 \delta _{m, 0}. \end{aligned}$$

(D.23)

Assembling all contributions for $j\ne 2$ we get,

$$\begin{aligned} \delta _\epsilon {{\underline{A}}}^j_n(E)&= - \epsilon \frac{n}{ \sqrt{2\alpha '} {{\underline{p}}}^+_0}\left[ \sum _{l\ne n} \frac{1}{n-l} {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^2_{n-l}(E)\right. \nonumber \\&\quad \left. + \frac{1}{ \sqrt{2\alpha '} {{\underline{p}}}^+_0} \underline{M}^{+ 2}(E) {{\underline{A}}}^j_n(E)\right] , \end{aligned}$$

(D.24)

where the contribution from ${{\underline{A}}}^-_n$ is absent since in $\delta _\epsilon {{\underline{A}}}^{j\ne 2}_n(E)$ there are no ${{\underline{\alpha }}}^-$. There are also no normal ordering issues in (D.24) since, $\underline{M}^{+ 2}(E)$ commutes with all DDFs as noted above.

1.1.2 D.1.2 Normal ordering subtlety in the $i=2$ case

In (D.7) we would like to consider the details of the variation $\delta _\epsilon A^2_m(E)$ since some normal ordering is required and because the integrand $\partial {{\underline{L}}}^- e^{i k^. {{\underline{L}}}^+}$ is not a good conformal operator - it has a cubic pole in the OPE with the stress energy tensor.

The key observation is that, $\lim _{z \rightarrow w} \partial {{\underline{L}}}^2(z) e^{i k^. {{\underline{L}}}^+(w)}$ is a perfectly well defined conformal operator and therefore its infinitesimal Lorentz rotation must be as well. We expect it to be given by the normal ordering of the single components rotations but other quantum i.e. vanishing classically terms may in principle appear.

We start by computing the infinitesimal Lorentz rotation,

$$\begin{aligned}&\left[ \sqrt{2} \epsilon M^{1 2} , \partial {{\underline{L}}}^2(z) e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w)}\right] = \delta _\epsilon (\partial {{\underline{L}}}^2(z)) e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w)}\nonumber \\&\qquad +\partial {{\underline{L}}}^2(z) \delta _\epsilon \left( e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w)} \right) \nonumber \\&\quad = - \epsilon ( \partial {{\underline{L}}}^+(z) -\partial {{\underline{L}}}^-(z) ) \, e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w)} \nonumber \\&\qquad + \partial {{\underline{L}}}^2(z) \, \left( \epsilon \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} \left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2(w)\right. \right. \nonumber \\&\qquad \left. \left. - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+(w) \right) e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w)} \right) . \end{aligned}$$

(D.25)

We can then evaluate the singular part of the OPE to get,

$$\begin{aligned}&\left. \left[ \sqrt{2} \epsilon M^{1 2} , \partial {{\underline{L}}}^2(z) e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w)} \right] \right| _{\text{ sing }}\nonumber \\&\quad = \frac{-\epsilon \alpha '}{2(z-w)} \left( i \frac{m}{\alpha ' {{\underline{p}}}_0^+} \eta ^{+-} + \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} {{\underline{p}}}_0^+ \right) e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w)} , \end{aligned}$$

(D.26)

which vanishes and therefore the infinitesimal Lorentz rotation is given by the normal ordering of the rotations of the single components.

Finally the infinitesimal Lorentz rotation is,

$$\begin{aligned}&\delta _\epsilon \left( \partial {{\underline{L}}}^2 e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+} \right) \nonumber \\&\quad = \epsilon \, : \left[ \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} \partial {{\underline{L}}}^2 \left( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2 - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+ \right) \right. \nonumber \\&\qquad \left. - (\partial {{\underline{L}}}^+ -\partial {{\underline{L}}}^- ) \right] e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+} :\nonumber \\&\quad = \epsilon \, : \left[ \partial {{\underline{L}}}^- + \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} ( {{\underline{p}}}_0^+ {{\underline{L}}}_{(\ne )}^2 - {{\underline{p}}}_0^2 {{\underline{L}}}_{(\ne )}^+ ) \partial {{\underline{L}}}^2 \right] e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+} : \nonumber \\&\qquad - \epsilon \partial \left( \frac{\alpha ' {{\underline{p}}}_0^+}{i m} e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+} \right) , \end{aligned}$$

(D.27)

which contains pieces which are not good conformal operators. However the sum of first two operators is free from cubic poles,

$T= - \frac{2}{\alpha '}: (\partial L)^2:$ to be

$$\begin{aligned}&-\frac{\alpha '}{2} T(z) \, \left( \partial {{\underline{L}}}^- e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+} \right) (w) \nonumber \\&\quad = \left. 2 e^{ \delta \cdot \partial L(z) } \right| _{\delta _\mu \delta _\nu \rightarrow \eta _{\mu \nu } } \, \left. e^{ \epsilon \cdot \partial L + i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+ } \right| _{\epsilon _-} \nonumber \\&\quad = 2 : e^{ \delta \cdot {{\underline{L}}}(z) + \epsilon \cdot \partial L(w) + i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+(w) } :\nonumber \\&\qquad \times \left. e^{ - \frac{\alpha '}{2} \frac{\epsilon \cdot \delta }{ (z-w)^2} - \frac{\alpha '}{2} \frac{i \delta \cdot k}{ z-w} } \right| _{\delta _\mu \delta _\nu \rightarrow \eta _{\mu \nu },~ \epsilon _-, k_+ \rightarrow \frac{m}{\alpha ' {{\underline{p}}}_0^+}} \nonumber \\&\quad = 2 \left( \frac{\alpha '}{2} \right) ^2 \frac{ i \epsilon \cdot k }{ (z-w)^3 } + \dots \nonumber \\&\quad = 2 \left( \frac{\alpha '}{2} \right) ^2 \frac{ i \frac{m}{\alpha ' {{\underline{p}}}_0^+} \eta ^{+-} }{ (z-w)^3 } + \cdots . \end{aligned}$$

(D.28)

In a similar way we get

$$\begin{aligned}&-\frac{\alpha '}{2} T(z) \, \left( \partial {{\underline{L}}}^2 L^2_{(\ne )} e^{i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+} \right) (w) \nonumber \\&\quad = \left. 2 e^{ \delta \cdot \partial L(z) } \right| _{\delta _\mu \delta _\nu \rightarrow \eta _{\mu \nu } } \, \left. e^{ \epsilon _{(1)} \cdot \partial L + \epsilon _{(2)} \cdot \partial L_{(\ne )} + i \frac{m}{\alpha ' {{\underline{p}}}_0^+} {{\underline{L}}}^+ } \right| _{\epsilon _{(1) 2}, \epsilon _{(2) 2}} \nonumber \\&\quad = 2 \left( \frac{\alpha '}{2} \right) ^2 \frac{ \epsilon _{(1)} \cdot \epsilon _{(2)} }{ (z-w)^3 } + \cdots \nonumber \\&\quad = 2 \left( \frac{\alpha '}{2} \right) ^2 \frac{ 1 }{ (z-w)^3 } + \cdots . \end{aligned}$$

(D.29)

Now we can examine the cubic pole contributions and see that,

$$\begin{aligned} -\frac{\alpha '}{2} T(z) \, \left( \dots \right)&= 2 \left( \frac{\alpha '}{2} \right) ^2 \frac{ -i \frac{m}{\alpha ' {{\underline{p}}}_0^+} }{ (z-w)^3 } + \frac{ i m }{\alpha '( {{\underline{p}}}^+_0)^2} \, {{\underline{p}}}_0^+\nonumber \\&\quad \times 2 \left( \frac{\alpha '}{2} \right) ^2 \frac{ 1 }{ (z-w)^3 } + \dots , \end{aligned}$$

(D.30)

which in fact vanishes.

1.1.3 D.1.3 Rewriting the variation $\delta _\epsilon A^j_m(E)$ for $j=2$

Finally we consider the $j=2$ case where there is a further contribution in eq. (D.7), when $i=2$. If we proceed as before, we can evaluate the full commutator as,

$$\begin{aligned} {[}A^i_m(E), \delta _\epsilon A^{j=2}_n]&= \epsilon \frac{i n}{ {{\underline{\alpha }}}^+_0} \left( - {{\underline{A}}}^2_{m+n} \frac{m}{m+n} + {{\underline{A}}}^2_{m+n} \right) \delta ^{i 2}\nonumber \\&\quad -m \delta ^{j 2} \frac{\epsilon }{ {{\underline{\alpha }}}^+_0} {{\underline{A}}}^i_{m+n} . \end{aligned}$$

(D.31)

We observe that the ansatz in (D.9) can generate the first term (in parentheses), wherein we get explicitly,

$$\begin{aligned} {c}_{-m} + {c}_{m+n} = -\frac{i \epsilon }{ {{\underline{\alpha }}}^+_0} \frac{n^2}{m(m+n)}. \end{aligned}$$

(D.32)

Once again, there are two basic possible linear homogeneous solutions to this recurrence relation, given by (D.15). Furthermore, we must choose the second possibility for the exact same reason as before, to obtain,

$$\begin{aligned} c_{l} = -\frac{i \epsilon }{ {{\underline{\alpha }}}^+_0} \frac{n}{n-l}. \end{aligned}$$

(D.33)

We see that this term is neatly absorbed in the expansion of $\delta _\epsilon A^{j}$, which now reads,

$$\begin{aligned} \delta _\epsilon {{\underline{A}}}^j_n(E)&= - i\epsilon \frac{n}{ \sqrt{2\alpha '} {{\underline{p}}}^+_0}\left[ \sum _{l\ne n} \frac{1}{n-l} {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^2_{n-l}(E)\right. \nonumber \\&\quad \left. + \frac{1}{ \sqrt{2\alpha '} {{\underline{p}}}^+_0} \underline{M}^{+ 2}(E) {{\underline{A}}}^j_n(E)\right] , \end{aligned}$$

(D.34)

and is no longer limited to $j\ne 2$.

Finally, we consider the last piece in (D.31) - since there are no more terms like $\delta ^{i \dots }$, we assume an expression like,

$$\begin{aligned}{} & {} \delta _\epsilon {{\underline{A}}}^{j=2}_n(E)|_{\text{ extra }} \nonumber \\{} & {} \quad = \delta ^{j 2} \left[ \sum _l d_l\, {{\underline{A}}}^k_l(E)\, {{\underline{A}}}^k_{n-l}(E) + {\bar{\mathcal{N} }} {\tilde{{\underline{A}}}}^-_n(E) \right] , \end{aligned}$$

(D.35)

where, ’extra’ labels the last term of (D.31). Comparison with the true expression gives,

$$\begin{aligned} d_{m+n} + d_{-m} = \frac{- i \epsilon }{ \sqrt{2\alpha '} {{\underline{p}}}_0^+}, \end{aligned}$$

(D.36)

whose simplest solution is,

$$\begin{aligned} d_l = \frac{1}{2}\frac{- i \epsilon }{ \sqrt{2\alpha '} {{\underline{p}}}_0^+}. \end{aligned}$$

(D.37)

Putting everything together, the complete variation is given by,

$$\begin{aligned} \delta _\epsilon {{\underline{A}}}^j_n(E)&= - i\epsilon \frac{n}{ \sqrt{2\alpha '} {{\underline{p}}}^+_0}\left[ \sum _{l\ne n} \frac{1}{n-l} {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^2_{n-l}(E)\right. \nonumber \\&\quad \left. + \frac{1}{ \sqrt{2\alpha '} {{\underline{p}}}^+_0} \underline{M}^{+ 2}(E) {{\underline{A}}}^j_n(E)\right] \nonumber \\&\quad - \delta ^{j 2} \left[ \frac{1}{2}\frac{i \epsilon }{ \sqrt{2\alpha '} {{\underline{p}}}_0^+} \sum _l {{\underline{A}}}^k_l(E)\, {{\underline{A}}}^k_{n-l}(E)\right. \nonumber \\&\quad \left. + {\bar{\mathcal{N} }} {{\tilde{ {{\underline{A}}}}}}^-_n(E) \right] . \end{aligned}$$

(D.38)

Appendix E: Details on Lorentz generators expressed DDF and Brower operators

In this appendix we report the details of how to write the action of Lorentz generators on DDF and Brower operators using DDF and Brower operators. This is not the same as expressing the Lorentz generators using DDF and Brower operators since they span a subset of all covariant operators.

1.1 E.1 Preliminaries

The covariant Lorentz generator

$$\begin{aligned} M^{\mu \nu } = x_0^\mu p_0^\nu - x_0^\nu p_0^\nu + i \sum _{n\ne 0} \frac{ \alpha ^\mu _n \alpha ^\nu _{-n} }{n}, \end{aligned}$$

(E.1)

satisfies

$$\begin{aligned} {[}M^{\mu \nu }, x_0^\lambda ]&= 2 i \delta ^{\lambda [\mu } x_0^{\nu ]} , \nonumber \\ {[}M^{\mu \nu }, \alpha _m^\lambda ]&= 2 i \delta ^{\lambda [\mu } \alpha _m^{\nu ]} , \end{aligned}$$

(E.2)

or for the chiral field

$$\begin{aligned} {[}M^{\mu \nu }, L(z)^\lambda ]&= 2 i \delta ^{\lambda [\mu } L(z)^{\nu ]} . \end{aligned}$$

(E.3)

1.1.1 E.1.1 A comment on commutators involving the not derived L(z)

Naively one would expect that the operator

$$\begin{aligned} {\widehat{{\hat{M}}}}^{\mu \nu }&= \mathcal{N} \oint _{z=0} \frac{d z}{2\pi i} \left( L^\mu (z) \partial L^\nu (z) - L^\nu (z) \partial L^\mu (z) \right) , \end{aligned}$$

(E.4)

be the Lorentz generator. But it is not so because of the term $-i \alpha 'p_0^\mu \, \ln z$ which implies a cut. Then one could think of

$$\begin{aligned} {\hat{M}}^{\mu \nu }&= \mathcal{N} \oint _{z=0} \frac{d z}{2\pi i} \left( L^\mu _{(\ne )}(z) \partial L^\nu (z) - L^\nu _{(\ne )}(z) \partial L^\mu (z) \right) , \nonumber \\&\quad \mathcal{N} = \frac{i}{\alpha '} \end{aligned}$$

(E.5)

be the Lorentz generator. But it is not so. We can evaluate directly the operator in terms of the oscillators as

$$\begin{aligned} \frac{1}{\mathcal{N} } {\hat{M}}^{\mu \nu }&= -i \frac{\alpha '}{2} \left( x_0^\mu p_0^\nu - x_0^\nu p_0^\mu \right) + \alpha '\sum _{m \ne 0} \frac{1}{m} \alpha _m^\mu \alpha _{-m}^\nu , \end{aligned}$$

(E.6)

and then the commutators

$$\begin{aligned} \frac{1}{\mathcal{N} } [{\hat{M}}^{\mu \nu } ,x^\lambda ]&= \alpha '\delta ^{ [\mu } x_0^{\nu ]} , \nonumber \\ \frac{1}{\mathcal{N} } [{\hat{M}}^{\mu \nu } ,\alpha _l^\lambda ]&= 2 \alpha '\delta ^{ [\mu } \alpha _l^{\nu ]} , \end{aligned}$$

(E.7)

which show explicitly what stated since the two variations differ by a factor 2.

We can evaluate the same using the fields. To this aim one has to go back to the original definitions and compute

$$\begin{aligned}&\frac{1}{\mathcal{N} } [{\hat{M}}^{\mu \nu } ,L^\lambda (w)] = \nonumber \\&\quad = \oint _{z=0, |z|>|w| } \frac{d z}{2\pi i} \left( L^\mu _{(\ne )}(z) \partial L^\nu (z)\right. \nonumber \\&\qquad \left. - L^\nu _{(\ne )}(z) \partial L^\mu (z) \right) \, L^\lambda (w)\nonumber \\&\qquad - L^\lambda (w) \, \oint _{z=0, |z|<|w| } \frac{d z}{2\pi i} \left( L^\mu _{(\ne )}(z) \partial L^\nu (z)\right. \nonumber \\&\qquad \left. - L^\nu _{(\ne )}(z) \partial L^\mu (z) \right) , \end{aligned}$$

(E.8)

and carefully use eq.s (D.5).

The proper Lorentz generator is then

$$\begin{aligned} M^{\mu \nu } = {\hat{M}}^{\mu \nu } + \frac{1}{2}( x^\mu _0 p^\nu _0 - x_0^\nu p_0^\mu ) . \end{aligned}$$

(E.9)

1.2 E.2 Generators $\underline{M}^{i j}$

It is immediate to compute

$$\begin{aligned} {[}\underline{M}^{i j}, \, {{\underline{L}}}^k]&= 2 i \eta ^{k [i} {{\underline{L}}}^{j]} , \nonumber \\ {[}\underline{M}^{i j}, \, {{\underline{L}}}^+]&= 0 , \nonumber \\ {[}\underline{M}^{i j}, \, {{\underline{L}}}^-]&= 0 , \end{aligned}$$

(E.10)

so that

$$\begin{aligned} {[}\underline{M}^{i j}, \, {{\underline{A}}}^k_n]&= 2 i \eta ^{k [i} {{\underline{A}}}_n^{j]} , \nonumber \\ {[}\underline{M}^{i j}, \, {{\underline{A}}}_n^-]&= [\underline{M}^{i j}, \, \tilde{{\underline{A}}}_n^-] = 0 . \end{aligned}$$

(E.11)

Then we are led to the expression for the physical states

$$\begin{aligned} \underline{M}^{i j}|_{DDF} = i \sum _{n\ne 0} \frac{1}{n} {{\underline{A}}}_n^{i} {{\underline{A}}}^j_{-n}. \end{aligned}$$

(E.12)

This expression reproduces the desired commutation relations for ${{\underline{A}}}_n$ with $n\ne 0$ but fails for ${{\underline{A}}}_0$. Let us see whether it is not possible to add some zero modes contributions to fix this issue. For example the natural guess $2 {{\underline{x}}}_0^{[i} {{\underline{p}}}_0^{j]}$ fails. In facts

$$\begin{aligned} {[} 2 {{\underline{x}}}_0^{[i} {{\underline{p}}}_0^{j]},\, {{\underline{A}}}^k_m] = 2 i \eta ^{k [i} {{\underline{A}}}_0^{j]}, \end{aligned}$$

(E.13)

for all n because of the ${{\underline{x}}}_0$ zero mode interacts with $\partial {{\underline{L}}}$. But this failure can be compensated as

$$\begin{aligned} \left[ -2 i \sum _{n\ne 0} \frac{1}{n} {{\underline{A}}}_n^{[i} {{\underline{A}}}_0^{j]},\, {{\underline{A}}}^k_m\right] =&-2 i \eta ^{k [i} {{\underline{A}}}_0^{j]} ,~~~m\ne 0 . \end{aligned}$$

(E.14)

The final expression which is valid also for ${{\underline{A}}}_0^i$ is then

$$\begin{aligned} \underline{M}^{i j} =&i \sum _{n\ne 0} \frac{1}{n} {{\underline{A}}}_n^{i} {{\underline{A}}}^j_{-n} \nonumber \\&- 2 i \sum _{n\ne 0} \frac{1}{n} {{\underline{A}}}_n^{[i} {{\underline{A}}}_0^{j]} + \frac{2}{ \sqrt{2\alpha '} } {{\underline{x}}}_0^{[i} {{\underline{A}}}_0^{j]} . \end{aligned}$$

(E.15)

1.3 E.3 Generators $\underline{M}^{+ i}$

It again is immediate to compute

$$\begin{aligned} {[}\underline{M}^{+ i}, \, {{\underline{L}}}^k]&= - i \delta ^{k i} {{\underline{L}}}^{+} , \nonumber \\ {[}\underline{M}^{+ i}, \, {{\underline{L}}}^+]&= 0 , \nonumber \\ {[}\underline{M}^{+ i}, \, {{\underline{L}}}^-]&= -i {{\underline{L}}}^i , \end{aligned}$$

(E.16)

so that

$$\begin{aligned} {[}\underline{M}^{+ i}, \, {{\underline{A}}}^k_n]&= - i \delta ^{k i} {{\underline{A}}}_0^{+}\, \delta _{n, 0} , \nonumber \\ {[}\underline{M}^{+ i}, \, {{\underline{A}}}_n^-]&= -i {{\underline{A}}}^i_n , \nonumber \\ {[}\underline{M}^{+ i}, \, {\tilde{ {{\underline{A}}}}}_n^-]&= 0 . \end{aligned}$$

(E.17)

Then we are led to the expression for the physical states which do not require the use of DDF zero modes ${{\underline{A}}}_0$

$$\begin{aligned} \underline{M}^{+ i}|_{DDF} = 0. \end{aligned}$$

(E.18)

Notice that this expression is valid only when using ${{\underline{A}}}^i$ and the improved Brower operators ${\tilde{ {{\underline{A}}}}}^-$. This means that when we have an expression with ${{\underline{A}}}^-$ it has to be reexpressed using ${\tilde{ {{\underline{A}}}}}^-$ and $\mathcal{L}$: this observation is important for the computation of the expectation values of the Casimir operators of the different physical states obtained by the DDF operators.

1.4 E.4 Generator $\underline{M}^{+ -}$

It again is immediate to compute

$$\begin{aligned} {[}\underline{M}^{+ -}, \, {{\underline{L}}}^k]&= 0 , \nonumber \\ {[}\underline{M}^{+ -}, \, {{\underline{L}}}^+]&= +i {{\underline{L}}}^- , \nonumber \\ {[}\underline{M}^{+ -}, \, {{\underline{L}}}^-]&= -i {{\underline{L}}}^+ , \end{aligned}$$

(E.19)

so that

$$\begin{aligned} {[}\underline{M}^{+ -}, \, {{\underline{A}}}^k_n]&= 0 , \nonumber \\ {[}\underline{M}^{+ -}, \, {{\underline{A}}}_n^-]&= -i {{\underline{A}}}^-_n , \nonumber \\ {[}\underline{M}^{+ -}, \, {\tilde{ {{\underline{A}}}}}_n^-]&= -i {\tilde{ {{\underline{A}}}}}_n^- . \end{aligned}$$

(E.20)

Notice that these results rely on the invariance of $\exp \left( i \frac{n}{\alpha '} \frac{ {{\underline{L}}}^+}{ {{\underline{p}}}_0^+} \right)$.

Exactly as for the generators $\underline{M}^{+ i}$ it cannot be represented using ${{\underline{A}}}^i$ and ${\tilde{ {{\underline{A}}}}}^-$, not even in the physical sector. The reason is that it can be only a functional of ${\tilde{ {{\underline{A}}}}}_n^-$ since $\underline{M}^{+ -}$ commutes with all ${{\underline{A}}}^i$ but ${\tilde{ {{\underline{A}}}}}_n^-$ satisfy a Virasoro algebra and therefore

1.5 E.5 Generators $\underline{M}^{- i}$

These are the most interesting generators since they combine a true rotation and a compensating conformal transformation in the transverse directions.

It again is immediate to compute

$$\begin{aligned} {[}\underline{M}^{- i}, \, {{\underline{L}}}^k]&= -i \delta ^{k i} {{\underline{L}}}^- , \nonumber \\ {[}\underline{M}^{- i}, \, {{\underline{L}}}^+]&= -i {{\underline{L}}}^i , \nonumber \\ {[}\underline{M}^{- i}, \, {{\underline{L}}}^-]&= 0 . \end{aligned}$$

(E.21)

Because of the non trivial rotation of ${{\underline{L}}}^+$ is not immediate to compute the effect on DDF and Brower operators.

The final result is that

$$\begin{aligned}&{[}\underline{M}^{- i}(E), \, {{\underline{A}}}^k_n(E)] \nonumber \\&\quad = i \frac{n}{ {{\underline{\alpha }}}^+_0} \sum _{l\ne n} \frac{1}{n-l} {{\underline{A}}}^k_l(E)\, {{\underline{A}}}^i_{n-l}(E)\nonumber \\&\qquad +\frac{n}{( {{\underline{\alpha }}}^+_0)^2} \underline{M}^{+ i}(E) {{\underline{A}}}^k_n(E)\nonumber \\&\qquad + \delta ^{k i} \left[ - \frac{i}{2 {{\underline{\alpha }}}_0^+} \sum _l {{\underline{A}}}^j_l(E)\, {{\underline{A}}}^j_{n-l}(E) - i {{\tilde{ {{\underline{A}}}}}}^-_n(E) \right] . \nonumber \\&\quad = i \frac{n}{ {{\underline{\alpha }}}^+_0} \sum _{l\ne n} \frac{1}{n-l} {{\underline{A}}}^k_l(E)\, {{\underline{A}}}^i_{n-l}(E)\nonumber \\&\qquad + \frac{n}{( {{\underline{\alpha }}}^+_0)^2} \underline{M}^{+ i}(E) {{\underline{A}}}^k_n(E) - i \delta ^{k i} { {{\underline{A}}}}^-_n(E), . \end{aligned}$$

(E.22)

The expression for the variation of ${{\underline{A}}}^k$ under $\underline{M}^{- i}(E)$ may seem quite strange since it involves $\underline{M}^{+ i}(E)$ but a simple consistency check is immediate to perform: the scaling w.r.t. $\underline{M}^{+ -}(E)$ works because of $1/ ( {{\underline{\alpha }}}_0^+)^2$. The same is true for the first term.

The previous result is obtained first computing the variation of ${{\underline{A}}}^k$ then inferring the expression of the generator $\underline{M}^{- i}$ and then using this expression we can compute the variation of ${{\underline{A}}}^-$ and ${\tilde{ {{\underline{A}}}}}^-$. The expression of the generator $\underline{M}^{- i}$ restricted to DDF operators is

$$\begin{aligned} \underline{M}^{- i}|_{DDF} =&i \sum _{m\ne 0} \frac{1}{m} {{\underline{A}}}_m^-(E) {{\underline{A}}}^i_{-m}(E) - \frac{1}{ ( {{\underline{\alpha }}}_0^+)^2} \underline{M}^{+ i}(E) \mathcal{L} _0(E) \nonumber \\ =&i \sum _{m\ne 0} \frac{1}{m} {{\tilde{ {{\underline{A}}}}}}_m^-(E) {{\underline{A}}}^i_{-m}(E)\nonumber \\&+\frac{i}{ 2 {{\underline{\alpha }}}_0^+} \sum _{m\ne 0} \frac{1}{m} \mathcal{L} _m(E) {{\underline{A}}}^i_{-m}(E) \nonumber \\&- \frac{1}{ ( {{\underline{\alpha }}}_0^+)^2} \underline{M}^{+ i} \mathcal{L} _0(E) , \end{aligned}$$

(E.23)

where $\mathcal{L} _n$ is defined in (2.13). The first term in the r.h.s. of the first line gives to the first and third terms of the final form of the ${{\underline{A}}}^k(E)$ variation.

Notice that there are apparently normal ordering problems since ${{\underline{A}}}_m^-(E)$ and ${{\underline{A}}}^i_{-m}(E)$ do not commute, however the ambiguity is proportional to the zero modes which are not DDF operators.

Appendix F: Details on extracting Poincaré irreps from Lorentz irreps

In this appendix we would like to give some details on how to perform the splitting of Lorentz polarizations into Poincaré irreps. This is useful if we want to compute the Casimirs in the covariant formalism.

We consider massive states with mass M and perform the decomposition in rest frame and then we covariantize the result.

We use $I,J, \ldots =1,2, \dots D-1$.

1.1 F.1 Lorentz vector splits

Let us consider the Lorentz vector $T_\mu$. In the rest frame from the $SO(D-1)$ point of view we have actually two irreps

$$\begin{aligned} T_\mu \rightarrow T_I\, \oplus \, T_0, \end{aligned}$$

(F.1)

which have two different Casimirs.

This split seems to break Lorentz invariance but it is nor so. We have in fact the following covariant characterization of the two irreps⁵

(F.2)

Finally we can write the decomposition of the Lorentz vector into Poincaré irreps as

(F.3)

1.2 F.2 Lorentz antisymmetric two tensor splits

Let us consider the Lorentz antisymmetric tensor $A_{[\mu \nu ]}$. From the $SO(D-1)$ point of view in the rest frame we have actually two irreps

$$\begin{aligned} A_{[\mu \nu ]} \rightarrow A_{[I J]}\, \oplus \, A_{[I 0]}, \end{aligned}$$

(F.4)

which again have two different Casimirs.

The covariant characterization is

(F.5)

All these relations can be verified either directly or in rest frame.

Finally we can write the decomposition of the Lorentz antisymmetric two tensor into Poincaré irreps as

(F.6)

1.2.1 F.2.1 Lorentz symmetric traceless two tensor splits

Let us consider the Lorentz symmetric traceless tensor $S_{\mu \nu }$. The traceless condition means that

$$\begin{aligned} S_{00} = S_{K K}. \end{aligned}$$

(F.7)

From the $SO(D-1)$ point of view in the rest frame we have actually three irreps

$$\begin{aligned} S_{\mu \nu } \rightarrow \left( S_{I J}- \frac{1}{D-1}\delta _{I J} S_{K K} \right) \, \oplus \, S_{I 0}\, \oplus \, S_{0 0} \equiv S_{K K},\nonumber \\ \end{aligned}$$

(F.8)

which have once again three different Casimirs.

The covariant characterization of the lowest irreps is easy and reads

(F.9)

Notice that not all tensors are traceless, i.e.

(F.10)

While the highest irrep is a little trickier since one would wrongly guess that the last irrep is the left over

(F.11)

but this is wrong since it misses $\frac{1}{D-1}\delta _{I J} S_{K K}$ in the irrep definition. The proper definition is

(F.12)

with $P_{\mu \nu } = \eta _{\mu \nu } - \frac{k_\mu k_\nu }{ k^2}$ so that $P_{\mu \nu } k^\nu =0$ and $P^\lambda _\lambda = D-1$. This expression can be verified in the rest frame and making use of $S_{00} = S_{K K}$. It has the wanted properties

(F.13)

Finally we can write the decomposition of the Lorentz symmetric traceless two tensor into Poincaré irreps as

(F.14)