Gluing II: boundary localization and gluing formulas

Abstract

We describe applications of the gluing formalism discussed in the companion paper. When a d-dimensional local theory $\hbox {QFT}_d$ is supersymmetric, and if we can find a supersymmetric polarization for $\hbox {QFT}_d$ quantized on a $(d-1)$-manifold W, gluing along W is described by a non-local $\hbox {QFT}_{d-1}$ that has an induced supersymmetry. Applying supersymmetric localization to $\hbox {QFT}_{d-1}$, which we refer to as the boundary localization, allows in some cases to represent gluing by finite-dimensional integrals over appropriate spaces of supersymmetric boundary conditions. We follow this strategy to derive a number of “gluing formulas” in various dimensions, some of which are new and some of which have been previously conjectured. First we show how gluing in supersymmetric quantum mechanics can reduce to a sum over a finite set of boundary conditions. Then we derive two gluing formulas for 3D ${\mathcal {N}}=4$ theories on spheres: one providing the Coulomb branch representation of gluing and another providing the Higgs branch representation. This allows to study various properties of their (2, 2)-preserving boundary conditions in relation to mirror symmetry. After that we derive a gluing formula in 4D ${\mathcal {N}}=2$ theories on spheres, both squashed and round. First we apply it to predict the hemisphere partition function, then we apply it to the study of boundary conditions and domain walls in these theories. Finally, we mention how to glue half-indices of 4D ${\mathcal {N}}=2$ theories.

Open-ended examples

Here we would like to present a few more examples where we can explicitly describe the supersymmetric gluing theory. However, due to additional subtleties, we do not attempt localizing it and leave it for the future work and hence the name “open-ended”. One such example is on gluing 2D ${{\mathcal {N}}}=(2,2)$ theories quantized on $S^1$, e.g., how to glue a sphere from two hemispheres. Another example is about gluing 3D ${{\mathcal {N}}}=2$ indices from half-indices.

1.1 2D ${{\mathcal {N}}}=(2,2)$ theories quantized on a circle

Let us consider 2D ${{\mathcal {N}}}=(2,2)$ theories quantized on a circle $S^1$. In this case, it would be straightforward to simply consider a flat space theory on $S^1\times {{\mathbb {R}}}$ and define a polarization in the phase space on $S^1$. Alternatively, we could consider a theory on the sphere $S^2$ in the vicinity of the equator (or on hemisphere $HS^2$ close to the boundary). The latter approach is slightly more technical than the $S^1\times {{\mathbb {R}}}$; however, we will follow it to point out a detail. We will comment on the $S^1\times {{\mathbb {R}}}$ case afterward.

Theories with ${{\mathcal {N}}}=(2,2)$ SUSY on $S^2$ are based on the algebra $\mathfrak {su}(2|1)$. We have already encountered them before as the gluing theories in 3D. Here we consider them as standalone theories, while the gluing will be represented by certain quantum mechanics on the boundary circle. We adhere to conventions of [40]. The sphere is cut into two hemispheres at $\theta =\pi /2$. The Killing spinor equations on $S^2$, as usual, are:

$$\begin{aligned} \nabla _i\epsilon =+\frac{1}{2r} \gamma _i \gamma ^{{{\hat{3}}}}\epsilon ,\quad \nabla _i{{\bar{\epsilon }}}=-\frac{1}{2r}\gamma _i \gamma ^{{{\hat{3}}}}{{\bar{\epsilon }}}. \end{aligned}$$

(173)

The supercharges on $S^2$, following [40], are denoted by $Q_{1,2}$ and $S_{1,2}$. We choose to preserve $Q_2$ and $S_1$ at the boundary. They correspond to Killing spinors:

$$\begin{aligned}&\epsilon =\exp \left( -\frac{i\theta }{2}\gamma _{{{\hat{2}}}}\right) \epsilon _o,\nonumber \\&\quad {{\bar{\epsilon }}}=\exp \left( +\frac{i\theta }{2}\gamma _{{{\hat{2}}}}\right) {{\bar{\epsilon }}}_o, \end{aligned}$$

(174)

where:

$$\begin{aligned} Q_2: \epsilon _o=\left( \begin{matrix} \alpha (\varphi )\\ 0 \end{matrix} \right) =e^{i\varphi /2}\left( \begin{matrix} a\\ 0 \end{matrix} \right) ,\quad {{\bar{\epsilon }}}_o=0,\nonumber \\ S_1: \epsilon _o=0,\quad {{\bar{\epsilon }}}_o=\left( \begin{matrix}0\\ {{\bar{\alpha }}}(\varphi ) \end{matrix} \right) =e^{-i\varphi /2}\left( \begin{matrix}0\\ {\bar{a}} \end{matrix} \right) . \end{aligned}$$

(175)

Notice that we keep the $\varphi$-dependent factor $e^{\pm i\varphi /2}$ as a part of $\epsilon _o$, ${{\bar{\epsilon }}}_o$: With such conventions, the 1D SUSY parameter $\alpha$ is anti-periodic in $\varphi$. We will see that it is natural in a moment.

We construct a Dirichlet polarization designed for cutting and gluing along the equator $\theta =\pi /2$. For the 2D chiral multiplet $(\phi , {{\bar{\phi }}}, \psi , {{\bar{\psi }}}, F, {\bar{F}})$, we define the boundary fields as follows:

$$\begin{aligned} u&=\phi \big |,\quad {\bar{u}}={{\bar{\phi }}}\big |,\nonumber \\ {{\bar{\alpha }}}\chi&={{\bar{\epsilon }}}\psi \big |,\quad \alpha {{\bar{\chi }}}=\epsilon {{\bar{\psi }}}\big |, \end{aligned}$$

(176)

and for the 2D vector multiplet $(A, \sigma _1, \sigma _2, \lambda , {{\bar{\lambda }}}, D)$, the boundary fields are (note that we use the same letter for the 2D and 1D gaugini):

$$\begin{aligned} a_\varphi&=A_\varphi \big |,\quad s=\sigma _2\big |,\nonumber \\ D_{\mathrm{1d}}&=-\frac{i}{r}{{\mathcal {D}}}_\theta \sigma _1\big | + D\big |,\nonumber \\ {{\bar{\alpha }}}\lambda&={{\bar{\epsilon }}}\gamma _{{{\hat{3}}}}\lambda \big |,\quad \alpha {{\bar{\lambda }}}=\epsilon \gamma _{{{\hat{3}}}}{{\bar{\lambda }}}\big |. \end{aligned}$$

(177)

One can easily check that these fields form a polarization, which is also supersymmetric as manifested by the following SUSY transformations. We find that the boundary values of the chiral multiplet transform under $Q_2, S_1$ as follows:

$$\begin{aligned} \delta u&={{\bar{\alpha }}}\chi ,\quad \delta {\bar{u}}=\alpha {{\bar{\chi }}},\nonumber \\ \delta \chi&=\alpha \left( \frac{1}{r} D_\varphi u + s u + i \frac{q}{2r} u\right) ,\nonumber \\ \delta {{\bar{\chi }}}&={{\bar{\alpha }}} \left( \frac{1}{r} D_\varphi {\bar{u}} - s{\bar{u}} -i\frac{q}{2r}{\bar{u}}\right) . \end{aligned}$$

(178)

The vector multiplet transformations are:

$$\begin{aligned} \delta s&= -\frac{i}{2} ({{\bar{\alpha }}}\lambda +\alpha {{\bar{\lambda }}}),\nonumber \\ \delta a_\varphi&= -\frac{r}{2} ({{\bar{\alpha }}}\lambda +\alpha {{\bar{\lambda }}}),\nonumber \\ \delta \lambda&=\alpha \left( \frac{i}{r} D_\varphi s - D_{\mathrm{1d}} \right) ,\nonumber \\ \delta {{\bar{\lambda }}}&= {{\bar{\alpha }}}\left( \frac{i}{r} D_\varphi s + D_{\mathrm{1d}} \right) ,\nonumber \\ \delta D_{\mathrm{1d}}&= -{{\bar{\alpha }}} \left( \frac{1}{2r} {{\mathcal {D}}}_\varphi \lambda - \frac{i}{4r}\lambda +\frac{1}{2} [s,\lambda ] \right) + \alpha \left( \frac{1}{2r} {{\mathcal {D}}}_\varphi {{\bar{\lambda }}} + \frac{i}{4r}{{\bar{\lambda }}} + \frac{1}{2} [s, {{\bar{\lambda }}}] \right) . \end{aligned}$$

(179)

One can immediately recognize these as SUSY transformations of the 1D ${{\mathcal {N}}}=2$ chiral and vector multiplets as described in [65], which already appeared in (8) and (5), with a small difference: Here we have an R-symmetry holonomy $A_\varphi ^{(R)}=1/2$ turned on. This is manifested by the terms $i\frac{q}{2r}u$ and $-i\frac{q}{2r}{\bar{u}}$ in $\delta \chi$ and $\delta {{\bar{\chi }}}$, as well as $\frac{i}{4r}{{\bar{\lambda }}}$ and $-\frac{i}{4r}{{\bar{\lambda }}}$ in $\delta D_{\mathrm{1d}}$.

To elaborate further on this point, notice that all fermions on $S^2$ take values in the corresponding spinor bundle ${{\mathcal {O}}}(-1)$. When restricted to the boundary $S^1 = \partial HS^2$, they become anti-periodic. So does the SUSY parameter $\epsilon$, and this is the reason we defined $\alpha$, the 1D SUSY parameter, to be anti-periodic as well. This also manifests, in yet another way, the fact that there is an R-charge holonomy 1/2 preset on $S^1$: since $\alpha$ has R-charge 1, it becomes anti-periodic in the presence of such a holonomy.

In fact, if we started with the flat space ${{\mathcal {N}}}=(2,2)$ SUSY on $S^1\times {{\mathbb {R}}}$ rather than the sphere, we would get the same boundary SUSY on $S^1$, except that these R-holonomy shifts would not be there. In this case, we would have to explicitly go to the twisted sector on $S^1\times {{\mathbb {R}}}$, in which the periodicity of R-charge-q fields is given by $e^{i\pi q}$, in order to the get the same boundary SUSY as in (178), (179). This observation means that if we want to glue a half-infinite cylinder $S^1\times {{\mathbb {R}}}_+$ to the hemisphere $HS^2$ along their $S^1$ boundary, the ${{\mathcal {N}}}=(2,2)$ theory on $S^1\times {{\mathbb {R}}}_+$ has to be in the twisted sector determined by the R-symmetry holonomy¹⁶$A^{(R)}_\varphi =1/2$. Otherwise, the boundary fields simply would not match. Note that this is specific to the $\mathfrak {su}(2|1)$-preserving background on $S^2$ that we have considered. Gluing in the topological background (as in the $tt^*$-geometry [25]) might work differently.

To have an even better interpretation of this, let us observe yet another fact. The algebra formed by the supercharges $Q_2$ and $S_1$ that we preserve at the boundary of $HS^2$ is $\mathfrak {su}(1|1)$, with the relation:

$$\begin{aligned} \{Q_2, S_1\} = P + \frac{R}{2}, \end{aligned}$$

(180)

where R is the original U(1) R-charge of the $S^2$ theory. Under this R-symmetry, s, $a_\varphi$ and $D_{\mathrm{1d}}$ are neutral, $\lambda$ has R-charge 1, scalar $\varphi$ has R-charge q, and $\chi$ has R-charge $q+1$. However, this algebra does not look like the 1D ${{\mathcal {N}}}=2$ algebra of [65]. In order to get their algebra, we have to redefine the translation generator as:

$$\begin{aligned} {\tilde{P}} = P + \frac{R}{2}, \end{aligned}$$

(181)

so that the algebra becomes $\{Q_2, S_1\}={\tilde{P}}$, the usual ${{\mathcal {N}}}=2$ SUSY in 1D. This shift of the translation generator by R/2 is precisely due to the R-symmetry holonomy.

To recapitulate, the $\mathfrak {su}(2|1)$-preserving SUSY background on $S^2$ in the vicinity of the equator looks like a twisted sector of the flat space SUSY on $S^1\times {{\mathbb {R}}}$, with the R-symmetry holonomy turned on. The vector R-symmetry corresponds to $\mathfrak {su}(2|1)_A$ on the sphere, while the axial R-symmetry corresponds to $\mathfrak {su}(2|1)_B$. The gluing is possible when the two pieces have the same R-symmetry holonomy.

1.1.1 Quarter-BPS polarization

We can describe one more potentially useful polarization at the boundary of $HS^2$: A quarter-BPS polarization that only preserves $Q_2 + S_1$. In terms of SUSY parameters, it means that $a={\bar{a}}$, so $\alpha =e^{i\varphi /2}a$, ${{\bar{\alpha }}}=e^{-\varphi /2}{\bar{a}}$. For chiral multiplets, the boundary fields are defined as:

$$\begin{aligned} u&=\phi \big |,\quad {\bar{u}}={{\bar{\phi }}}\big |,\nonumber \\ a\chi&={{\bar{\epsilon }}}\psi \big |,\quad a{{\bar{\chi }}}=\epsilon {{\bar{\psi }}}\big |. \end{aligned}$$

(182)

For the vector multiplet, we use the same names for the boundary scalars as for their bulk counterparts:

$$\begin{aligned} \sigma _1&=\sigma _1\big |,\quad \sigma _2=\sigma _2\big |,\quad A_\varphi =A_\varphi \big |. \end{aligned}$$

(183)

By acting with ${{\mathcal {Q}}}=Q_2+S_1$ on these, we define the corresponding fermions:

$$\begin{aligned}{}[{{\mathcal {Q}}},\sigma _1]&=\mu _1,\quad [{{\mathcal {Q}}},\sigma _2]=\mu _2,\quad [{{\mathcal {Q}}},A_\varphi ]=-ir\mu _2, \end{aligned}$$

(184)

whose explicit expressions in terms of $\lambda$’s are:

$$\begin{aligned} \mu _1&=\frac{1}{2\sqrt{2}}\left[ e^{-i\varphi /2}(\lambda _1-\lambda _2) - e^{i\varphi /2}({{\bar{\lambda }}}_1 - {{\bar{\lambda }}}_2) \right] ,\nonumber \\ \mu _2&=-\frac{i}{2\sqrt{2}}\left[ e^{-i\varphi /2}(\lambda _1+\lambda _2) + e^{i\varphi /2}({{\bar{\lambda }}}_1 + {{\bar{\lambda }}}_2) \right] . \end{aligned}$$

(185)

A trivial computation shows that their Poisson bracket is zero, i.e., they define a good polarization for the fermions.

Below we summarize SUSY of the boundary fields:

$$\begin{aligned} \delta u&=a\chi ,\quad \delta {\bar{u}}=a{{\bar{\chi }}},\nonumber \\ \delta \chi&=a\left( \frac{1}{r}{{\mathcal {D}}}_\varphi u +\sigma _2 u + i\frac{q}{2r}u \right) ,\nonumber \\ \delta {{\bar{\chi }}}&=a\left( \frac{1}{r}{{\mathcal {D}}}_\varphi {\bar{u}}-{\bar{u}}\sigma _2 -i\frac{q}{2r}{\bar{u}} \right) ,\nonumber \\ \delta \sigma _1&=a\mu _1,\quad \delta \sigma _2=a\mu _2,\quad \delta A_\varphi = -i\alpha r\mu _2,\nonumber \\ \delta \mu _1&=\alpha \left( \frac{1}{r}{{\mathcal {D}}}_\varphi \sigma _1 + [\sigma _2,\sigma _1] \right) ,\nonumber \\ \delta \mu _2&=\alpha \cdot \frac{1}{r}{{\mathcal {D}}}_\varphi \sigma _2, \end{aligned}$$

(186)

where q is the R-charge of $\phi$. These fields form a $\frac{1}{4}$-BPS polarization which can be used to glue ${{\mathcal {Q}}}$-closed states in the usual way, i.e., $\langle \varPsi _2|\varPsi _1\rangle =\int {{\mathscr {D}}}{{\mathscr {B}}}\, \langle \varPsi _2|{{\mathscr {B}}}\rangle \langle {{\mathscr {B}}}|\varPsi _1\rangle$, with ${{\mathscr {B}}}=(u, {\bar{u}}, \chi , {{\bar{\chi }}}, \sigma _1, \sigma _2, A_\varphi , \mu _1, \mu _2)$.

1.2 3D ${{\mathcal {N}}}=2$ index and half-index

Discussion of the previous subsection can be uplifted to three dimensions, for ${{\mathcal {N}}}=2$ theories quantized on $S^1\times S^1$. The most important application would be to understanding half-indices [44], that is partition functions on $S^1\times HS^2$, and how they are glued into full indices on $S^1\times S^2$.

In complete analogy with the 2D ${{\mathcal {N}}}=(2,2)$ case, the vector and chiral multiplets of 3D ${{\mathcal {N}}}=2$ theory quantized on $T^2$ give vector and chiral multiplets in the gluing theory. The gluing theory is an ${{\mathcal {N}}}=(0,2)$ gauge theory on $T^2$, and we mostly think of it as the device to glue two half-indices on $S^1\times HS^2$. Fields of this theory are also defined in the twisted sector. The origin of this twisted sector is twofold: On the one hand, like in the $S^1$ case of previous subsection, it comes from the supersymmetric background on $S^2$; on the other hand, when we study indices on $S^1\times S^2$, $\epsilon$ is not periodic in the $S^1$ direction. Following [70], and as already discussed in Sect. 6, one has to turn on proper holonomies in the $S^1$ direction that would generate appropriate fugacities in the index.

As usual, denoting the boundary multiplets collectively by ${{\mathscr {B}}}$, the gluing is performed by:

$$\begin{aligned} \int {{\mathscr {D}}}{{\mathscr {B}}}\langle \varPsi _1|{{\mathscr {B}}}\rangle \langle {{\mathscr {B}}}|\varPsi _2\rangle . \end{aligned}$$

(187)

This is a 2D ${{\mathcal {N}}}=(0,2)$ theory on the torus with appropriate holonomies turned on. The matter content of this 2D theory is uniquely determined by the matter content of the 3D theory: Each 3D ${{\mathcal {N}}}=2$ vector multiplet gives a 2D ${{\mathcal {N}}}=(0,2)$ vector multiplet, and each 3D ${{\mathcal {N}}}=2$ chiral of R-charge $\varDelta$ gives a 2D ${{\mathcal {N}}}=(0,2)$ chiral of the same R-charge. Notice that unlike in theories with twice as many SUSY (which were discussed earlier in this paper), the R-charges of matter multiplets are not canonically fixed. The 2D theory by itself appears anomalous due to the unbalanced chiral matter; however, as a gluing theory, it is non-anomalous thanks to the bulk contribution $\langle \varPsi _1|{{\mathscr {B}}}\rangle \langle {{\mathscr {B}}}|\varPsi _2\rangle$ providing the necessary anomaly inflow. This was discussed from the general point of view in [29].

One should further localize this theory [13, 14, 43]. The Coulomb branch localization locus is parametrized by holonomies of the gauge fields around the two 1-cycles of $T^2$. They are combined into a single complex variable u valued in the complexified maximal torus of the gauge group. In addition to that, there are holonomies for the flavor symmetries and the R-symmetry turned on. The wavefunctions $\langle \varPsi _1|{{\mathscr {B}}}\rangle$ and $\langle {{\mathscr {B}}}|\varPsi _2\rangle$ evaluated at this localization locus are half-indices [44] with Dirichlet boundary conditions:

$$\begin{aligned} \langle \varPsi _1|{{\mathscr {B}}}\rangle \big |_{L.L.} = \langle {{\mathscr {B}}}|\varPsi _2\rangle \big |_{L.L.} = I_D(x;q;u). \end{aligned}$$

(188)

We leave detailed treatment of this problem for the future work.