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.
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Notes
The quantization in [20, 108] is achieved by placing a theory in Omega-background; such approach goes back to the work of [89] where the four-dimensional Omega background [80, 88] was used. On the other hand, the quantization of Higgs and Coulomb branches in [30, 32] was achieved by placing a theory on \(S^3\) background, which is related by conformal map to the quantization in SCFT [10, 11, 27].
In principle, it should be possible to extend our formalism to complex bosonic polarizations. It would require more work because for bosonic fields, one has to care about the convergence of the path integral and the choice of integration cycle.
After a SUSY variation, the partial gauge condition, whether it is \(A_t\big |=0\) or \(\partial _t A_t\big |=0\), breaks down. One needs to perform a compensating gauge transformation with parameter \(\kappa \) such that \(\kappa \big |=0\), so that it would restore the gauge condition without affecting boundary values of other field.
Recall that, in a slight abuse of terminology, by “polarization” we mean both the Lagrangian distribution on the phase space and the maximal Poisson-commuting set of coordinates that parametrize the space of leaves of this distribution.
Only the maximal torus of \(SU(2)_R\) is preserved on \(S^4\).
Note the unusual reality condition \({{\bar{\phi }}}=- \phi ^\dagger \).
For the computation of Poisson brackets, one may assume that \(A_\perp \big | =0\). However, \(A_\perp \big |\) does not carry any physical data, and any condition on it is merely part of gauge fixing. If we work in Lorenz gauge in the bulk, it enforces \({{\mathcal {D}}}_\perp A_\perp \big |=0\) at the boundary, while using temporal gauge in the vicinity of the boundary implies \(A_\perp \big | =0\).
Note that the 3D chiral multiplet contribution \(Z_{\mathrm{ch}}(a)\) of (110) is often dropped in the literature. This is allowed as long as the matter representation \({{\mathcal {R}}}\) is self-conjugate. Otherwise, it is a non-trivial a-dependent phase factor that has to be included.
Only for real scalars, the answer can possibly involve square roots of the determinants, such as \(\prod _{w\in {{\mathcal {R}}}}\prod _{n>0}\left[ (n + iw(a))^{-n}(n-iw(a))^{-n}\right] ^{1/2}\). However, the Dirichlet boundary conditions do not violate complex structure of the hypermultiplet scalars, thus both on \(HS^4\) and \(S^4\) one has to compute determinants for complex scalars only. This can be considered as an argument in favor of (122).
One might worry that \(\frac{1}{\mu _b(a)}\delta _{{\mathfrak {t}}}(a,{\tilde{a}})\) is poorly defined at \({\tilde{a}}=0\), since \(\mu _b(a)\) vanishes at \(a=0\). In fact, it is well-defined. The wave function on the full algebra \({\mathfrak {g}}\) would be \(\frac{\det _{\mathrm{Adj}}' a}{\mu _b(a)}\delta _{{\mathfrak {g}}}(a)\), and this is clearly well-defined. Upon passing to integration over \({\mathfrak {t}}\subset {\mathfrak {g}}\), the Vandermonde factor \(\det {}_{\mathrm{Adj}}' a\) disappears. This is analogous to the delta function on \({{\mathbb {R}}}^2\) being well-defined despite having a dangerous-looking expression \(\frac{1}{2\pi r}\delta (r)\) in polar coordinates due to the Jacobian.
Notice that each \(Q_{\mathrm{Loc}}\)-cohomology class determined by \(f_{{\mathcal {B}}}(a)\) contains a half-BPS boundary condition. To construct it, impose the Dirichlet boundary condition (112) parametrized by a and then integrate the result against \(\mu (a)f_{{\mathcal {B}}}(a)\). Therefore, general \(Q_{\mathrm{Loc}}\)-invariant boundary conditions, while probably highly non-trivial in the full theory, are equivalent to half-BPS boundary conditions at the level of \(Q_{\mathrm{Loc}}\)-cohomology.
For \(S^3_b\), this is broken down to \(\mathfrak {su}(1|1)_\ell \oplus \mathfrak {su}(1|1)_r\).
It is not obvious what is the appropriate space of physically allowed functions \(f_{{\mathcal {B}}}(a)\). However, it is very natural to conjecture that it can be identified with the \({{\mathbb {Z}}}_2\)-invariant subspace of complex tempered distributions, \(({{\mathcal {S}}}'({{\mathbb {R}}})\otimes {{\mathbb {C}}})^{{{\mathbb {Z}}}_2}\): 1) we have seen that for Dirichlet boundary conditions, \(f_{{\mathcal {B}}}(a)\) is a delta function; 2) the form of S-transformation suggests that \(\sinh (\pi a)f_{{\mathcal {B}}}(a)\) should have a Fourier transform.
In their case, the answer does not depend on \(\beta _1\), so we might as well choose \(\beta _1=\beta _2\), which leads to \(e^{-\beta _1 R}=e^{-\beta \frac{R}{2}}\) in their equation (17). This R/2 in 3d is precisely our R in 4d.
Since the flat space \({{\mathcal {N}}}=(2,2)\) SUSY has two R-symmetries, the vector and the axial one, we can turn on holonomy for either of them. This corresponds to the \(\mathfrak {su}(2|1)_A\) or \(\mathfrak {su}(2|1)_B\)-preserving background on the sphere. The cylinder with the vector R-symmetry holonomy can be glued to the hemisphere with the \(\mathfrak {su}(2|1)_A\)-background on it, while the cylinder with the axial holonomy can be glued to the hemisphere with the \(\mathfrak {su}(2|1)_B\) background on it.
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Acknowledgements
The author thanks Tudor Dimofte, Yale Fan, Bruno Le Floch, Davide Gaiotto, Sergei Gukov, Victor Mikhaylov, Alexei Morozov, Natalie Paquette, Silviu Pufu, Mauricio Romo, David Simmons-Duffin, Gustavo J. Turiaci, Ran Yacoby for comments and discussions, and in particular Tudor Dimofte and Sergei Gukov for comments on the draft.
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This work was supported by the Walter Burke Institute for Theoretical Physics and the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, as well as the Sherman Fairchild Foundation.
Open-ended examples
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:
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:
where:
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:
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):
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:
The vector multiplet transformations are:
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 holonomyFootnote 16\(A^{(R)}_\varphi =1/2\). 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:
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:
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:
For the vector multiplet, we use the same names for the boundary scalars as for their bulk counterparts:
By acting with \({{\mathcal {Q}}}=Q_2+S_1\) on these, we define the corresponding fermions:
whose explicit expressions in terms of \(\lambda \)’s are:
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:
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:
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:
We leave detailed treatment of this problem for the future work.
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Dedushenko, M. Gluing II: boundary localization and gluing formulas. Lett Math Phys 111, 18 (2021). https://doi.org/10.1007/s11005-021-01355-8
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DOI: https://doi.org/10.1007/s11005-021-01355-8