Hemisphere Partition Functions in SUSY Gauge Theories

• Hemisphere partition functions are exact supersymmetric observables computed on manifolds with boundary, encoding D-brane data and monodromy structures.
• They enable precise evaluation in 2D GLSMs and 4D gauge theories by linking brane factors, Mellin-Barnes integrals, and central charge calculations.
• They also facilitate the factorization of 3D interval partition functions and connect equivariant K-theory with affine representation theory in mixed-dimensional setups.

Hemisphere partition functions are supersymmetric partition functions on manifolds with boundary whose exact evaluation by localization makes them useful across several distinct settings: as exact, non-perturbative central charges of B-type D-branes in two-dimensional $\mathcal N=(2,2)$ GLSMs on the disk; as wave-functions depending on boundary data for four-dimensional $\mathcal N=2$ gauge theories on $HS^4$; as generating functions for transport and operator data in mixed-dimensional abelian gauge theories with boundary matter; and as building blocks in factorizations of three-dimensional interval partition functions (Knapp et al., 2016, Gava et al., 2016, Gupta et al., 2019, Zhao et al., 28 Sep 2025). In all of these realizations, boundary conditions or brane data enter explicitly in the localized integrand, and the resulting quantities organize analytic continuation, monodromy, gluing, or factorization structures.

1. Definition and localization framework

In two-dimensional $U(1)$ GLSMs with D-branes, the hemisphere partition function $Z_{D^2}(\mathcal B)$ is an exact, non-perturbative function computed by supersymmetric localization on a 2d disk. The GLSM data consist of the gauge group, chiral matter representations, the superpotential $W$, and an R-symmetry, while B-branes are realized as matrix factorizations of the superpotential enhanced with equivariance data. The localized answer depends on a brane factor $f_{\mathcal B}(\sigma)$, which encodes the boundary data (Knapp et al., 2016).

For a general GLSM with gauge group $G$, matter fields of gauge charges $Q_i$ and R-charges $R_i$, the hemisphere partition function takes the form

$\mathcal N=2$0

with

$\mathcal N=2$1

For branes described via complexes of Wilson line branes, the brane factor is a sum of exponentials,

$\mathcal N=2$2

These expressions make explicit that the boundary object enters only through the brane factor in the integrand (Erkinger et al., 2017).

A closely related localization paradigm appears in four-dimensional $\mathcal N=2$3 gauge theories on the hemisphere $\mathcal N=2$4. There the path integral is computed with either Dirichlet or Neumann supersymmetric boundary conditions, and the resulting quantities are wave-functions of the theory depending on the boundary data. The one-loop determinants are computed using either an $\mathcal N=2$5 harmonics basis or a full $\mathcal N=2$6 harmonics basis, with the computation reduced to solving kernel and co-kernel equations or evaluating $\mathcal N=2$7 eigenvalues and multiplicities (Gava et al., 2016).

2. Two-dimensional GLSMs, brane factors, and central charges

For the $\mathcal N=2$8 GLSMs describing degree-$\mathcal N=2$9 hypersurfaces in $HS^4$0, the hemisphere partition function is

$HS^4$1

where $HS^4$2 is the complexified FI-theta parameter and $HS^4$3 encodes R-charge assignments and can be set to zero in the Calabi-Yau case. For a Wilson line brane of charge $HS^4$4, the brane factor is

$HS^4$5

In the large radius phase, residue evaluation gives the quantum-corrected central charge, and for specifically constructed branes $HS^4$6 coincides with mirror period integrals (Knapp et al., 2016).

A central structural ingredient is the grade restriction rule. To define the integral globally over the full parameter space, including singular points, D-branes need to be grade restricted, with charges constrained by

$HS^4$7

This charge window depends on the phase of the GLSM and the value of $HS^4$8. The grade restriction rule is also what allows one to compare descriptions of the same brane across different regions in Kähler moduli space (Knapp et al., 2016).

After the change of variables $HS^4$9, the hemisphere partition function for grade-restricted branes corresponding to Koszul complexes becomes a Mellin-Barnes integral,

$U(1)$0

with

$U(1)$1

The full partition function is proportional to this integral. These Mellin-Barnes representations satisfy the Picard-Fuchs equation

$U(1)$2

This identifies the hemisphere partition function as a direct GLSM realization of the period problem, but without explicit use of the mirror Calabi-Yau (Knapp et al., 2016).

3. Analytic continuation, conifold behavior, and monodromy

For one-parameter Calabi-Yau hypersurfaces, the Kähler moduli space has three special points: large volume at $U(1)$3, Landau-Ginzburg at $U(1)$4, and the conifold or singular point at $U(1)$5. The Mellin-Barnes integrals are adapted to analytic continuation from large volume to the conifold because the contour can be deformed to follow the desired singularity. Near the conifold, $U(1)$6, the continued solutions are expanded by using Bühring’s method for holomorphic solutions and a generalization of Nørlund’s method for logarithmic and higher solutions, yielding series in powers of $U(1)$7 together with, in resonant cases, possible logarithmic terms (Knapp et al., 2016).

The quintic provides the canonical example. For $U(1)$8,

$U(1)$9

and the grade-restricted GLSM brane corresponding to the structure sheaf is represented by

$Z_{D^2}(\mathcal B)$0

At the conifold, the analytic continuation gives $Z_{D^2}(\mathcal B)$1, reproducing the statement that the D6-brane becomes massless there. A common misconception is that this continuation requires an explicit mirror construction; the GLSM calculation performs the continuation directly and does not have to refer to the mirror Calabi-Yau (Knapp et al., 2016).

Monodromies can also be extracted directly from the hemisphere partition function. The prescription is to choose a reference basis of GLSM branes, grade restrict them to a window, perform the monodromy transformation, grade restrict back if necessary by binding empty branes, and then express the resulting brane factor in terms of the original basis. For a theta shift $Z_{D^2}(\mathcal B)$2, the integrand is multiplied by $Z_{D^2}(\mathcal B)$3, so

$Z_{D^2}(\mathcal B)$4

The monodromy matrix $Z_{D^2}(\mathcal B)$5 is then read off from

$Z_{D^2}(\mathcal B)$6

This method is purely algebraic: no need to solve Picard-Fuchs equations, evaluate residue integrals, or perform analytic continuation of periods. For $Z_{D^2}(\mathcal B)$7, the large-radius, Landau-Ginzburg, and conifold monodromy matrices were recomputed in this way and shown to agree, up to known basis transformations, with the classic mirror-symmetry results (Erkinger et al., 2017).

4. Hemisphere wave-functions on $Z_{D^2}(\mathcal B)$8 and gluing

In four-dimensional $Z_{D^2}(\mathcal B)$9 gauge theories on $W$0, supersymmetric boundary conditions are organized into Dirichlet and Neumann types. Dirichlet boundary conditions freeze components of the gauge multiplet scalar at the equator, while Neumann boundary conditions keep the normal derivative fixed and allow the field itself to fluctuate at the boundary. The boundary conditions are chosen so that the boundary terms in the supersymmetry variation of the action vanish, preserving half of the supersymmetries (Gava et al., 2016).

Localization reduces the path integral to saddle points in which the gauge field is pure gauge and the scalar field takes constant values $W$1. The fluctuation problem is encoded in an operator $W$2 mapping between bosonic and fermionic field spaces, and the one-loop determinant is

$W$3

The $W$4 harmonics approach solves the kernel and co-kernel equations mode by mode, while the $W$5 approach reduces the computation to $W$6 eigenvalues and multiplicities. With proper regularization and parity assignments, the net multiplicities agree in the two methods (Gava et al., 2016).

For the vector multiplet with Dirichlet boundary conditions, the one-loop determinant is

$W$7

while for Neumann boundary conditions,

$W$8

For the hypermultiplet,

$W$9

Here $f_{\mathcal B}(\sigma)$0, with $f_{\mathcal B}(\sigma)$1 the Barnes $f_{\mathcal B}(\sigma)$2-function (Gava et al., 2016).

The hemisphere quantities are wave-functions, and gluing reconstructs the full $f_{\mathcal B}(\sigma)$3 partition function. For Dirichlet boundary conditions,

$f_{\mathcal B}(\sigma)$4

where the 3d vector multiplet partition function at the equator re-gauges the global symmetry at the interface. For Neumann boundary conditions,

$f_{\mathcal B}(\sigma)$5

If one hemisphere has Dirichlet and the other Neumann boundary conditions, the one-loop determinants multiply directly to give the full $f_{\mathcal B}(\sigma)$6 result. Point-like instantons localized at the pole of each hemisphere contribute a factor $f_{\mathcal B}(\sigma)$7 (Gava et al., 2016).

5. Mixed-dimensional abelian theories, transport, and squashing

A different class of hemisphere partition functions arises in four-dimensional $f_{\mathcal B}(\sigma)$8 abelian gauge theory on a hemisphere coupled to charged matter on the boundary. Localization reduces the path integral to a single ordinary integral over a real variable, and the exact answer is

$f_{\mathcal B}(\sigma)$9

with

$G$0

The main result is that this partition function is identical to that of $G$1 abelian Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where the quantized Chern-Simons level is replaced by an arbitrary complexified gauge coupling $G$2 (Gupta et al., 2019).

Because the underlying theory has conformal symmetry, the current two-point functions determine the zero temperature conductivity of the Lorentzian versions of these theories at any value of the coupling. The two-point function coefficients are obtained from

$G$3

and the conductivity is encoded in

$G$4

At certain self-dual points, the complexified conductivity associated to the $G$5 gauge symmetry is

$G$6

The same integral also allows the calculation of scaling dimensions of certain protected operators by minimizing $G$7 with respect to the trial charges (Gupta et al., 2019).

On a squashed hemisphere $G$8, the partition function of $G$9 supersymmetric mixed dimensional QED depends on the complex gauge coupling $Q_i$0, the choice of R-symmetry, and the squashing deformation. For $Q_i$1 positive and $Q_i$2 negative boundary chirals,

$Q_i$3

with

$Q_i$4

where $Q_i$5, $Q_i$6, and $Q_i$7. The superconformal R-symmetry is determined using 3-dimensional F-maximization, and the coefficient $Q_i$8 in the 2-point function of the boundary energy-momentum tensor is extracted from

$Q_i$9

At weak coupling, each boundary chiral multiplet contributes $R_i$0. As $R_i$1 decreases, $R_i$2 decreases in the non-chiral case $R_i$3, whereas in the chiral case $R_i$4, $R_i$5, $R_i$6 first increases slightly and then decreases (Gupta et al., 2020).

6. Three-dimensional interval factorizations and affine-character structures

In three-dimensional $R_i$7 theories on $R_i$8, interval partition functions admit factorizations into sums of products of hemisphere partition functions with additional normalization factors. Here the hemisphere partition functions are computed on $R_i$9 and depend not only on boundary data but also on a choice of vacuum or parameter $\mathcal N=2$00. The general factorization pattern is

$\mathcal N=2$01

The paper proves this factorization explicitly for supersymmetric quantum electrodynamics and Chern-Simons-Yang-Mills theories (Zhao et al., 28 Sep 2025).

For $\mathcal N=2$02 SQED with $\mathcal N=2$03 flavors, $\mathcal N=2$04 gauge group, and Chern-Simons level $\mathcal N=2$05, the perturbative part of the hemisphere partition function for vacuum $\mathcal N=2$06 is

$\mathcal N=2$07

The interval partition function then factorizes as

$\mathcal N=2$08

The normalization factor is interpreted as the inverse of the K-theoretic norm squared of the class at the $\mathcal N=2$09-th fixed point of the Higgs branch $\mathcal N=2$10, up to sign. In the IR, the theory flows to a non-linear sigma model into $\mathcal N=2$11, and the factorization mirrors the localization formula in equivariant $\mathcal N=2$12-theory (Zhao et al., 28 Sep 2025).

For $\mathcal N=2$13 $\mathcal N=2$14 Chern-Simons-Yang-Mills at level $\mathcal N=2$15 coupled to $\mathcal N=2$16 boundary fundamental Fermi multiplets per boundary, the hemisphere partition functions with a Wilson line in representation $\mathcal N=2$17 are

$\mathcal N=2$18

The main theorem identifies these hemisphere partition functions exactly with affine characters of $\mathcal N=2$19. The interval partition function factorizes as

$\mathcal N=2$20

with normalization coefficients given by inner products of affine characters. This places hemisphere partition functions at the intersection of supersymmetric localization, equivariant $\mathcal N=2$21-theory, and affine representation theory (Zhao et al., 28 Sep 2025).