Hemisphere Partition Function Techniques

Updated 6 January 2026

The hemisphere partition function approach is a set of localization and factorization techniques that reduce complex path integrals in supersymmetric theories to tractable lower-dimensional integrals.
It computes exact observables such as protected operator dimensions, current-current correlators, and D-brane central charges by exploiting boundary conditions on hemisphere-like geometries.
The method underpins duality tests, TQFT constructions, and precision calculations in collider physics by analytically encoding monodromy and non-global QCD observables.

The hemisphere partition function approach is a suite of localization and factorization techniques for computing exact partition functions of supersymmetric quantum field theories defined on manifolds with hemispherical or disk topology, often with additional data prescribed on the boundary. These methods enable the reduction of complicated path integrals to lower-dimensional matrix integrals or sums, allowing exact computation of observables such as protected operator dimensions, current-current correlators, monodromies, and transport coefficients across various dimensions and physical regimes. The approach has found critical applications in studying supersymmetric gauge theories, D-brane monodromies in Calabi–Yau compactifications, and non-global observables in collider physics through soft functions.

1. Fundamental Construction and Localization

The hemisphere partition function arises from placing a supersymmetric quantum field theory on a hemisphere-like geometry, such as $HS^4$ (the 4d Euclidean hemisphere with boundary $S^3$ ), or lower-dimensional analogs such as the disk $D^2$ in two dimensions or $HS^2$ in three dimensions. The key technical tool is supersymmetric localization: after selecting a compatible boundary condition that preserves a subset of supercharges, the full path integral localizes onto a much simpler set of supersymmetric field configurations, typically constant field profiles for scalar fields in vector multiplets.

For 4d $\mathcal N=2$ abelian gauge theory on $HS^4$ coupled to 3d chiral matter on the boundary, the localization locus reduces to a constant $S$ for the bulk scalar, and the partition function reduces to a single real integral involving the complexified gauge coupling $\tau$ , Fayet-Iliopoulos terms, and 3d one-loop determinants given by hyperbolic gamma (or $\ell$ -function) factors. The general hemisphere partition function is

$Z_{HS^4}(\tau; \{q_+, q_-, q_t\}) = \int_{-i\infty}^{i\infty} \! dS \; \exp\Bigl[i\pi \tau S^2 + 2\pi q_t S\Bigr] \prod_\pm \exp\Bigl[n_\pm \ell(1-q_\pm \pm S)\Bigr],$

where $q_\pm$ are the R-charges of boundary chirals and $n_\pm$ their multiplicities (Gupta et al., 2019).

In 2d GLSMs, the hemisphere partition function similarly localizes to integrals over Cartan zero-modes of the gauge multiplet, with bulk and boundary contributions from chiral matter and brane factors encoding boundary matrix factorizations (Erkinger et al., 2017, Knapp et al., 2016).

2. Analytic Structure, Duality, and Monodromy

The analytic properties of the hemisphere partition function carry detailed information about dualities and monodromies in the underlying quantum field theory or associated geometric moduli space.

In 2d GLSMs, the hemisphere (disk) partition function with B-type boundary conditions computes the central charge of the corresponding D-brane. Its contour representation in terms of Gamma functions allows analytic continuation across the FI-theta parameter space, directly encoding the monodromy data of brane charges as one crosses phase boundaries. The procedure to extract monodromy matrices involves tracking how brane factors transform under $\theta \to \theta + 2\pi$ (large-radius monodromy), as well as nontrivial wall-crossing associated with phase transitions (Landau–Ginzburg, conifold) (Erkinger et al., 2017).
In 4d/3d “supersymmetric graphene” systems, the hemisphere partition function admits explicit Fourier-transform identities that implement S-duality, exchanging strong and weak coupling ( $\tau \to -1/\tau$ ) and mixing real mass parameters. At self-dual points, protected transport coefficients such as the conductivity can be computed exactly in terms of $\tau$ , e.g., $\Sigma_{gg}(\tau = i) = \tau/2$ (Gupta et al., 2019).
For Calabi–Yau models, hemisphere partition functions can be analytically continued to compute D-brane central charges at singular loci (e.g., the conifold), entirely within the GLSM formulation via Mellin–Barnes techniques (Knapp et al., 2016).

3. Extraction of Physical Observables

The hemisphere partition function framework provides exact algorithms for extracting a wide range of physical observables:

Scaling dimensions of protected boundary operators are obtained as the R-charge values that extremize the absolute value of the partition function, reflecting the underlying 3d F-maximization principle. The condition $\frac{\partial}{\partial q_i} |Z(q)| = 0$ yields the exact scaling dimensions for chiral operators (Gupta et al., 2019, Gupta et al., 2020).
Current-current correlators and transport: The second derivatives of $\ln Z$ with respect to background charges give exact correlation functions of conserved currents. In 3d, the conductivity (at zero temperature) is related to

$\sigma = \frac{1}{4\pi} \Re\left(\partial_q^2 \ln Z\right),$

while the Hall conductivity is extracted from the imaginary part, through the structure

$\Sigma = 2\pi (\sigma_H + i\sigma) = \frac{i}{2\pi} \partial_q^2 \ln Z.$

These quantities can be traced along the RG flow from weak to strong coupling, allowing identification of exact universal values at duality-invariant couplings (Gupta et al., 2019).

Boundary energy-momentum tensor correlators are obtained, in squashed hemisphere settings, as the second derivative of the free energy with respect to the squashing parameter. This yields stress-tensor two-point function coefficients such as $\tau_R$ , which interpolate between free-field and strongly coupled values depending on $\tau$ (Gupta et al., 2020).
D-brane central charges and periods: In 2d GLSMs, the hemisphere partition function computes the quantum-corrected central charges of B-branes, matches to Mellin–Barnes integral representations for mirror periods, and allows direct analytic continuation into all geometric phases and across singular points, bypassing the need for explicit mirror maps (Knapp et al., 2016).

4. Factorization and Gluing Structures

The hemisphere partition function admits natural factorization and gluing properties:

In 3d $\mathcal N=2$ gauge theories, interval partition functions on $T^2 \times [0,1]$ factorize into sums of products of left and right hemisphere partition functions with universal normalization factors (Zhao et al., 28 Sep 2025). For 3d SQED[ $N$ ], the interval function is

$Z_{\rm int}(q, x) = \sum_{\alpha=1}^N Z_\alpha(q, x) \ N_\alpha \ Z_\alpha(q, x^{-1}),$

where $Z_\alpha$ are hemisphere partition functions associated to Higgs vacua and $N_\alpha$ are explicit normalization factors reflecting equivariant $K$ -theory inner products.

The same structure manifests for 3d Chern–Simons–Yang–Mills theories, where hemisphere partition functions correspond to affine $\widehat{\mathfrak{su}}(N)_k$ characters and the interval partition function is their diagonal sum with deformed inner product coefficients.

This factorization mirrors the gluing axioms of TQFT and allows hemisphere partition functions to serve as building blocks for constructing more complicated amplitude configurations.

5. Hemisphere Partition Functions in Collider and Field Theory Observables

The hemisphere partition function approach underpins precision calculations of non-global QCD observables through the analysis of soft functions:

In Soft-Collinear Effective Theory, hemisphere soft functions $S(\omega_L, \omega_R)$ measure the total soft radiation into hemispheres defined by event-shape axes, and are essential for factorizing cross sections in jet physics (Becher et al., 2016, Kelley et al., 2011). The hemisphere partitioning decomposes phase space, and the corresponding soft functions encode the effects of soft Wilson lines restricted to each hemisphere.
The approach facilitates the computation, up to N $^3$ LO, of soft functions by partitioning emissions according to hemisphere constraints, enabling the analytic extraction of master integrals with full transcendental structure (Baranowski et al., 2022).
The resulting hemisphere soft functions display non-trivial logarithmic structure, with non-global logarithms resummed by infinite-dimensional anomalous dimension matrices and reflected in the explicit non-polynomiality of soft function finite terms at two and three loops (Becher et al., 2016, Kelley et al., 2011). The hemisphere partition strategy is essential for isolating these effects and matching to RG-related resummations.

6. Mathematical and Geometric Interpretations

The hemisphere partition function framework interfaces with deep mathematical concepts:

Equivariant $K$ -theory: In 3d settings, interval partition functions decompose into equivariant $K$ -theory classes of target spaces, with hemisphere partition functions corresponding to fixed-point contributions. The normalization factors are inverses of norms of orthogonal idempotents in equivariant $K$ -theory, giving a geometric interpretation to partition function factorizations (Zhao et al., 28 Sep 2025).
Mirror symmetry and periods: In 2d GLSMs with B-type boundary conditions, hemisphere partition functions exactly reproduce Mellin–Barnes integral representations of periods of the mirror Calabi–Yau. Analytic continuation within the GLSM localizes the full monodromy and period structure of the target space (Knapp et al., 2016).
Representation theory: In Chern–Simons–Yang–Mills constructions, hemisphere partition functions coincide with affine Lie algebra characters, embedding the analysis in the framework of infinite-dimensional representation theory and modular transformations (Zhao et al., 28 Sep 2025).

7. Extensions, Specializations, and Open Directions

The hemisphere partition function approach generalizes across a wide class of theories and geometries:

For supersymmetric gauge theories on squashed spheres or ellipsoids, partition functions retain deformation parameters (squashing), providing additional probes for operator correlation functions via derivatives with respect to geometric moduli (Gupta et al., 2020).
The approach admits straightforward extension to theories with boundaries and defects, as well as to multi-parameter moduli spaces, where the associated contours and "band restrictions" generalize naturally to higher-dimensional charge lattices (Erkinger et al., 2017).
In nonabelian generalizations, boundary matrix factorizations and the corresponding partition function localization are under development, indicating potential for extracting monodromy and other data from higher-rank GLSMs and related target spaces.

The method continues to evolve, with connections to TQFT gluing, categorified representation theory, and exact computation in quantum field theory. Its algebraic, analytic, and geometric character makes it a versatile instrument across high-energy theory, mathematical physics, and string theory.