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Factorizations of 3d Interval Partition Functions

This paper demonstrates that interval partition functions in three-dimensional $\mathcal{N}=2$ theories can be factorized into sums of products of hemisphere partition functions with Wilson loop insertions, a result explicitly proven for supersymmetric quantum electrodynamics and Chern-Simons-Yang-Mills theories where the gluing factors are interpreted via $S^2 \times S^1$ partition functions or affine characters.

Original authors: Boan Zhao, Panos Betzios, Paul Luis Roehl

Published 2026-08-19

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CC BY 4.0

Original authors: Boan Zhao, Panos Betzios, Paul Luis Roehl

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the landscape of modern theoretical physics, there exists a branch dedicated to understanding the behavior of matter and energy under the strict rules of supersymmetry. This framework, which posits a deep symmetry between particles of matter and particles of force, allows physicists to calculate quantities that would otherwise be impossible to determine. One such quantity is the partition function, a mathematical object that acts like a comprehensive summary of all the possible ways a physical system can arrange itself. When these systems are placed on specific, curved shapes—such as a sphere or a cylinder—their partition functions reveal hidden structures and relationships. For decades, researchers have known that the partition function of a system on a closed sphere can be broken down, or factorized, into simpler pieces calculated on hemispheres, which are then glued back together. This process is akin to understanding a whole by studying its halves and the rules that join them. However, a more complex scenario has remained less explored: what happens when the system is not a closed sphere, but an open interval, a shape with two distinct ends?

A team of researchers has now tackled this specific challenge, demonstrating that the partition functions for three-dimensional systems on an interval can also be broken down into simpler components. They focused on two distinct types of physical theories. The first involves a simple gauge theory with a single force carrier and several matter fields, while the second involves a more complex theory with a larger gauge group and boundary matter fields. In both cases, the researchers proved that the complex calculation for the entire interval can be reconstructed by summing up products of calculations performed on a single hemisphere. These hemisphere calculations are modified by the insertion of specific loops, known as Wilson loops, which act as probes at the tip of the hemisphere. The pieces are then reconnected using "gluing factors," which are mathematical weights that ensure the final result matches the physics of the full interval.

The first part of their work focused on a system containing a single force carrier and several matter fields, each carrying a unit of charge. In this scenario, the researchers showed that the gluing factors used to reassemble the hemisphere pieces are directly related to the partition function of the system on a closed sphere. They established a precise mathematical link, proving that if one knows the behavior of the system on a sphere, one can derive the rules for gluing the interval pieces together. They demonstrated that the set of possible Wilson loops they chose forms a complete basis, meaning no information is lost in the process. By inverting the known relationship between the sphere and the hemisphere, they successfully derived the factorization for the interval, confirming that the interval's behavior is entirely determined by the hemisphere's behavior and the sphere's gluing rules.

The second part of the study addressed a more intricate system involving a larger gauge group and matter fields living specifically on the boundaries of the interval. Here, the researchers encountered a situation where the standard rules of the first example did not immediately apply due to differences in how the fields behave at the boundaries. Despite these complications, they found that the interval partition function still admits a factorization. In this case, the hemisphere pieces correspond to mathematical objects known as affine characters, which describe the symmetries of the system. The researchers proved that the interval function is a sum of products of these characters, weighted by specific factors. This result is significant because it connects the physics of the interval to the well-understood properties of affine characters, suggesting a deep underlying order. They provided two different methods to prove this: one relying on a set of difference equations that the characters must satisfy, and another using a known relationship between different types of symmetry groups.

The findings confirm that the principle of factorization, previously understood for closed spheres, extends robustly to open intervals with boundaries. The researchers did not merely suggest this possibility; they provided explicit proofs for both examples, showing exactly how the hemisphere calculations combine to form the interval result. In the first case, the gluing factors were identified as the inverse of the sphere partition function, while in the second, the gluing factors were determined to be specific weights related to the affine characters. The work leaves open the physical interpretation of these gluing factors in the second case, noting that a deeper understanding of their meaning will require future investigation. Nevertheless, the study establishes a clear and rigorous framework for decomposing complex interval partition functions into simpler, more manageable hemisphere components, offering a new tool for exploring the mathematical structure of supersymmetric theories.

Technical Summary: Factorizations of 3d Interval Partition Functions

Problem Statement
Supersymmetric partition functions on manifolds with boundaries are known to satisfy specific mathematical relations, most notably the factorization of topologically twisted $S^2 \times S^1$ partition functions into products of hemisphere partition functions glued by interval partition functions. However, the factorization of the 3d interval partition functions themselves (defined on $T^2 \times [0, 1]$ ) has received less attention. While previous work established that interval partition functions can be factorized by inserting a complete basis of states (effectively splitting the interval into two), this paper investigates a different type of factorization. The goal is to express interval partition functions as sums of products of hemisphere partition functions (with Wilson loop insertions at the tips) glued by suitable factors, rather than by other interval partition functions.

Methodology
The authors analyze two distinct examples of 3d $\mathcal{N}=2$ theories using supersymmetric localization techniques. The analysis involves computing partition functions on three geometries: the sphere $S^2 \times S^1$ , the hemisphere $HS^2 \times S^1$ , and the interval/cylinder $T^2 \times [0, 1]$ .

Example 1: $U(1)$ Vector Multiplet with $N$ Chiral Multiplets
- Setup: A 3d $\mathcal{N}=2$ $U(1)$ vector multiplet coupled to $N$ chiral multiplets of unit charge. The authors assume $N$ is even to avoid parity anomalies.
- Boundary Conditions: A specific basis of boundary conditions is chosen where only one scalar field $X_\alpha$ acquires a non-zero vacuum expectation value (VEV) on the boundary torus, while others vanish. This breaks the gauge symmetry completely on the boundary.
- Technique: The authors utilize Higgs branch localization. The hemisphere partition functions localize to vortex configurations, while the interval partition functions localize to a point (a constant solution).
- Derivation: They first establish the factorization of the $S^2 \times S^1$ partition function into hemisphere partition functions. By inverting the matrix of these hemisphere functions (weighted by interval partition functions), they derive the factorization of the interval partition function into a sum over Wilson loops inserted at the tips of the hemispheres.
Example 2: $U(k)$ Chern-Simons-Yang-Mills with Boundary Fermi Multiplets
- Setup: A 3d $\mathcal{N}=2$ $U(k)$ Chern-Simons-Yang-Mills theory with level $-k-N$ , coupled to $N$ fundamental Fermi multiplets on each boundary torus. The level is tuned to cancel boundary gauge anomalies.
- Technique: The authors employ the conformal branching rule for the embedding $\widehat{U(k)}_N \oplus \widehat{SU(N)}_k \to \widehat{U(kN)}_1$ . They utilize integral representations of affine characters and difference equations satisfied by these characters.
- Derivation: They demonstrate that the hemisphere partition functions of this system compute the characters of the affine Lie algebra $\widehat{SU(N)}_k$ . By showing that the interval partition function satisfies the same difference equations as the affine characters in the fugacities of the boundary matter, they prove that the interval partition function admits a diagonal factorization into products of these affine characters.

Key Contributions and Results

General Factorization Formula: The paper explicitly proves the factorization formula:
$I_{\alpha\beta} = \sum_{i,j} H_{\alpha i} S_{ij} \tilde{H}_{\beta j}$
where $I_{\alpha\beta}$ is the interval partition function, $H_{\alpha i}$ are hemisphere partition functions with Wilson loop $i$ at the tip, and $S_{ij}$ are gluing factors.
Interpretation of Gluing Factors:
- In the $U(1)$ case, the gluing factor $S_{ij}$ is identified as the inverse of the $S^2 \times S^1$ partition function with Wilson loops $i$ and $j$ inserted at the poles. The authors prove the invertibility of this matrix for small vortex grading parameters.
- In the $U(k)$ case, the interval partition function factorizes diagonally into affine characters:
  $I_{N,k} = \sum_{\mu} f_{N,k}^{\mu\mu}(q) \chi_{\mu}^{\widehat{SU(N)}_k}(q, x) \chi_{\mu}^{\widehat{SU(N)}_k}(q, y)$
  Here, the hemisphere partition functions are shown to be equal to the affine characters $\chi_{\mu}^{\widehat{SU(N)}_k}$ (up to a prefactor).
Mathematical Connections:
- For the $U(1)$ theory, the factorization is linked to the equivariant K-theory of the Higgs branch $\mathbb{P}^{N-1}$ , where the Wilson loops form a basis and the gluing matrix corresponds to a non-degenerate inner product.
- For the $U(k)$ theory, the results rely on level-rank duality ( $U(k)_{-k-N} \leftrightarrow SU(N)_{k+N}$ ) and the conformal branching rules of affine Lie algebras. The authors provide a detailed proof that the hemisphere partition functions compute $\widehat{SU(N)}_k$ characters, a result previously conjectured but not explicitly derived in the literature.

Significance and Claims
The paper claims to establish a new type of factorization for 3d interval partition functions, moving beyond the standard decomposition into other interval functions. The significance lies in:

Explicit Proofs: Providing rigorous proofs for these factorizations in two distinct classes of theories (Abelian and non-Abelian Chern-Simons-Yang-Mills).
Geometric Interpretation: Clarifying the role of Wilson loops at the hemisphere tips as the basis for the factorization sum.
Connection to Affine Characters: Demonstrating that hemisphere partition functions in Chern-Simons-Yang-Mills theories are affine characters, thereby linking 3d supersymmetric partition functions directly to the representation theory of affine Lie algebras via the Chern-Simons/WZW duality.
Generalization: The authors suggest that their analysis is expected to generalize straightforwardly to other supersymmetric setups across different dimensions, though they do not provide explicit calculations for these cases.

The paper maintains a modest tone regarding the physical interpretation of the gluing factors in the non-Abelian case, noting that while the diagonal factorization is proven, the full physical interpretation of the coefficients $f_{N,k}^{\mu\mu}$ is left for future work.

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