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Holomorphic Blocks for 3d Non-abelian Partition Functions

Masato Taki

TL;DR

The paper proves that 3d partition functions of non-abelian $ ext{N}=2$ theories on $S^3_b$ factorize into holomorphic blocks, extending the known abelian cases. Using abelianization via the Cauchy formula, the authors express the non-abelian partition function as a sum over vortex sectors, with each block given by the $K$-theoretic uplift of vortex partition functions; the blocks also match contributions to the 3d superconformal index. They provide explicit block formulas for both vector-like and chiral $U(N)$ theories, and show these blocks have an open topological string interpretation on strip geometries with $N$ A-branes. This work substantiates Beem et al.’s conjecture on 3d factorization, links vortex counting to open string amplitudes, and suggests broader implications for dimensional uplifts and dualities in supersymmetric gauge theories. $

Abstract

The most recent studies on the supersymmetric localization reveal many non-trivial features of supersymmetric field theories in diverse dimensions, and 3d gauge theory provides a typical example. It was conjectured that the index and the partition function of a 3d N=2 theory are constructed from a single component: the holomorphic block. We prove this conjecture for non-abelian gauge theories by computing exactly the 3d partition functions and holomorphic blocks.

Holomorphic Blocks for 3d Non-abelian Partition Functions

TL;DR

The paper proves that 3d partition functions of non-abelian theories on factorize into holomorphic blocks, extending the known abelian cases. Using abelianization via the Cauchy formula, the authors express the non-abelian partition function as a sum over vortex sectors, with each block given by the -theoretic uplift of vortex partition functions; the blocks also match contributions to the 3d superconformal index. They provide explicit block formulas for both vector-like and chiral theories, and show these blocks have an open topological string interpretation on strip geometries with A-branes. This work substantiates Beem et al.’s conjecture on 3d factorization, links vortex counting to open string amplitudes, and suggests broader implications for dimensional uplifts and dualities in supersymmetric gauge theories. $

Abstract

The most recent studies on the supersymmetric localization reveal many non-trivial features of supersymmetric field theories in diverse dimensions, and 3d gauge theory provides a typical example. It was conjectured that the index and the partition function of a 3d N=2 theory are constructed from a single component: the holomorphic block. We prove this conjecture for non-abelian gauge theories by computing exactly the 3d partition functions and holomorphic blocks.

Paper Structure

This paper contains 12 sections, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. 3d partition functions and factorization
  3. Factorization of ellipsoid partition functions
  4. $\mathcal{N}=2$ $U(N)$ vector-like theory
  5. $\mathcal{N}=2$ $U(N)$ chiral theory
  6. Vortex partition functions and topological strings
  7. Discussion
  8. Double-sine function
  9. 3d partition function
  10. Details of computation
  11. vector-like theory
  12. chiral theory

Figures (4)

  • Figure 1: (a) The $id$-gluing of two $\bar{T}^2$'s: the trivial decomposition of $S^1\times S^2$. (b) The $S$-gluing of two $\bar{T}^2$'s: the Heegaard decomposition of three-sphere $S^3$ through the S element of the mapping class group $SL(2, \mathbb{Z})$.
  • Figure 2: The integral contour for the partition function. The crosses denote the poles of the integrant of the partition function.
  • Figure 3: This strip geometry is half of the toric geometry which leads to 4d $\mathcal{N}=2$$U(N_f)$ gauge theory with $2N_f$ flavors through the geometric engineering. We set $R_j=\emptyset$ for $j\not\in \{i_\alpha\}$.
  • Figure 4: This strip geometry is half of the toric geometry which leads to 4d $\mathcal{N}=2$$U(2N_f)$ pure Yang-Mills theory through the geometric engineering.