Factorisation of N = 2 theories on the squashed 3-sphere

TL;DR

This paper demonstrates that partition functions of N=2 theories on the squashed 3-sphere can be factorized into vortex theory blocks, revealing a modular structure and non-perturbative completion related to the ellipsoid geometry.

Contribution

The paper explicitly evaluates the matrix integral for abelian theories, showing a novel factorization into vortex blocks and identifying their modular relations.

Findings

Partition functions can be expressed as sums of products of vortex blocks.

The first block corresponds to the vortex theory on R^2 x S_1 with parameter hbar.

The second block corresponds to the dual vortex theory with parameter hbar^L.

Abstract

Partition functions of N=2 theories on the squashed 3-sphere have been recently shown to localise to matrix integrals. By explicitly evaluating the matrix integral we show that abelian partition functions can be expressed as a sum of products of two blocks. We identify the first block with the partition function of the vortex theory, with equivariant parameter hbar=2 Pi i b^2, defined on R^2 x S_1 corresponding to the b->0 degeneration of the ellipsoid. The second block gives the partition function of the vortex theory with equivariant parameter hbar^L=2 Pi i/b^2, on the dual R^2 x S_1 corresponding to the 1/b ->0 degeneration. The ellipsoid partition appears to provide the hbar -> hbar^L modular invariant non-perturbative completion of the vortex theory.