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

TL;DR

The paper analyzes how $\mathcal{N}=2$ theories on the squashed sphere $S^3_b$ localise to matrix integrals and, in the degenerate limits $b\to 0$ or $1/b\to 0$, factorise into coupled vortex and antivortex sectors with equivariant parameters $\hbar=2\pi i b^2$ and $\hbar^L=2\pi i/b^2$. For abelian $U(1)$ theories with flavors, the authors derive a factorised expression $Z_{S^3_b}=\sum_i Z^{(i)}_{cl} Z^{(i)}_{1\text{loop}} Z^{(i)}_V \bar{Z}^{(i)}_{1\text{loop}} \bar{Z}^{(i)}_V$, where the vortex blocks $Z^{(i)}_V$ are basic hypergeometric series identified with open topological-string partition functions. This ellipsoid form provides a modular invariant, non-perturbative completion of the vortex theory and is naturally interpreted through geometric engineering via open strings on strip geometries, with mirror-curve/M-theory insights guiding the correspondence. The results hint at a general factorisation mechanism extendable to non-abelian theories via abelianisation and connect to broader 3d-3d/Chern-Simons perspectives, suggesting new routes to exact non-perturbative data in 3d gauge theories.

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.

Figures (2)

• Figure 1: The strip geometry.
• Figure 2: The "half $SU(N)$ geometry".