Factorisation of $ \mathcal{N} = 2 $ theories on the squashed 3-sphere

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.