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

Abstract

Partition functions of $\mathcal{N} = 2$ 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 ${\mathbb{R}^2} \times {S_1}$ R corresponding to the b→0 degeneration of the ellipsoid. The second block gives the partition function of the vortex theory, with equivariant parameter ${\hbar^L} = {{{2\pi i}} \left/ {{{b^2}}} \right.}$, on the dual ${\mathbb{R}^2} \times {S_1}$ corresponding to the 1/b→0 degeneration. The ellipsoid partition appears to provide the $\hbar \to {\hbar^L}$ modular invariant non-perturbative completion of the vortex theory.