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

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 h = 2πib 2, defined on ℝ 2 × 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 h L = 2πi/b 2, on the dual ℝ 2 × S 1 corresponding to the 1/b → 0 degeneration. The ellipsoid partition appears to provide the h → h L modular invariant non-perturbative completion of the vortex theory. © SISSA 2012