Factorisation of 3d $$ \mathcal{N} $$ = 4 twisted indices and the geometry of vortex moduli space

Abstract

We study the twisted indices of N = 4 supersymmetric gauge theories in three dimensions on spatial S 2 with an angular momentum refinement. We demonstrate factorisation of the index into holomorphic blocks for the T [SU(N )] theory in the presence of generic fluxes and fugacities. We also investigate the relation between the twisted index, Hilbert series and the moduli space of vortices. In particular, we show that each holomorphic block coincides with a generating function for the χ t genera of the moduli spaces of "local" vortices. The twisted index itself coincides with a corresponding generating function for the χ t genera of moduli spaces of "global" vortices in agreement with a proposal of Bullimore et al. We generalise this geometric interpretation of the twisted index to include fluxes and Chern-Simons levels. For the T [SU(N )] theory, the relevant moduli spaces are the local and global versions of Laumon space respectively and we demonstrate the proposed agreements explicitly using results from the mathematical literature. Finally, we exhibit a precise relation between the Coulomb branch Hilbert series and the Poincaré polynomials of the corresponding vortex moduli spaces.

Blocks and vortices in the 3d ADHM quiver gauge theory

“…where H A,B S 2 denote respectively states in each twist on S 2 that are annihilated by Q 11 + and Q 22 − . It has been shown in various cases that these indices factorise into the A-and B-shifted holomorphic blocks [27,28]. We will show our hemisphere partition functions associated to vacua λ (3.16) provide an exact factorisation:…”

“…The A-and B-twisted indices of 3d N = 4 theories on S 1 × S 2 are in fact independent of q [28] and coincide with the M C and M H Hilbert series of the theory. In particular…”

“…Quasimap moduli spaces QM d can occasionally be realised as smooth varieties. For example the quasimap moduli space for the T ρ [SU(N )] theory coincides with Laumon space Q d [28,60]. In this case the equivariant Euler characteristic is the χ t genus of Laumon space.…”

“…In this case the equivariant Euler characteristic is the χ t genus of Laumon space. In the limit q → 0, the authors argue in [28], that χ t (Q d ) becomes the Poincaré polynomial of the compact core π −1 (0) ⊂ Q d . In this section we study the same "Poincaré polynomial" limit of the ADHM theory and, by analogy with T ρ [SU(N )], 8 we expect that we are again computing a Poincaré polynomial of the compact core of a putative resolution of the space of quasimaps to Hilb N (C 2 ):…”

“…Remarkably, a wide class of these partition functions can be constructed from a common constituent, the holomorphic blocks of [24]. The factorisation into blocks has been demonstrated in a number of examples including the three-sphere partition function [25], the superconformal index [26] and topologically twisted indices [27,28]. The factorisation can also be understood from the Higgs branch localisation perspective [29,30].…”