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Factorisation of 3d \(\mathcal{N} = 4\) twisted indices and the geometry of vortex moduli space. (English) Zbl 1454.81220

Summary: We study the twisted indices of \(\mathcal{N} = 4\) 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 \(T[ \mathrm{SU} (N)]\) 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 \(\chi_t\) genera of the moduli spaces of “local” vortices. The twisted index itself coincides with a corresponding generating function for the \(\chi_t\) 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 \(T[ \mathrm{SU} (N)]\) 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.

MSC:

81T60 Supersymmetric field theories in quantum mechanics
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
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