Twisted Index on Hyperbolic Four-Manifolds

TL;DR

This paper introduces a topologically twisted index for 4D N=1 gauge theories on AdS2 x S1, computed via supersymmetric localization, revealing dependence on R-charge and expressed through elliptic gamma functions.

Contribution

It presents a novel method to compute the twisted index on hyperbolic four-manifolds using localization, including boundary conditions and R-symmetry coupling.

Findings

Index expressed in terms of elliptic gamma functions

Dependence of the index on R-charge established

Classification of boundary conditions for gauge and matter fields

Abstract

We introduce the topologically twisted index for four-dimensional $\mathcal N=1$ gauge theories quantized on ${\rm AdS}_2 \times S^1$. We compute the index by applying supersymmetric localization to partition functions of vector and chiral multiplets on ${\rm AdS}_2 \times T^2$, with and without a boundary: in both instances we classify normalizability and boundary conditions for gauge, matter and ghost fields. The index is twisted as the dynamical fields are coupled to a R-symmetry background 1-form with non-trivial exterior derivative and proportional to the spin connection. After regularization the index is written in terms of elliptic gamma functions, reminiscent of four-dimensional holomorphic blocks, and crucially depends on the R-charge.