Twisted index on hyperbolic four-manifolds

We introduce the topologically twisted index for four-dimensional \({\mathcal {N}}=1\) gauge theories quantized on \({\textrm{AdS}_2}\times S^1\). We compute the index by applying supersymmetric localization to partition functions of vector and chiral multiplets on \({\textrm{AdS}_2}\times T^2\), 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.