Twisted index on hyperbolic four-manifolds

We introduce the topologically twisted index for four-dimensional N=1 N = 1 gauge theories quantized on AdS₂× S¹ AdS 2 × S 1 . We compute the index by applying supersymmetric localization to partition functions of vector and chiral multiplets on AdS₂× T² AdS 2 × 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.