  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.
