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.

我们引入了在\({\textrm{AdS}_2}\times S^1\)上量化的四维\({\mathcal {N}}=1\)规范理论的拓扑扭曲指数。我们通过将超对称局部化应用于\({\textrm{AdS}_2}\times T^2\)上的向量和手性多重态的配分函数来计算索引，有边界和无边界：在这两种情况下，我们对归一化性和边界条件进行分类用于规范场、物质场和幽灵场。当动力场耦合到具有非平凡外导数且与自旋连接成比例的R对称背景 1 形式时，索引会发生扭曲。正则化后，索引以椭圆伽玛函数的形式编写，让人想起四维全纯块，并且关键取决于R电荷。