Generalized Indices for $\mathcal{N}=1$ Theories in Four-Dimensions

Understand

We use localization techniques to calculate the Euclidean partition functions for $\mathcal{N}=1$ theories on four-dimensional manifolds $M$ of the form $S^1 \times M_3$, where $M_3$ is a circle bundle over a Riemann surface.

• These are generalizations of the $\mathcal{N}=1$ indices in four-dimensions including the lens space index.
• We show that these generalized indices are holomorphic functions of the complex structure moduli on $M$.
• We exhibit the deformation by background flat connections.