Localization of 4d $\mathcal{N}=1$ theories on $\mathbb{D}^2\times \mathbb{T}^2$

Understand

We consider 4d $\mathcal{N}=1$ gauge theories with R-symmetry on a hemisphere times a torus.

• We apply localization techniques to evaluate the exact partition function through a cohomological reformulation of the supersymmetry transformations.
• Our results represent the natural elliptic lifts of the lower dimensional analogs as well as a field theoretic derivation of the conjectured 4d holomorphic blocks, from which partition functions of compact spaces with diverse topology can be recovered through gluing.
• We also analyze the different boundary conditions which can naturally be imposed on the chiral multiplets, which turn out to be either Dirichlet or Robin-like.