Localization on Hopf surfaces

TL;DR

This work develops a complete localization framework for four-dimensional ${\cal N}=1$ gauge theories with an R-symmetry on Hermitian manifolds, focusing on Hopf surfaces ${\rm H}_{p,q}\simeq S^1\times S^3$. By exploiting two opposite-R supercharges, the authors reduce the partition function to a matrix model and compute exact one-loop determinants, expressing the result as $Z[{\rm H}_{p,q}]=e^{-F(p,q)}\mathcal{I}(p,q)$, where $\mathcal{I}(p,q)$ is the supersymmetric index and $F(p,q)$ depends on the complex structure parameters via the anomaly coefficients $(a,c)$. Anomaly cancellation is shown to enforce consistency, with $F(p,q)$ capturing a supersymmetric Casimir energy that governs the leading large-$N$ behavior and matches holographic predictions for dual gravity theories. The analysis also clarifies the reduction to three dimensions and the role of special functions (Jacobi theta and elliptic gamma) in the exact determinants, providing a precise bridge between 4d localization, 3d indices, and holography. This framework advances exact results for observables on curved supersymmetric backgrounds and informs gravity dual computations through the holographic renormalized action on Sasaki-Einstein boundaries.

Abstract

We discuss localization of the path integral for supersymmetric gauge theories with an R-symmetry on Hermitian four-manifolds. After presenting the localization locus equations for the general case, we focus on backgrounds with S^1 x S^3 topology, admitting two supercharges of opposite R-charge. These are Hopf surfaces, with two complex structure moduli p,q. We compute the localized partition function on such Hopf surfaces, allowing for a very large class of Hermitian metrics, and prove that this is proportional to the supersymmetric index with fugacities p,q. Using zeta function regularisation, we determine the exact proportionality factor, finding that it depends only on p,q, and on the anomaly coefficients a, c of the field theory. This may be interpreted as a supersymmetric Casimir energy, and provides the leading order contribution to the partition function in a large N expansion.