Equivariant localization and holography

TL;DR

This paper explores equivariant localization techniques applied to holography, focusing on geometries like toric orbifolds and Calabi-Yau singularities, with applications in supersymmetric theories and supergravity solutions.

Contribution

It introduces methods for evaluating equivariant volumes and applies them to compute anomaly polynomials, orbifold indices, and supergravity free energies.

Findings

Derived new formulas for equivariant volumes using fixed-point and Molien-Weyl methods.

Applied localization to compute supersymmetric partition functions and anomaly polynomials.

Proved factorization of supergravity free energies into gravitational blocks.

Abstract

We discuss the theory of equivariant localization focussing on applications relevant for holography. We consider geometries comprising compact and non-compact toric orbifolds, as well as more general non-compact toric Calabi-Yau singularities. A key object in our constructions is the equivariant volume, for which we describe two methods of evaluation: the Berline-Vergne fixed-point formula and the Molien-Weyl formula, supplemented by the Jeffrey-Kirwan prescription. We present two applications in supersymmetric field theories. Firstly, we describe a method for integrating the anomaly polynomial of SCFTs on compact toric orbifolds. Secondly, we discuss equivariant orbifold indices that are expected to play a key role in the computation of supersymmetric partition functions. In the context of supergravity, we propose that the equivariant volume can be used to characterise universally the geometry of a large class of supersymmetric solutions. As an illustration, we employ equivariant localization to prove the factorization in gravitational blocks of various supergravity free energies, recovering previous results as well as obtaining generalizations.