Equivariant localization and holography

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.

arXiv comment69 pages, 10 figures; published version, few typos corrected

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• Equivariant localization for AdS/CFT
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• Toric gravitational instantons in gauged supergravity
• Super Yang-Mills on Branched Covers and Weighted Projective Spaces
• NUTs, Bolts, and Spindles
• Exact Results for SYM on $Y^{p,q}$ and $S^2\times S^2$ with Conical Singularities
• Branes wrapped on quadrilaterals
• $\mathcal{I}$-extremization with baryonic charges
• Constant maps in equivariant topological strings and geometric modeling of fluxes
• Giant Gravitons and Volume Minimisation
• Kodaira-Spencer Anomalies with Stora-Zumino Method