Equivariant localization for AdS/CFT

TL;DR

This paper introduces a topological method using equivariant localization to compute BPS observables in AdS/CFT gravity solutions, bypassing the need for explicit supergravity solutions.

Contribution

It develops a formalism applying equivariant localization to AdS/CFT, enabling the calculation of physical observables from topological data alone.

Findings

Computed central charges for new supergravity solutions.

Recovered known results without explicit solutions.

Demonstrated the formalism on AdS5 and AdS3 backgrounds.

Abstract

We explain how equivariant localization may be applied to AdS/CFT to compute various BPS observables in gravity, such as central charges and conformal dimensions of chiral primary operators, without solving the supergravity equations. The key ingredient is that supersymmetric AdS solutions with an R-symmetry are equipped with a set of equivariantly closed forms. These may in turn be used to impose flux quantization and compute observables for supergravity solutions, using only topological information and the Berline--Vergne--Atiyah--Bott fixed point formula. We illustrate the formalism by considering $AdS_5\times M_6$ and $AdS_3\times M_8$ solutions of $D=11$ supergravity. As well as recovering results for many classes of well-known supergravity solutions, without using any knowledge of their explicit form, we also compute central charges for which explicit supergravity solutions have not been constructed.