Equivariant volume extremization and holography

TL;DR

This paper demonstrates that gravitational extremization problems in holography can be formulated using equivariant volume, successfully recovering known results and validating conjectured formulas for various brane configurations.

Contribution

It provides a geometric framework linking equivariant volume to holographic extremization, extending its applicability to new solutions and confirming previous conjectures.

Findings

Reproduces known gravitational central charges and free energies.

Validates gravitational block formulas for M5 and D4 branes.

Connects equivariant volume approach with localization methods.

Abstract

In a previous paper two of us (D.M. and A.Z.) proposed that a vast class of gravitational extremization problems in holography can be formulated in terms of the equivariant volume of the internal geometry, or of the cone over it. We substantiate this claim by analysing supergravity solutions corresponding to branes partially or totally wrapped on a four-dimensional orbifold, both in M-theory as well as in type II supergravities. We show that our approach recovers the relevant gravitational central charges/free energies of several known supergravity solutions and can be used to compute these also for solutions that are not known explicitly. Moreover, we demonstrate the validity of previously conjectured gravitational block formulas for M5 and D4 branes. In the case of M5 branes we make contact with a recent approach based on localization of equivariant forms, constructed with Killing spinor bilinears.