Equivariant localization in supergravity in odd dimensions

TL;DR

The work develops a localization framework for odd-dimensional manifolds with boundary and applies it to five-dimensional minimal gauged supergravity. By recasting the on-shell action as an integral of a Chern–Simons-type form plus a boundary term and employing an equivariant localization on toric geometries, the authors express the renormalized action in terms of purely topological and toric data, independent of the explicit metric. The key results include a metric-agnostic derivation of the AdS$_5$ rotating black hole entropy function from topology and a general formula linking the on-shell action to the analytic continuation of Sasakian volumes via the GMS master volume. The framework offers broad potential extensions to more general flux configurations, other dimensions, and holographic contexts, providing a powerful topological handle on holographic actions and entropy functionals.

Abstract

We discuss a localization formula for certain integrals on odd-dimensional manifolds with boundaries, equipped with a Killing vector, and employ this to localize the regularised on-shell action of a large class of supersymmetric solutions of five dimensional minimal gauged supergravity. Specifically, we consider asymptotically AdS_5 solutions in the time-like class, in which the transverse Kähler foliation is assumed to be toric. We find that the background subtraction regularization method leads to an intriguing formula for the on-shell action, in terms of an analytic continuation of the Martelli-Sparks-Yau Sasakian volume. In particular, we show that the regularised on-shell action is a function of the toric data of an effective compact five-dimensional manifold, as well as of the supersymmetric Killing vector, outside the corresponding dual cone. As our main example we provide a derivation of the well-known entropy function of supersymmetric and rotating black holes in AdS_5, using only topological data.

Figures (3)

• Figure 1: The two-dimensional polytope of $S^5$ obtained by cutting the first octant with the plane $\xi_i y_i=\frac{1}{2}$ and the three localization loci $L_a$. Notice that to have a compact polytope, the Reeb vector, which is the normal to the plane, must lie inside the octant.
• Figure 2: The two-dimensional polytopes of the Euclidean black hole and of AdS$_5$ are obtained by cutting the first octant, the polytope of $C(S^5)$, with the plane $\xi_i y_i=\frac{1}{2}$ where $\xi$ lies outside the octant itself. Different positions of the plane give different topologies. In the figure we are assuming that the components of $\xi$ are real. To obtain the complex geometry of the non-extremal Euclidean black holes an analytic continuation is needed.
• Figure 3: The two-dimensional polytope of the non-extremal $\mathbb{C}\text{BS}$ solution (on the left) and of the extremal solution (on the right). The latter is obtained by collapsing the segment corresponding to the compact divisor to a point.