Localizing Romans supergravity

TL;DR

This work derives a general fixed-point localization formula for the holographically renormalized on-shell action in Euclidean Romans F(4) supergravity, expressing I entirely in terms of the R-symmetry Killing vector and topological data of the six-manifold M_6. By constructing equivariantly closed forms and applying the BV/AB fixed-point theorem, the authors reduce the action to contributions from fixed-point sets of dimensions 0, 2, and 4, with weights determined by local spinor projections and normal-bundle data. They provide extensive checks by applying the formula to a broad class of product- and fibred-space examples, recovering known AdS_6/CFT_5 results and making new predictions for unknown solutions, while also linking to dual 5d USp(2N) field theories on various backgrounds. The analysis leverages toric geometry to compute weights and fluxes, and shows how topological data alone (Euler characteristic, signatures, Chern classes) govern the on-shell action and charges, offering a powerful, PDE-free route to holographic free energies and entropies in six-dimensional supergravity. The results pave the way for extensions to more general matter couplings and broader classes of supersymmetric backgrounds."

Abstract

We derive a general formula for the on-shell action of six-dimensional Euclidean Romans supergravity using equivariant localization. Our results are obtained without the need for solving any of the equations of motion, instead working on the assumption of the existence of a solution. We show that the on-shell action is completely determined in terms of the R-symmetry Killing vector and topological data. We easily recover known results in the literature, make predictions for hitherto unknown solutions, and also match to holographic field theory duals.

Figures (8)

• Figure 1: The toric diagram for $\mathbb{R}^4\times S^2$. It is infinite in extent in two directions denoted by the dashed lines.
• Figure 2: The toric diagram for $\mathbb{R}^2\times S^2_{\varepsilon_1}\times S^2_{\varepsilon_2}$. It is infinite in extent in one direction, following the dashed vertical lines.
• Figure 3: Toric diagram for $\mathcal{O}(-1)\oplus \mathcal{O}(-2)\rightarrow S^2$. The red faces are the north and south pole of the sphere, while the green and blue are the centres of the two copies of $\mathbb{C}$.
• Figure 4: Toric diagram for $\mathcal{O}(-1,-2) \rightarrow S^2 \times S^2$.
• Figure 5: Toric diagram for $\mathcal{O}(-2)\rightarrow \mathbb{CP}^2$.