We use the results of our companion paper [$\textit{J. Phys. A}$ (2025) 495201], which explores the equivariant generalization of the constant-map contribution in topological string theory on toric Calabi-Yau manifolds $\textit{X}$, to establish a holographic correspondence with M2-brane partition functions. We rigorously test this conjecture within the perturbative regime, incorporating all finite $\textit{N}$ corrections on the field theory side. Our approach involves additional key details, such as incorporating effective 4d higher-derivative supergravity corrections to introduce refinement. A central result is the derivation of the Airy function representation for the squashed $\textit{S}$ partition function of the field theory, for an arbitrary squashing parameter. We demonstrate that this Airy function structure is universal across all M2-brane models and provide a general expression in terms of the equivariant volume of $\textit{X}$, incorporating the mesonic deformations corresponding to complexified masses. This expression is then evaluated explicitly for several examples, including ABJM theory, its flavored generalizations, circular quivers, and beyond, demonstrating agreement with the available field theory localization results. We extend the analysis to the superconformal and twisted indices of M2-brane models, and their spindle generalizations, leaving their full perturbative completion for future work. Finally, we explore avenues for generalizing these results to other brane systems, explicitly applying the idea to D3-branes.

Australia

Luca Cassia

Bulgaria

Kiril Hristov

M2-brane partition functions and HD supergravity from equivariant volumes

We study Einstein-Maxwell theory in $\textit{D}$ ≥ 3 spacetime dimensions including $\textit{all}$ Lorentz-invariant parity-even four-derivative couplings. Building on the results of DOI:10.1007/s11433-024-2398-1, we consider static, charged, asymptotically flat black hole solutions to first order in the higher-derivative expansion. In $\textit{D}$ = 4 and $\textit{D}$ = 5, we compute the corrected black hole thermodynamics and compare with the Reall-Santos prescription based on the two-derivative background, highlighting a subtlety when both inner and outer horizons are involved. By introducing natural variables, as in DOI:10.1007/JHEP08(2023)003, we recast the on-shell actions in terms of left- and right-moving chemical potentials, which significantly simplifies the analysis.

China

Peng-Ju Hu

Yi Pang

Black hole thermodynamics at 4 derivatives, natural variables and BPS limits

We construct solutions in the form of $\textit{AdS}$ × Σ, where Σ denotes a spindle geometry, within a particular 4d $$ \mathcal{N} $$ = 2 gauged supergravity with vector multiplets and a charged hypermultiplet stemming from a consistent truncation of 6d matter coupled $\textit{F}$(4) gauged supergravity on a Riemann surface. We also compute the Bekenstein-Hawking entropy for the presumed asymptotically $\textit{AdS}$ black hole solutions, providing a dual prediction for the spindle index of 3d superconformal field theories falling under the class $$ \mathcal{F} $$ . Notably, the gravitational block description of the solutions accurately replicates the entropy and the values of scalar fields at the two orbifold points of the spindle, validating our findings.

South Korea

Minwoo Suh

Spindle black holes and theories of class $$ \mathcal{F} $$

It was recently observed in [1] for thermal Kerr-Newman black holes in 4d flat space that one can rewrite the conventional thermodynamics on the inner and outer horizons in terms of left- and right-moving variables with a remarkable simplification of the corresponding expressions. With the goal of illustrating the wide applicability of these newly proposed natural variables, we extend the original observation in four independent directions that can be further superimposed on each other. These four generalizations can be thought of as different deformations of the original 4d Einstein-Maxwell theory, all within the framework of supergravity: higher derivative (HD) corrections in minimal 4d supergravity; additional scalar and vector couplings in matter-coupled 4d supergravity; higher dimensions, in particular 5d minimal supergravity; and a cosmological constant in 4d minimal gauged supergravity with Anti-de Sitter (AdS) vacuum. Each of these generalizations offers a different lesson about the novel thermodynamics, and we pay special attention to the respective BPS limits that can be understood from fixed point formulae, demonstrating the power of the natural variables to capture the full phase space.

Black hole thermodynamics in natural variables: quadrophenia

We construct new classes of supersymmetric AdS × Σ solutions of 4d gauged supergravity in presence of charged hypermultiplet scalars, with Σ the complex weighted projective space known as a $\textit{spindle}$. These solutions can be viewed as near-horizon geome- tries of asymptotically Anti de-Sitter (AdS ) black holes with magnetic fluxes that admit embedding in 11d on Sasaki-Einstein (SE ) manifolds, which renders them of holographic interest. We show that in each case the Bekenstein-Hawking entropy follows from the procedure of gluing two gravitational blocks, ultimately determined by SE data. This allows us to establish the general form of the gravitational blocks in gauged 4d $$ \mathcal{N} $$ = 2 supergravity with charged scalars and massive vectors. Holographically, our results provide a large N answer for the spindle index with anti-twist and additional mesonic or baryonic fluxes of a number of $$ \mathcal{N} $$ = 2 Chern-Simons-matter theories.

Spindle black holes in AdS

We consider the general thermal asymptotically flat Kerr-Newman black holes in 4d Einstein-Maxwell theory. Even though their thermodynamics has been understood for decades, the Gibbs free energy and on-shell action are only known implicitly as functions of the standard chemical potentials. Using the so-called left and right moving (or holomorphic and anti-holomorphic) variables related to the chemical potentials on both the outer and the inner horizons, we are able to present explicit and very simple expressions for all quantities. We discuss various limits in the parameter space, remarkably finding a smooth BPS limit allowing direct access to the extremal surface. In the BPS limit the anti-holomorphic part of the on-shell action vanishes identically, leading automatically to the holomorphic expression expected microscopically. This gives us confidence that the newly defined thermal partition function in terms of these variables is the natural candidate for a full microscopic description.

Explicit black hole thermodynamics in natural variables

We initiate a systematic study of supersymmetric backgrounds in 4d 𝒩 = 2 Euclidean supergravity in the presence of infinite towers of higher derivative corrections. Adopting a Gibbons-Hawking view towards the evaluation of the action in terms of nuts and bolts, we consider the two maximally symmetric vacua ℝ4 and ℍ4 (Euclidean AdS4) and their unique supersymmetric deformations with (anti-) self-dual Maxwell tensors corresponding to a single nut at the center. These are the Omega background of Nekrasov-Okounkov, Ωℝ4, and its generalization with a cosmological constant of Martelli-Passias-Sparks, denoted Ωℍ4 (also known as the gravity dual of the U(1) × U(1) squashed sphere). We write down the BPS configurations in the superconformal formalism in the presence of vector multiplets and derive the corresponding off- and on-shell actions. Our results provide a rigorous proof for important parts of the conjecture in [1] and its holographic corollary in [2], which we discuss in detail along with extensions such as the addition of hypermultiplets and the presence of conical defects.

Maximally symmetric nuts in 4d 𝒩 = 2 higher derivative supergravity

We apply the conjecture of [1] for gravitational building blocks to the effective supergravity description of M-theory on S7 /ℤ k . Utilizing known localization results for the holographically dual ABJM theory, we determine a complete tower of higher derivative corrections to the AdS4 supergravity and a further set of quantum corrections. This uniquely fixes the gravitational block, leading to holographic predictions for a number of exact ABJM observables, excluding only constant and non-perturbative corrections in the gauge group rank N. The predicted S3 partition function is an Airy function that reproduces previous results and generalizes them to include arbitrary squashing and mass deformations/R-charge assignments. The topologically twisted and superconformal indices are instead products of two different Airy functions, in agreement with direct numeric calculations in the unrefined limit of the former object. The general fixed-point formula for an arbitrary supersymmetric background is similarly given as a product of Airy functions.

ABJM at finite N via 4d supergravity

We explore the BPS limit of asymptotically flat black hole thermodynamics, drawing analogies with recent studies of black holes in anti-de Sitter space. Although ultimately motivated by supersymmetry and quantum gravity, the BPS limit is classically well-defined for the dyonic Kerr-Newman black holes in pure Einstein-Maxwell theory and various generalizations with additional scalars and gauge fields that admit embedding in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supergravity (for concreteness we consider the STU model). This limit is manifestly different from extremality as it includes a family of rotating Euclidean saddles that represent a (fictitious) thermal branch, smoothly connected with the extremal static Reissner-Nordström black hole and its generalizations with extra matter. We are able to evaluate unambiguously all BPS chemical potentials, finding constant imaginary angular velocity as a consequence of Wick rotation. In the mixed ensemble of fixed magnetic and varying electric charges, we derive the OSV formula that is crucial for the microscopic understanding of the system, and provide evidence that it should be interpreted as a fixed-point formula.

The dark (BPS) side of thermodynamics in Minkowski4

We study four-derivative corrections to 5d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 minimal gauged supergravity using tools from conformal supergravity. There are two supersymmetric invariants at the four-derivative order and we show explicitly how to write down the action in the Poincaré frame in terms of their coefficients. We apply our results in the context of holography where the four-derivative terms in the supergravity action translate into corrections to the conformal anomalies of the dual 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SCFT. We use these results to study corrections to the entropy of the supersymmetric black string solutions of minimal supergravity. We also discuss explicit embeddings of the 5d supergravity in string and M-theory and use them to fix the higher-derivative coefficients in terms of the microscopic parameters.