We construct singly and doubly spinning non-supersymmetric F1–P black ring solutions in five-dimensional supergravity. These black rings have regular horizons and non-zero temperature. The singly spinning configuration lies in the duality orbit of the black ring constructed by Elvang, Emparan, and Figueras, while the doubly spinning configuration is a charged extension of the black ring constructed by Chen, Hong, and Teo. We analyze the physical properties of these solutions and the various limits they admit. In particular, the doubly spinning solution admits an extremal limit in which the entropy satisfies the relation $\textit{S}$ = 2$\textit{πJ}$ , thereby linking it directly to the angular momentum on the $\textit{S}$ .

India

Pavan Dharanipragada

China

Gurmeet Punia

Amitabh Virmani

Non-supersymmetric F1-P black rings

We construct the index saddle for the supersymmetric F1-P black ring. Our construction proceeds by taking a supersymmetric limit of a non-supersymmetric doubly spinning F1-P black ring. We express the resulting saddle as a three-center Bena-Warner solution. The black ring saddle possesses a finite-area event horizon, yet the two-derivative index vanishes. The solution is singular on certain subspaces of the horizon, where higher-derivative corrections are expected to become important. We argue that, once such corrections are taken into account, the solution can yield a finite result. In particular, we present a scaling analysis showing that the index agrees with the microscopic result, up to an overall numerical constant that cannot be fixed by the scaling argument alone. This analysis applies only within a restricted region of parameter space, whose full significance is not yet fully understood.

Index saddle for supersymmetric F1-P black ring

We study fully BPS and a broad class of half-BPS stationary configurations of four-dimensional Euclidean $$ \mathcal{N}=2 $$ supergravity with higher-derivative interactions. Working within the off-shell conformal supergravity framework of de Wit and Reys (arXiv:1706.04973), we analyse the complete set of Killing spinor equations and obtain the corresponding algebraic and differential constraints. We further derive the Euclidean attractor equations and evaluate the Wald entropy for the fully BPS AdS × $\textit{S}$ background. For half-BPS stationary configurations, we obtain the generalized stabilization equations expressing all fields in terms of harmonic functions on three-dimensional flat base space, extending the Lorentzian analysis of Cardoso et al. ( ) to the Euclidean signature. Our results provide a framework for studying supersymmetric saddles and computing the gravitational indices entirely within Euclidean higher-derivative supergravity, without recourse to analytic continuation.

South Korea

Soumya Adhikari

Abhinava Bhattacharjee

BPS solutions of 4d Euclidean $$ \mathcal{N}=2 $$ supergravity with higher derivative interactions

Certain supersymmetric elementary string states with angular momentum can be viewed as small black rings in a five-dimensional string theory. These black rings have zero area event horizon. The 4D-5D connection relates these small rings to small black holes without angular momentum in one less dimension. Recent works have proposed saddle solutions that compute the supersymmetric index for small black holes using gravitational path integral. In this paper, we propose an analogous saddle solution for a five-dimensional small black ring. The dominant contribution comes from a black ring saddle that rotates in both independent planes in five dimensions and has a finite area event horizon. We also write the saddle solution as a three center Bena-Warner solution.

Subhodip Bandyopadhyay

Yogesh Srivastava

The gravitational index of a small black ring

In a recent paper, Anupam, Chowdhury, and Sen [arXiv:2308.00038] constructed the non-extremal saddle that reproduces the supersymmetric index of the BMPV black hole with three independent charges in the classical limit. This saddle solution is a finite temperature complex solution saturating the BPS bound. In this paper, we write this solution in a canonical form in terms of harmonic functions on three-dimensional flat base space, thereby showing that it is supersymmetric. We also show that it exhibits the new form of attraction.

Kaberi Goswami

Attractor saddle for 5D black hole index

We compute the supersymmetric index of half BPS black holes in N=2 supergravity with higher curvature corrections and show that the result agrees with the degeneracy of supersymmetric extremal black holes carrying the same charges. Both sides of the computation are done gravitationally.

Germany

Subramanya Hegde

Ashoke Sen

P. Shanmugapriya

Supersymmetric index for half BPS black holes in N=2 supergravity with higher curvature corrections

The 4D-5D connection allows us to view the same near horizon geometry as part of a 4D black hole or a 5D black hole. A much studied example of this phenomenon is the BMPV black hole uplifted to 6D with flat base space versus Taub-NUT base space. These black holes have identical near horizon AdS × $\textit{S}$ geometry. In this paper, we study modes in AdS × $\textit{S}$ and identify those that correspond to supersymmetric hair modes in the full black hole spacetimes. We show that these modes satisfy non-normalisable boundary conditions in AdS . The non-normalisable boundary conditions are different for different hair modes and for different asymptotic completion. We also discuss how the supersymmetric hair modes on BMPV black holes fit into the classification of supersymmetric solutions of 6D supergravity.

Supersymmetric black hole hair and AdS

Supersymmetric elementary string states in the compactified heterotic string theory are described by small black holes that have zero area event horizon. In this paper we compute the supersymmetric index of such elementary string states using gravitational path integral. The dominant contribution to the path integral comes from an Euclidean rotating black hole solution of the supergravity theory with a finite area event horizon, but the logarithm of the index, computed from the saddle point, vanishes. Nevertheless we show that the solution is singular on certain subspaces of the horizon where higher derivative corrections can be important, and once the higher derivative corrections are taken into account the solution could yield a finite result for the logarithm of the index whose form agrees with the microscopic results up to an overall numerical constant. While the numerical constant is not determined in our analysis, we show that it is independent of the details of the compactification and even the number of non-compact dimensions, in agreement with the microscopic results.

United Kingdom

Chandramouli Chowdhury

Supersymmetric index for small black holes

In a recent paper [arXiv:2308.00038], Anupam, Chowdhury, and Sen conjectured that the finite temperature Euclidean five-dimensional Cvetic-Youm solution saturating the BPS bound is supersymmetric. In this paper, we explicitly construct Killing spinors for this solution in five-dimensional minimal supergravity. We also expand on the previous discussions of Killing spinors for the finite temperature Euclidean Kerr-Newman solution saturating the BPS bound. For both these cases, we show that the total charge gets divided into two harmonic sources on three-dimensional flat base space.

Killing spinors for finite temperature Euclidean solutions at the BPS bound

Black holes carry more information about the microstates than just the total degeneracy. As a concrete example, the <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math>-twined helicity trace indices for <math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math>-BPS black holes of the CHL models allow extracting information about the distribution of the <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> charges among the black hole microstates. The number of black hole microstates carrying a definite eigenvalue under the generator of the <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>N</mi></mrow></msub></math> twining group must be positive. This leads to a specific prediction for the signs of certain linear combinations of Fourier coefficients of Siegel modular forms. We explicitly test these predictions for low charges. In the D1-D5-P duality frame, we compute the appropriate hair removed partition functions and show the positivity of the appropriate Fourier coefficients for low charges. We present various consistency checks on our computations.