<p>We investigate the thermodynamics of Reissner-Nordström Gauss-Bonnet (RN-GB) black holes in anti–de Sitter (AdS) space with three horizon geometries (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>=</mo><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn></math>) within the grand canonical ensemble. Using the recently developed topological approach to black hole thermodynamics, inspired by Duan’s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi></math>-mapping theory, we analyze the black holes by treating both critical points in the phase diagram and black hole solutions as defects in the thermodynamic parameter space. Our results show that the Gauss-Bonnet coupling significantly alters the topological classification of RN-GB AdS black holes, distinguishing them from their RN AdS counterparts in the grand canonical ensemble while aligning with their canonical ensemble counterparts. Complementary analyses of local stability using specific heat validate the implication of topological analysis. Furthermore, an evaluation of global stability via Gibbs free energy provides a comprehensive understanding on system phase structure. Notably, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>=</mo><mo>+</mo><mn>1</mn></math>, topological analysis suggests liquid-gas-type phase transitions, whereas global analysis favors Hawking-Page transitions. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, topology indicates a single stable black hole branch, yet the global analysis reveals the presence of Hawking-Page transitions.</p>

South Korea

Imtak Jeon

Bum-Hoon Lee

Wonwoo Lee

Madhu Mishra

Stability and topological nature of charged Gauss-Bonnet AdS black holes in five dimensions

Quantum study of supersymmetric theories on Euclidean two dimensional anti-de Sitter space (EAdS2) requires complexified spectrum. For a chiral multiplet, we showed that the spectrum of the Dirac operator acquires a universal shift of i/2 from the real spectrum to make the supersymmetry between boson and fermion manifest, where both the bosonic and fermionic eigenfunctions are normalizable using an appropriate definition of Euclidean inner product. We extend this analysis to the vector multiplet, where we show that the gaugino requires both +i/2 and i/2 shift from the real spectrum, and there is additional isolated point at vanishing spectral parameter which is mapped by supersymmetry to the boundary zero modes of the vector field. Furthermore, this spectral analysis shows that not every bosonic fields in the vector multiplet can satisfy normalizable boundary condition. Nevertheless, aided by a reorganization of fields into a cohomological form, we find the supersymmetry mapping between bosons and fermions in terms of the expansion coefficients with respect to the newly constructed basis.

Alfredo Lezcano

Augniva Ray

Supersymmetric spectrum for vector multiplet on Euclidean AdS2

A correction to this paper has been published: https://doi.org/10.1007/JHEP07(2023)056

Erratum to: Supersymmetric localization:   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (2, 2) theories on S2 and AdS2

Quantum study of supersymmetric theories on Euclidean two dimensional anti–de Sitter space (<math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math>) is invalid if we use the standard normalizable functional basis due to its incompatibility with supersymmetry. We cure this problem by demonstrating that supersymmetry requires a complexified spectrum and constructing the supersymmetric basis for scalar and spinor fields. Our new basis is free of fermionic zero modes, delta-function normalizable with respect to a newly defined inner product, and compatible with the supersymmetric asymptotic boundary condition. We also explore the one-loop evaluation using this basis and show that it agrees with the standard nonsupersymmetric basis up to a global contribution arising from the fermion zero mode.

Alfredo González Lezcano

Supersymmetry and complexified spectrum on Euclidean <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math>

We begin to develop the formalism of localization for the functional integral of supergravity on AdS3 × S2. We show how the condition of supersymmetry in the Euclidean ℍ3 /ℤ × S2 geometry naturally leads to a twist of the S2 around the time direction of AdS3. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of (0, 4) SCFT2 at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on ℍ3 /ℤ × S2 localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on AdS2 × S2. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of  arXiv:1112.5371  , which could be of independent interest.

United Kingdom

Axel Ciceri

Sameer Murthy

Localization on AdS3 × S2. Part I. The 4d/5d connection in off-shell Euclidean supergravity

Application of the supersymmetric localization method to theories on anti-de Sitter spacetime has received recent interest, yet still remains as a challenging problem. In this paper, we focus on (global) Euclidean AdS2, on which we consider an Abelian 𝒩 = (2, 2) theory and implement localization computation to obtain the exact partition function. For comparison, we also revisit the theory on S2 and perform a parallel computation. We refine the notion of equivariant supersymmetry and use appropriate functional integration measure. For AdS2 we choose a supersymmetric boundary condition which is compatible with the principle of variation. To evaluate the 1-loop determinant about the localization saddle, we use index theory and fixed point formula, where we pay attention to the effect of zero modes and their superpartners. The existence of fermionic superpartner of 1-form boundary zero modes is proven. Obtaining the 1-loop determinant requires expansion of the index that presents an ambiguity, which we resolve using boundary condition. The resulting partition function reveals an overall dependence on the size of the background manifold, AdS2 as well as S2, as a sum of two types of contributions: a local one from local conformal anomaly through the index computation and a global one coming from zero modes. This overall size dependence matches with the perturbative 1-loop evaluation using heat kernel method.

Supersymmetric localization: 𝒩 = (2, 2) theories on S2 and AdS2

We use localization to compute the partition function of a four dimensional, supersymmetric, abelian gauge theory on a hemisphere coupled to charged matter on the boundary. Our theory has eight real supercharges in the bulk of which four are broken by the presence of the boundary. The main result is that the partition function is identical to that of     <math> <mi>N</mi> </math> $$ \mathcal{N} $$  = 2 abelian Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where the quantized Chern-Simons level is replaced by an arbitrary complexified gauge coupling τ. The localization reduces the path integral to a single ordinary integral over a real variable. This integral in turn allows us to calculate the scaling dimensions of certain protected operators and two-point functions of abelian symmetry currents at arbitrary values of τ. Because the underlying theory has conformal symmetry, the current two-point functions tell us the zero temperature conductivity of the Lorentzian versions of these theories at any value of the coupling. We comment on S-dualities which relate different theories of supersymmetric graphene. We identify a couple of self-dual theories for which the complexified conductivity associated to the U(1) gauge symmetry is τ/2.

India

Rajesh Gupta

Christopher Herzog

Duality and transport for supersymmetric graphene from the hemisphere partition function

We develop the formalism of supersymmetric localization in supergravity using the deformed BRST algebra defined in the presence of a supersymmetric background as recently formulated in [1]. The gravitational functional integral localizes onto the cohomology of a global supercharge Q eq, obeying Q  eq 2  = H, where H is a global symmetry of the background. Our construction naturally produces a twisted version of supergravity whenever supersymmetry can be realized off-shell. We present the details of the twisted graviton multiplet and ghost fields for the superconformal formulation of four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supergravity. As an application of our formalism, we systematize the computation of the exact quantum entropy of supersymmetric black holes. In particular, we compute the one-loop determinant of the Q eq   <math> <mi>V</mi> </math>  $$ \mathcal{V} $$  deformation operator for the off-shell fluctuations of the Weyl multiplet around the AdS2 × S 2 saddle. This result, which is consistent with the corresponding large-charge on-shell analysis, is needed to complete the first-principles computation of the quantum entropy.

Twisting and localization in supergravity: equivariant cohomology of BPS black holes

Logarithmic terms are quantum corrections to black hole entropy determined completely from classical data, thus providing a strong check for candidate theories of quantum gravity purely from physics in the infrared. We compute these terms in the entropy associated to the horizon of a magnetically charged extremal black hole in AdS ×4S7 using the quantum entropy function and discuss the possibility of matching against recently derived microscopic expressions.

France

Shailesh Lal

Logarithmic corrections to entropy of magnetically charged AdS 4 black holes

We use the techniques of supersymmetric localization to compute the BPS black hole entropy in N = 2 $$ \mathcal{N}=2 $$ supergravity. We focus on the n v + 1 vector multiplets on the black hole near horizon background which is AdS 2 × S 2 space. We find the localizing saddle point of the vector multiplets by solving the localization equations, and compute the exact one-loop partition function on the saddle point. Furthermore, we propose the appropriate functional integration measure. Through this measure, the one-loop determinant is written in terms of the radius of the physical metric, which depends on the localizing saddle point value of the vector multiplets. The result for the one-loop determinant is consistent with the logarithmic corrections to the BPS black hole entropy from vector multiplets.