We study the sphere partition function of 3d $$ \mathcal{N} $$ = 2 holographic SCFTs arising on the worldvolume of $\textit{N}$ coincident M2-branes in the presence of squashing and real mass deformations. We argue that the all-order large $\textit{N}$ perturbative expansion of the partition function resums into a compact expression in terms of an Airy function. We provide ample evidence for this conjecture using numerical methods, saddle point analysis in the ’t Hooft limit, and the relation of the squashed sphere partition function to the Cardy-like limit of the superconformal index. We also discuss the relation of our results to topological string theory and the statistical mechanics of Fermi gases. Our conjecture has a bearing on the AdS/CFT correspondence and the structure of perturbative M-theory corrections to 11d supergravity which we discuss.

Belgium

Nikolay Bobev

Pieter-Jan Smet

South Korea

Junho Hong

France

Valentin Reys

China

Xuao Zhang

An Airy tale at large $\textit{N}$

We study the superconformal index of 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories on S 1 × ω  S 2 in the Cardy-like limit where the radius of the S 1 is much smaller than that of the S 2. We show that the first two leading terms in this Cardy-like expansion are dictated by the Bethe Ansatz formulation of the topologically twisted index of the same theory. We apply this relation to 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 holographic superconformal field theories describing the low-energy dynamics of N M2-branes and derive closed form expressions, valid to all orders in the 1/N expansion, for the two leading terms in the Cardy-like expansion of the superconformal index. We also discuss the implications of our results for the entropy of supersymmetric Kerr-Newman black holes in AdS4 and the four-derivative corrections to 4d gauged supergravity.

Sunjin Choi

Superconformal indices of 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs and holography

We study the logarithmic corrections to various CFT partition functions in the context of the AdS4/CFT3 correspondence for theories arising on the worldvolume of M2-branes. We utilize four-dimensional gauged supergravity and heat kernel methods and present general expressions for the logarithmic corrections to the gravitational on-shell action and black hole entropy for a number of different supergravity backgrounds. We outline several subtle features of these calculations and contrast them with a similar analysis of logarithmic corrections performed directly in the eleven-dimensional uplift of a given four-dimensional supergravity background. We find results consistent with AdS/CFT provided that the infinite sum over KK modes on the internal space is regularized in a specific manner. This analysis leads to an explicit expression for the logarithmic correction to the Bekenstein-Hawking entropy of large Kerr-Newmann and Reissner-Nordström black holes in AdS4. Our results also have important implications for effective field theory coupled to gravity in AdS4 and for the existence of scale-separated AdS4 vacua in string theory, which come in the form of new constraints on the field content and mass spectrum of matter fields.

Marina David

A compendium of logarithmic corrections in AdS/CFT

We use holography in conjunction with recent results from supersymmetric localization to compute certain thermal observables for 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 holographic SCFTs arising on the worldvolume of N M2-branes. We obtain results for the thermal free energy density on S 1 × ℝ 2, the Casimir energy on T 2 × ℝ, and the three leading coefficients in the large temperature limit of the free energy on S 1 × S 2 valid to subleading order in the large N limit. As a byproduct of our holographic analysis we also present a conjecture for the structure of the large temperature expansion of the thermal free energy of general 3d CFTs on S 1 × S 2.

Holographic thermal observables and M2-branes

We study the S 1 ×   <math> <msub> <mi>Σ</mi> <mi>g</mi> </msub> </math>  $$ {\Sigma}_{\mathfrak{g}} $$  topologically twisted index and the squashed sphere partition function of various 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  ≥ 2 holographic superconformal field theories arising from M2-branes. Employing numerical techniques in combination with well-motivated conjectures we provide compact closed-form expressions valid to all orders in the perturbative 1/N expansion for these observables. We also discuss the holographic implications of our results for the topologically twisted index for the dual M-theory Euclidean path integral around asymptotically AdS4 solutions of 11d supergravity. In Lorentzian signature this leads to a prediction for the corrections to the Bekenstein-Hawking entropy of a class of static asymptotically AdS4 BPS black holes.

Large N partition functions of 3d holographic SCFTs

We study a limit of the superconformal index of the ABJM theory on S 1 × S 2 in which the size of the circle is much smaller than the radius of the two-sphere. We derive closed form expressions for the two leading terms in this Cardy-like limit which are valid to all orders in the 1/N expansion. These results are facilitated by a judicious rewriting of the superconformal index which establishes a connection with the Bethe Ansatz Equations that control the topologically twisted index. Using the same technique we extend these results to the superconformal index of another holographic theory: 3d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM coupled to one adjoint and N  f  fundamental hypermultiplets. We discuss the implications of our results for holography and the physics of charged rotating black holes in AdS4.

Large N superconformal indices for 3d holographic SCFTs

We study the large N limit of some supersymmetric partition functions of the U(N) k  × U(N) −k  ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large N limit, for the partition function on the U(1) × U(1) invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed k to all orders in the 1/N expansion. We use these results to derive the topologically twisted index and the sphere partition function in the ’t Hooft limit which correspond to genus g type IIA string theory free energies to all orders in the α′ expansion. We discuss the implications of our results for holography and the physics of AdS4 black holes.

Large N partition functions of the ABJM theory

Using recent results in four-derivative 5d <math><mi>N</mi><mo>=</mo><mn>2</mn></math> minimal gauged supergravity, we evaluate the regularized on-shell action of the Euclidean solution in this theory that admits a Lorentzian continuation to an <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></math> black hole with one electric charge and two angular momenta. We focus on the supersymmetric limit of this solution and employ holography to show that the result can be expressed purely in terms of angular momentum fugacities and the ’t Hooft anomalies for the <math><mi>U</mi><mo>(</mo><mn>1</mn><msub><mo>)</mo><mi>R</mi></msub></math> R symmetry of the dual 4d <math><mi>N</mi><mo>=</mo><mn>1</mn></math> superconformal field theory. This holographic calculation is in perfect agreement with recent studies of the 4d <math><mi>N</mi><mo>=</mo><mn>1</mn></math> superconformal index “on the second sheet.” We illustrate the utility of these results in two classes of 4d <math><mi>N</mi><mo>=</mo><mn>1</mn></math> holographic SCFTs that have ’t Hooft anomalies with suitable large-<math><mi>N</mi></math> behavior that leads to nontrivial corrections at first subleading order in the <math><mn>1</mn><mo>/</mo><mi>N</mi></math> expansion. We also explicitly calculate the Wald entropy of the black hole solution and delineate the leading four-derivative corrections to the Bekenstein-Hawking entropy.

Vasil Dimitrov

Annelien Vekemans

Higher derivative corrections and <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></math> black holes

We present a simple closed form expression for the topologically twisted index of the ABJM theory as a function of the magnetic fluxes and complex chemical potentials valid at fixed <math><mi>k</mi></math> and to all orders in the <math><mn>1</mn><mo>/</mo><mi>N</mi></math> expansion. This in turn leads to analytic expressions for the topologically twisted index at fixed genus in the ’t Hooft limit to all orders in the <math><mn>1</mn><mo>/</mo><msqrt><mi>λ</mi></msqrt></math> expansion. These results have important implications for holography and the microscopic entropy counting of supersymmetric static <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math> black holes. Generalizations to other SCFTs arising from M2-branes are also briefly discussed.

Large <math><mi>N</mi></math> Partition Functions, Holography, and Black Holes

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.