Proceedings of the American Mathematical Society

We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with N ≫ 1 flavors and a global U(1) charge. We provide a general definition of the charge in the (G, Σ) formalism, and compute its universal relation to the infrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the many-body density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.

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Notes on the complex Sachdev-Ye-Kitaev model

We derive the Bekenstein-Hawking entropy for a class of BPS black holes in the massive type IIA supergravity background AdS(4) x S-6 from a microscopic counting of supersymmetric ground states in a ...

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Holographic microstate counting for AdS4 black holes in massive IIA supergravity

We study the entropy of static dyonic BPS black holes in AdS4 in 4d gauged supergravities with vector and hyper multiplets, and how the entropy can be reproduced with a microscopic counting of states in the AdS/CFT dual field theory. We focus on the particular example of BPS black holes in AdS in massive Type IIA, whose dual three-dimensional boundary description is known and simple. To count the states in field theory we employ a supersymmetric topologically twisted index, which can be computed exactly with localization techniques. We find a perfect match at leading order.

Black hole entropy in massive Type IIA

We study the canonical $\mathrm{AdS}/\mathrm{CFT}$ correspondence between 4d $SU(N)$ $\mathcal{N}=4$ super Yang-Mills theory (SYM) and type IIB superstring theory on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. We analyze the supersymmetric index of the $\mathcal{N}=4$ SYM on ${S}^{1}\ifmmode\times\else\texttimes\fi{}{M}_{3}$ which counts supersymmetric states with fixed quantum numbers. We compute an asymptotic behavior of the index in the limit of shrinking ${S}^{1}$ for any $N$ by a refinement of the 4d supersymmetric Cardy formula. The asymptotic behavior for the superconformal index case (${M}_{3}={S}^{3}$) at large $N$ agrees with the Bekenstein-Hawking entropy of a rotating electrically charged Bogomolnyi-Prasad-Sommerfeld (BPS) black hole in ${\mathrm{AdS}}_{5}$ via a Legendre transformation as recently shown in the literature. We also find that the agreement formally persists for finite $N$ if we slightly modify the $\mathrm{AdS}/\mathrm{CFT}$ dictionary between the Newton constant and $N$. This implies the existence of a nonrenormalization property of the black hole entropy against quantum corrections. We also study the cases with other gauge groups and additional matter and the orbifold $\mathcal{N}=4$ SYM. It turns out that the entropies of all the CFT examples in this paper are universally given by $2\ensuremath{\pi}\sqrt{{Q}_{1}{Q}_{2}+{Q}_{1}{Q}_{3}+{Q}_{2}{Q}_{3}\ensuremath{-}2c({J}_{1}+{J}_{2})}$ with charges ${Q}_{1,2,3}$, angular momenta ${J}_{1,2}$, and central charge $c$. The results for other ${M}_{3}$ make predictions to the gravity side.

Quantum black hole entropy from 4d supersymmetric Cardy formula

A new analysis of quantum-mechanical models that describe interactions in large ensembles of Majorana fermions reveals richer phenomena in these complex systems, providing a potential path towards physical applications.

Symmetry Breaking in Coupled SYK or Tensor Models

The topologically twisted index for ABJM theory with gauge group U(N)
 k
 × U(N)−k has recently been shown, in the large-N limit, to reproduce the BekensteinHawking entropy of certain magnetically charged asymptotically AdS4 black holes. We numerically study the index beyond the large-N limit and provide evidence that it contains a subleading logarithmic term of the form −1/2 log N. On the holographic side, this term naturally arises from a one-loop computation. However, we find that the contribution coming from the near horizon states does not reproduce the field theory answer. We give some possible reasons for this apparent discrepancy.

/pdf/toward-microstate-counting-beyond-large-n-in-localization-3dndfchmm7.pdf

Toward microstate counting beyond large N in localization and the dual one-loop quantum supergravity

Within 11-dimensional supergravity we compute the logarithmic correction to the entropy of magnetically charged asymptotically ${\mathrm{AdS}}_{4}$ black holes with arbitrary horizon topology. We find perfect agreement with the expected microscopic result arising from the dual field theory computation of the topologically twisted index. Our result relies crucially on a particular limit to the extremal black hole case and clarifies some aspects of quantum corrections in asymptotically AdS spacetimes.

One-Loop Test of Quantum Black Holes in anti-de Sitter Space.

Due to the failure of thermodynamics for low temperature near-extremal black holes, it has long been conjectured that a "thermodynamic mass gap" exists between an extremal black hole and the lightest near-extremal state. For non-supersymmetric near-extremal black holes in Einstein gravity, with an AdS$_2$ throat, no such gap was found. Rather, at that energy scale, the spectrum exhibits a continuum of states, up to non-perturbative corrections. In this paper, we compute the partition function of near-BPS black holes in supergravity where the emergent, broken, symmetry is $PSU(1,1|2)$. To reliably compute this partition function, we show that the gravitational path integral can be reduced to that of a $\mathcal N=4$ supersymmetric extension of the Schwarzian theory, which we define and exactly quantize. In contrast to the non-supersymmetric case, we find that black holes in supergravity have a mass gap and a large extremal black hole degeneracy consistent with the Bekenstein-Hawking area. Our results verify several string theory conjectures, concerning the scale of the mass gap and the counting of extremal micro-states.

The statistical mechanics of near-BPS black holes

As shown in [1], two copies of the large N Majorana SYK model can produce spontaneous breaking of a Z2 symmetry when they are coupled by appropriate quartic terms. In this paper we similarly study two copies of the complex SYK model coupled by a quartic term preserving the U(1) × U(1) symmetry. We also present a tensor counterpart of this coupled model. When the coefficient α of the quartic term lies in a certain range, the coupled large N theory is nearly conformal. We calculate the scaling dimensions of fermion bilinear operators as functions of α. We show that the operator $$ {c}_{1i}^{\dagger }{c}_{2i} $$
 , which is charged under the axial U(1) symmetry, acquires a complex dimension outside of the line of fixed points. We derive the large N Dyson-Schwinger equations and show that, outside the fixed line, this U(1) symmetry is spontaneously broken at low temperatures because this operator acquires an expectation value. We support these findings by exact diagonalizations extrapolated to large N.

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Wenli Zhao

Papers

Symmetry Breaking in Coupled SYK or Tensor Models

Toward microstate counting beyond large N in localization and the dual one-loop quantum supergravity

One-Loop Test of Quantum Black Holes in anti-de Sitter Space.

The statistical mechanics of near-BPS black holes

Spontaneous breaking of U(1) symmetry in coupled complex SYK models