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Generalizing previous results for$$ \mathcal{N} $$$N$= 0 and$$ \mathcal{N} $$$N$= 1, we analyze$$ \mathcal{N} $$$N$= 2 JT supergravity on asymptotically AdS₂spaces with arbitrary topology and show that this theory of gravity is dual, in a holographic sense, to a certain random matrix ensemble in which supermultiplets of differentR-charge are statistically independent and each is described by its own$$ \mathcal{N} $$$N$= 2 random matrix ensemble. We also analyze the case with a time-reversal symmetry, either commuting or anticommuting with theR-charge. In order to compare supergravity to random matrix theory, we develop an$$ \mathcal{N} $$$N$= 2 analog of the recursion relations for Weil-Petersson volumes originally discovered by Mirzakhani in the bosonic case.

 

We construct a Type II_∞von Neumann algebra that describes the largeNphysics of single-trace operators in AdS/CFT in the microcanonical ensemble, where there is no need to include perturbative 1/Ncorrections. Using only the extrapolate dictionary, we show that the entropy of semiclassical states on this algebra is holographically dual to the generalized entropy of the black hole bifurcation surface. From a boundary perspective, this constitutes a derivation of a special case of the QES prescription without any use of Euclidean gravity or replicas; from a purely bulk perspective, it is a derivation of the quantum-corrected Bekenstein-Hawking formula as the entropy of an explicit algebra in theG →0 limit of Lorentzian effective field theory quantum gravity. In a limit where a black hole is first allowed to equilibrate and then is later potentially re-excited, we show that the generalized second law is a direct consequence of the monotonicity of the entropy of algebras under trace-preserving inclusions. Finally, by considering excitations that are separated by more than a scrambling time we construct a “free product” von Neumann algebra that describes the semiclassical physics of long wormholes supported by shocks. We compute Rényi entropies for this algebra and show that they are equal to a sum over saddles associated to quantum extremal surfaces in the wormhole. Surprisingly, however, the saddles associated to “bulge” quantum extremal surfaces contribute with a negative sign.

 

A bstract We describe an algebra of observables for a static patch in de Sitter space, with operators gravitationally dressed to the worldline of an observer. The algebra is a von Neumann algebra of Type II 1 . There is a natural notion of entropy for a state of such an algebra. There is a maximum entropy state, which corresponds to empty de Sitter space, and the entropy of any semiclassical state of the Type II 1 algebras agrees, up to an additive constant independent of the state, with the expected generalized entropy S gen = ( A/ 4 G N ) + S out . An arbitrary additive constant is present because of the renormalization that is involved in defining entropy for a Type II 1 algebra. 

A bstract We discuss aspects of the possible transition between small black holes and highly excited fundamental strings. We focus on the connection between black holes and the self gravitating string solution of Horowitz and Polchinski. This solution is interesting because it has non-zero entropy at the classical level and it is natural to suspect that it might be continuously connected to the black hole. Surprisingly, we find a different behavior for heterotic and type II cases. For the type II case we find an obstruction to the idea that the two are connected as classical solutions of string theory, while no such obstruction exists for the heterotic case. We further provide a linear sigma model analysis that suggests a continuous connection for the heterotic case. We also describe a solution generating transformation that produces a charged version of the self gravitating string. This provides a fuzzball-like construction of near extremal configurations carrying fundamental string momentum and winding charges. We provide formulas which are exact in α ′ relating the thermodynamic properties of the charged and the uncharged solutions. 

A bstract Recently Leutheusser and Liu [1, 2] identified an emergent algebra of Type III 1 in the operator algebra of $$ \mathcal{N} $$ N = 4 super Yang-Mills theory for large N . Here we describe some 1/ N corrections to this picture and show that the emergent Type III 1 algebra becomes an algebra of Type II ∞ . The Type II ∞ algebra is the crossed product of the Type III 1 algebra by its modular automorphism group. In the context of the emergent Type II ∞ algebra, the entropy of a black hole state is well-defined up to an additive constant, independent of the state. This is somewhat analogous to entropy in classical physics. 

A bstract In the AdS/CFT correspondence, amplitudes associated to connected bulk manifolds with disconnected boundaries have presented a longstanding mystery. A possible interpretation is that they reflect the effects of averaging over an ensemble of boundary theories. But in examples in dimension D ≥ 3, an appropriate ensemble of boundary theories does not exist. Here we sharpen the puzzle by identifying a class of “fixed energy” or “sub-threshold” observables that we claim do not show effects of ensemble averaging. These are amplitudes that involve states that are above the ground state by only a fixed amount in the large N limit, and in particular are far from being black hole states. To support our claim, we explore the example of D = 3, and show that connected solutions of Einstein’s equations with disconnected boundary never contribute to these observables. To demonstrate this requires some novel results about the renormalized volume of a hyperbolic three-manifold, which we prove using modern methods in hyperbolic geometry. Why then do any observables show apparent ensemble averaging? We propose that this reflects the chaotic nature of black hole physics and the fact that the Hilbert space describing a black hole does not have a large N limit. 

Abstract This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020) 

Recently, it has been found that Jackiw-Teitelboim (JT) gravity, which is a two-dimensional theory with bulk action − 1 / 2 ∫ d 2 x g ϕ ( R + 2 ) , is dual to a matrix model, that is, a random ensemble of quantum systems rather than a specific quantum mechanical system. In this article, we argue that a deformation of JT gravity with bulk action − 1 / 2 ∫ d 2 x g ( ϕ R + W ( ϕ ) ) is likewise dual to a matrix model. With a specific procedure for defining the path integral of the theory, we determine the density of eigenvalues of the dual matrix model. There is a simple answer if W (0) = 0, and otherwise a rather complicated answer. 

A bstract Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1) 2 D Chern-Simons theory than like Einstein gravity. 

We derive a holomorphic anomaly equation for the Vafa-Wittenpartition function for twisted four-dimensional \mathcal{N} =4 𝒩 = 4 super Yang-Mills theory on \mathbb{CP}^{2} ℂ ℙ 2 for the gauge group SO(3) S O ( 3 ) from the path integral of the effective theory on the Coulomb branch.The holomorphic kernel of this equation, which receives contributionsonly from the instantons, is not modular but ‘mock modular’. Thepartition function has correct modular properties expected from S S -dualityonly after including the anomalous nonholomorphic boundary contributionsfrom anti-instantons. Using M-theory duality, we relate this phenomenonto the holomorphic anomaly of the elliptic genus of a two-dimensionalnoncompact sigma model and compute it independently in two dimensions.The anomaly both in four and in two dimensions can be traced to atopological term in the effective action of six-dimensional (2,0) ( 2 , 0 ) theory on the tensor branch. We consider generalizations to othermanifolds and other gauge groups to show that mock modularity is genericand essential for exhibiting duality when the relevant field space isnoncompact.