The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Superstring theoryをめぐって(放談室)

We present a uniform treatment of rigid supersymmetric field theories in a curved spacetime $ \mathcal{M} $, focusing on four-dimensional theories with four supercharges. Our discussion is significantly simpler than earlier treatments, because we use classical background values of the auxiliary fields in the supergravity multiplet. We demonstrate our procedure using several examples. For $ \mathcal{M} = Ad{S_4} $ we reproduce the known results in the literature. A supersymmetric Lagrangian for $ \mathcal{M} = {\mathbb{S}^4} $ exists, but unless the field theory is conformal, it is not reflection positive. We derive the Lagrangian for $ \mathcal{M} = {\mathbb{S}^3} \times \mathbb{R} $ and note that the time direction $ \mathbb{R} $ can be rotated to Euclidean signature and be compactified to $ {\mathbb{S}^1} $ only when the theory has a continuous R-symmetry. The partition function on $ \mathcal{M} = {\mathbb{S}^3} \times {\mathbb{S}^1} $ is independent of the parameters of the flat space theory and depends holomorphically on some complex background gauge fields. We also consider R-invariant $ \mathcal{N} = 2 $ theories on $ {\mathbb{S}^3} $ and clarify a few points about them.

Rigid Supersymmetric Theories in Curved Superspace

We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function |Z(β + it)|2 as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.

https://link.springer.com/content/pdf/10.1007%2FJHEP05%282017%29118.pdf

Black Holes and Random Matrices

General scaling arguments, and the behavior of the thermal entropy density, are shown to lead to an infrared metric holographically representing a compressible state with hidden Fermi surfaces. This metric is characterized by a general dynamic critical exponent, z, and a specic hyperscaling violation exponent, . The same metric exhibits a logarithmic violation of the area law of entanglement entropy, as shown recently by Ogawa et al. (arXiv:1111.1023). We study the dependence of the entanglement entropy on the shape of the entangling region(s), on the total charge density, on temperature, and on the presence of additional visible Fermi surfaces of gauge-neutral fermions; for the latter computations, we realize the needed metric in an Einstein-Maxwell-dilaton theory. All our results support the proposal that the holographic theory describes a metallic state with hidden Fermi surfaces of fermions carrying gauge charges of deconned gauge elds.

/pdf/hidden-fermi-surfaces-in-compressible-states-of-gauge-2qidpgbbai.pdf

Hidden Fermi surfaces in compressible states of gauge-gravity duality

Einstein-Maxwell theory coupled to a dilaton is known to give rise to extremal solutions with hyperscaling violation. We study the behaviour of these solutions in the presence of a small magnetic field. We find that in a region of parameter space the magnetic field is relevant in the infra-red and completely changes the behaviour of the solution which now flows to an AdS
 2 × R
 2 attractor. As a result there is an extensive ground state entropy and the entanglement entropy of a sufficiently big region on the boundary grows like the volume. In particular, this happens for values of parameters at which the purely electric theory has an entanglement entropy growing with the area, A, like A log(A) which is believed to be a characteristic feature of a Fermi surface. Some other thermodynamic properties are also analysed and a more detailed characterisation of the entanglement entropy is also carried out in the presence of a magnetic field. Other regions of parameter space not described by the AdS
 2 × R
 2 end point are also discussed.

/pdf/entangled-dilaton-dyons-3gt3cmx40t.pdf

Entangled dilaton dyons

We discuss monopole operators in $U(N_c)$ Chern-Simons-matter theories in three space-time dimensions. We mention an apparent problem in the matching of such operators in dualities between non-supersymmetric theories, and suggest a possible resolution. A similar apparent problem exists in the mapping of chiral monopole operators in theories with ${\cal N}=2$ supersymmetry. We show that in many theories the lowest naive chiral monopole operator is actually not chiral, and we find the lowest monopole operator that is actually chiral in these theories. It turns out that there are several different forms of this operator, depending on the number of colors, the number of flavours, and the Chern-Simons level. Since we use the supersymmetric index to find the lowest chiral monopoles, our results for these monopoles are guaranteed to be invariant under the dualities in supersymmetric theories. The theories we discuss are believed to be dual in the 't~Hooft large $N_c$ limit to classical high-spin gravity theories. We argue that these theories (supersymmetric or not) should not have classical solutions charged under the $U(1)$ gauge field in the high-spin multiplet.

On monopole operators in supersymmetric Chern-Simons-matter theories

The exact 2-point function of certain physically motivated operators in SYK-like spin glass models is computed, bypassing the Schwinger-Dyson equations. The models possess an IR low energy conformal window, but our results are exact at all time scales. The main tool developed is a new approach to the combinatorics of chord diagrams, allowing to rewrite the spin glass system using an auxiliary Hilbert space, and Hamiltonian, built on the space of open chord diagrams. We argue the latter can be interpreted as the bulk description and that it reduces to the Schwarzian action in the low energy limit.

/pdf/chord-diagrams-exact-correlators-in-spin-glasses-and-black-51y9pbj2hu.pdf

Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction

We describe general features of thermal correlation functions in quantum systems, with specific focus on the fluctuation-dissipation type relations implied by the KMS condition. These end up relating correlation functions with different time ordering and thus should naturally be viewed in the larger context of out-of-time-ordered (OTO) observables. In particular, eschewing the standard formulation of KMS relations where thermal periodicity is combined with time-reversal to stay within the purview of Schwinger-Keldysh functional integrals, we show that there is a natural way to phrase them directly in terms of OTO correlators. We use these observations to construct a natural causal basis for thermal n-point functions in terms of fully nested commutators. We provide several general results which can be inferred from cyclic orbits of permutations, and exemplify the abstract results using a quantum oscillator as an explicit example.

/pdf/thermal-out-of-time-order-correlators-kms-relations-and-43rwlf8bf9.pdf

Thermal out-of-time-order correlators, KMS relations, and spectral functions

In this paper, we introduce an $\mathcal{N}=1$ supersymmetric SYK model with $SO(q)$ global symmetry. We study the large $N$ expansion of the bi-local collective action of our model. At strong coupling limit, this model exhibits a super-reparametrization symmetry, and the $SO(q)$ global symmetry is enhanced to a $\widehat{SO}(q)$ local symmetry. The corresponding symmetry algebra is the semi-direct product of the super-Virasoro and the super-Kac-Moody algebras. These emergent symmetries are spontaneously and explicitly broken, which leads to a low energy effective action: super-Schwarzian action plus an action of a super-particle on the $SO(q)$ group manifold. We analyze the zero mode contributions to the chaotic behavior of four point functions in various $SO(q)$ channels. In singlet channel, we show that the out-of-time-ordered correlators related to bosonic bi-locals exhibit the saturation of the chaos bound as in the non-SUSY SYK model. On the other hand, we find that the ones with fermionic bi-locals in the singlet channel have ${\pi\over\beta}$ Lyapunov exponent. In the anti-symmetric channel, we demonstrate that the out-of-time-ordered correlator related to a $SO(q)$ generator grows linearly in time. We also compute the non-zero mode contributions which give consistent corrections to the leading Lyapunov exponents from the zero modes.

/pdf/supersymmetric-syk-model-with-global-symmetry-1i3xdszmut.pdf

Prithvi Narayan

Papers

Entangled dilaton dyons

On monopole operators in supersymmetric Chern-Simons-matter theories

Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction

Thermal out-of-time-order correlators, KMS relations, and spectral functions

Supersymmetric SYK Model with Global Symmetry