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

The microscopic formula for the degeneracy of BMPV black hole microstates contains a series of exponentially suppressed corrections to the leading Bekenstein Hawking expression. We identify saddle points of the quantum entropy function for the BMPV black hole which are natural counterparts to these corrections and discuss the matching of leading and next-to-leading terms from the microscopic and macroscopic sides in a limit where the black hole charges are large.

/pdf/on-exponentially-suppressed-corrections-to-bmpv-black-hole-xz2p7q15nm.pdf

Prithvi Narayan

Papers

On exponentially suppressed corrections to BMPV black hole entropy