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

We study the two-point function for fermionic operators in a class of strongly coupled systems using the gauge-gravity correspondence. The gravity description includes a gauge field and a dilaton which determines the gauge coupling and the potential energy. Extremal black brane solutions in this system typically have vanishing entropy. By analyzing a charged fermion in these extremal black brane backgrounds we calculate the two-point function of the corresponding boundary fermionic operator. We find that in some region of parameter space it is of Fermi liquid type. Outside this region no well-defined quasi-particles exist, with the excitations acquiring a non-vanishing width at zero frequency. At the transition, the two-point function can exhibit non-Fermi liquid behaviour.

Holographic Fermi and Non-Fermi Liquids with Transitions in Dilaton Gravity

/pdf/holographic-fermi-and-non-fermi-liquids-with-transitions-in-10tvvduax6.pdf

Holographic Fermi and non-Fermi liquids with transitions in dilaton gravity

We discuss a 1+1 dimensional generalization of the Sachdev-Ye-Kitaev model. The model contains N Majorana fermions at each lattice site with a nearest-neighbour hopping term. The SYK random interaction is restricted to low momentum fermions of definite chirality within each lattice site. This gives rise to an ordinary 1+1 field theory above some energy scale and a low energy SYK-like behavior. We exhibit a class of low-pass filters which give rise to a rich variety of hyperscaling behaviour in the IR. We also discuss another set of generalizations which describes probing an SYK system with an external fermion, together with the new scaling behavior they exhibit in the IR.

/pdf/higher-dimensional-generalizations-of-the-syk-model-2pamfaaecp.pdf

Higher Dimensional Generalizations of the SYK Model

Extremal black branes are of interest because they correspond to the ground states of field theories at finite charge density in gauge/gravity duality. The geometry of such a brane need not be translationally invariant in the spatial direc- tions along which it extends. A less restrictive requirement is that of homogeneity, which still allows points along the spatial directions to be related to each other by symmetries. In this paper, we find large new classes of homogeneous but anisotropic extremal black brane horizons, which could naturally arise in gauge/gravity dual pairs. In 4 + 1 dimensional spacetime, we show that such homogeneous black brane solutions are classified by the Bianchi classification, which is well known in the study of cosmology, and fall into nine classes. In a system of Einstein gravity with nega- tive cosmological term coupled to one or two massive Abelian gauge fields, we find solutions with an additional scaling symmetry, which could correspond to the near- horizon geometries of such extremal black branes. These solutions realize many of the Bianchi classes. In one case, we construct the complete extremal solution which asymptotes to AdS space.

/pdf/bianchi-attractors-a-classification-of-extremal-black-brane-2u161gyv3u.pdf

Bianchi attractors: a classification of extremal black brane geometries

We discuss a 1+1 dimensional generalization of the Sachdev-Ye-Kitaev model. The model contains $N$ Majorana fermions at each lattice site with a nearest-neighbour hopping term. The SYK random interaction is restricted to low momentum fermions of definite chirality within each lattice site. This gives rise to an ordinary 1+1 field theory above some energy scale and a low energy SYK-like behavior. We exhibit a class of low-pass filters which give rise to a rich variety of hyperscaling behaviour in the IR. We also discuss another set of generalizations which describes probing an SYK system with an external fermion, together with the new scaling behavior they exhibit in the IR.

Prithvi Narayan

Papers

Holographic Fermi and Non-Fermi Liquids with Transitions in Dilaton Gravity

Holographic Fermi and non-Fermi liquids with transitions in dilaton gravity

Higher Dimensional Generalizations of the SYK Model

Bianchi attractors: a classification of extremal black brane geometries

Higher Dimensional Generalizations of the SYK Model