On thermalization in the SYK and supersymmetric SYK models

The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well as its $\mathcal{N}=1$ supersymmetric extension and find evidence that these models satisfy ETH. We discuss the implications of ETH for a gravitational dual and the quantum information-theoretic properties of SYK it suggests.Comment: Published versio

Note on global symmetry and SYK model

The goal of this note is to explore the behavior of effective action in the SYK model with general continuous global symmetries. A global symmetry will decompose the whole Hamiltonian of a many-body system to several single charge sectors. For the SYK model, the effective action near the saddle point is given as the free product of the Schwarzian action part and the free action of the group element moving in the group manifold. With a detailed analysis in the free sigma model, we prove a modified version of Peter-Weyl theorem that works for generic spin structure. As a conclusion, we could make a comparison between the thermodynamics and the spectral form factors between the whole theory and the single charge sector, to make predictions on the SYK model and see how symmetry affects the chaotic behavior in certain timescales.Comment: 44 page

Note on the Green's function formalism and topological invariants

It has been discovered previously that the topological order parameter could be identified from the topological data of the Green's function, namely the (generalized) TKNN invariant in general dimensions, for both non-interacting and interacting systems. In this note, we show that this phenomenon has a clear geometric derivation. This proposal could be regarded as an alternative proof for the identification of the corresponding topological invariant and the topological order parameter

No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit

We introduce high-energy limits which allow us to derive recursion relations fixing the various couplings of Lagrangians of two-dimensional relativistic quantum field theories with no tree-level particle production in a very straightforward way. The sine-Gordon model, the Bullough-Dodd theory, Toda theories of various kinds and the U(N) non-linear sigma model can all be rediscovered in this way. The results here were the outcome of our explorations at the 2017 Perimeter Institute Winter School.Comment: 20 page

Twisted Holography: The Examples of 4d and 5d Chern-Simons Theories

Twisted holography is a duality between a twisted supergravity, and a twisted supersymmetric gauge theory living on the D-branes in the supergravity. The main objectives of this duality is the comparison between the algebra of observables in the bulk twisted supergravity and the algebra of observables in the boundary twisted supersymmetric gauge theory. In this thesis, two example of the twisted holography duality are explored. The bulk theory for the first example is the 4d topological-holomorphic Chern-Simons theory, which is expected to be dual to 2d BF theory with line defects. The algebra of observables in the 2d BF theory is computed by two methods: perturbation theory (Feynman diagrams), and phase space quantization. By holography duality this algebra is expected to be isomorphic to the algebra of bulk-boundary scattering process, and the latter is computed in this thesis using perturbative method. The bulk theory for the second example is the 5d topological-holomorphic Chern-Simons theory, which is expected to be dual to the large-N limit of a family of 1d quantum mechanics built from the ADHM quivers. The generators and relations of the large-N limit algebra of observables in the 1d quantum mechanics are studied from algebraic point view. By holography duality, this algebra is expected to be the algebra of observables on the universal line defect coupled to the 5d Chern-Simons theory, and some nontrivial relations of the latter algebra are computed in this thesis using perturbative method. The surface defects and various fusion process between line and surface defects are also explored

Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical R-Matrices for Superspin Chains from The Bethe/Gauge Correspondence

We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d $\mathcal N=2$ quiver gauge theories. The Bethe/Gauge Correspondence relates such a gauge theory to an anisotropic/elliptic superspin chain, and the stable envelopes compute the R-matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using the corresponding 3d $\mathcal N=2$ SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on $I \times \mathbb E$ for an interval $I$ and an elliptic curve $\mathbb E$ compute the stable envelopes, and in turn the geometric R-matrix, of the $\mathfrak{sl}(1|1)$ spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the R-matrix. The reduction to 2d gives the K-theoretic stable envelopes and the trigonometric R-matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational R-matrix. The latter recovers the results of Rim\'anyi and Rozansky.Comment: 89 Pages + Appendices + References = 125 Page

Supersymmetric SYK model and random matrix theory

In this paper, we investigate the effect of supersymmetry on the symmetry classification of random matrix theory ensembles. We mainly consider the random matrix behaviors in the N = 1 supersymmetric generalization of Sachdev-Ye-Kitaev (SYK) model, a toy model for two-dimensional quantum black hole with supersymmetric constraint. Some analytical arguments and numerical results are given to show that the statistics of the supersymmetric SYK model could be interpreted as random matrix theory ensembles, with a different eight-fold classification from the original SYK model and some new features. The time-dependent evolution of the spectral form factor is also investigated, where predictions from random matrix theory are governing the late time behavior of the chaotic hamiltonian with supersymmetry