Supersymmetric SYK Model with Global Symmetry

TL;DR

This paper introduces a supersymmetric SYK model with global symmetry, explores its emergent symmetries, and analyzes chaotic behavior, revealing saturation of chaos bounds and unique Lyapunov exponents.

Contribution

It presents the first supersymmetric SYK model with $SO(q)$ symmetry, studies its large $N$ expansion, and derives the associated super-Schwarzian effective action.

Findings

Emergent super-reparametrization and local $SO(q)$ symmetries.

Saturation of chaos bound in bosonic correlators.

Distinct Lyapunov exponents for fermionic correlators.

Abstract

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.