Supersymmetric SYK Model with Global Symmetry

TL;DR

The paper constructs and analyzes an N=1 supersymmetric SYK model with SO(q) flavor symmetry, uncovering an emergent SDiff super-reparametrization and a local \\widehat{SO}(q) symmetry in the strong-coupling, large-N limit. The authors derive a bi-local supermatrix collective action and a low-energy effective action comprising a super-Schwarzian term and a super-particle action on the SO(q) manifold, then study four-point functions via conformal eigenfunctions to obtain the spectrum and OPE data. They analyze chaos through out-of-time-ordered correlators, finding maximal Lyapunov growth λ_L = 2π/β in bosonic singlet channels, a π/β growth for certain fermionic nontopological modes, and linear growth in the antisymmetric channel from a bosonic SO(q) zero mode, with 1/βJ corrections from non-zero modes. The results illuminate how SUSY and non-Abelian flavor symmetries shape chaotic behavior and suggest holographic interpretations involving boundary gravitons and gauginos beyond the standard Schwarzian framework.

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 ${π\overβ}$ 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.

Figures (1)

• Figure 1: Schematic diagrams for the contributions of the zero modes. The single solid line denotes the fermion $\chi$ while the single dashed line represents the auxiliary boson $b$. In addition, the double wavy line represents the $\eta$ zero mode for the singlet channel or the $k$ zero mode for the anti-symmetric channel. The double dashed line denotes the $\epsilon$ zero mode for the singlet channel or the $\rho$ zero mode for the anti-symmetric channel. However, the bi-locals $b^ib^i$ is not coupled to the zero mode $\rho$.