Note on global symmetry and SYK model
Junyu Liu, Yehao Zhou
TL;DR
The paper develops a framework for SYK models with continuous global symmetries, showing that the low-energy effective action factors into a Schwarzian sector and a free sigma-model on the symmetry group. A generalized Peter–Weyl theorem for twisted line bundles is proven, enabling a precise decomposition of the partition function across charge sectors and spin structures. By working out explicit results for U(1), SU, and SO groups, the authors reveal how symmetry restructures thermodynamics and spectral statistics, with single-charge sectors displaying group-dependent power-law decays in form factors and full sectors exhibiting additional exponential suppressions. Despite these sectoral modifications, the Schwarzian-driven maximal chaos persists, suggesting robust chaotic behavior in SYK-like models with global symmetries. The findings point to concrete predictions for scrambling and thermodynamics in symmetry-augmented SYK models and offer a path toward holographic interpretations of such systems.
Abstract
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.