# Note on global symmetry and SYK model

## TL;DR

This paper investigates how continuous global symmetries influence the effective action, thermodynamics, and spectral properties of the SYK model, revealing the role of symmetry in quantum chaos and sector decomposition.

## Contribution

It introduces a modified Peter-Weyl theorem for the SYK model with symmetries and analyzes the impact of symmetry on the model's effective action and chaotic behavior.

## Key findings

- Effective action decomposes into Schwarzian and group manifold parts.
- Modified Peter-Weyl theorem applicable to generic spin structures.
- Symmetry affects thermodynamics and spectral form factors in the SYK model.

## 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.