On the Replica Problem in Supersymmetric SYK Models

TL;DR

This paper explores the replica problem in supersymmetric SYK models, developing new methods to analyze their phase structure, symmetries, and holographic duals, with implications for understanding supersymmetric wormholes.

Contribution

It introduces a superconformal symmetry-based approach and an ordered super-Schwarzian action to study replica structures and holography in supersymmetric SYK models.

Findings

Solvable conditions for n-replica non-supersymmetric SYK models.

Development of a multi-ordered trick for supersymmetric replica analysis.

Numerical results on modular thermodynamics and holographic constraints.

Abstract

We investigate the replica problem for Sachdev-Ye-Kitaev (SYK) models. First, we consider $n-$replicas of the non-supersymmetric SYK model, finding that this $n$-replica model is solvable only under specific conditions. We then introduce the $\mathcal{N}=1$ supersymmetry and utilize the superconformal symmetry to develop a ``multi-ordered trick" that covers the replica structure. By incorporating ordered off-diagonal couplings, we study the resulting thermal phase structure under higher-order interactions. The Lorentzian time dynamics is analyzed, and we plot the time evolution of the effective action. Furthermore, we investigate emergent superconformal symmetry in the low-energy limit of the replicated theory. In the superconformal limit, we propose an ordered super-Schwarzian action and derive reparameterization relations for the ordered coordinates. Corresponding constraints are derived for holographic matching to $\mathcal{N}=1$ super-Jackiw-Teitelboim(SJT) gravity. We numerically calculate the modular thermodynamics (modular entropy, the $n$-dependent relative entropy, and the entanglement capacity) using fully diagonal methods. Our result provide a framework for studying $n$-replica wormholes with supersymmetric SYK model.