Factorization and complex couplings in SYK and in Matrix Models

TL;DR

This paper investigates the factorization problem in holographic toy models like SYK and matrix models, introducing a fictitious ensemble averaging to resolve factorization issues and analyzing the contributions of wormholes and half-wormholes.

Contribution

It proposes a new perspective on factorization by decomposing the squared partition function into wormhole and half-wormhole terms, connecting to ensemble averaging over couplings.

Findings

The half-wormhole term corresponds to averaging over the imaginary part of couplings.

In SYK, the approach reproduces known results from a novel perspective.

In matrix models, the form of the linked half-wormholes is proposed and validated for GUE.

Abstract

We consider the factorization problem in toy models of holography, in SYK and in Matrix Models. In a theory with fixed couplings, we introduce a fictitious ensemble averaging by inserting a projector onto fixed couplings. We compute the squared partition function and find that at large $N$ for a typical choice of the fixed couplings it can be approximated by two terms: a "wormhole" plus a "pair of linked half-wormholes". This resolves the factorization problem. We find that the second, half-wormhole, term can be thought of as averaging over the imaginary part of the couplings. In SYK, this reproduces known results from a different perspective. In a matrix model with an arbitrary potential, we propose the form of the "pair of linked half-wormholes" contribution. In GUE, we check that errors are indeed small for a typical choice of the hamiltonian. Our computation relies on a result by Brezin and Zee for a correlator of resolvents in a "deterministic plus random" ensemble of matrices.