Factorization and complex couplings in SYK and in Matrix Models

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.

Automated summary

We study the factorization problem in holographic toy models, SYK and Matrix Models. In theories with fixed couplings, we introduce a fictitious ensemble averaging using a projector onto fixed couplings. By computing the squared partition function, we find that for a typical choice of fixed couplings, it can be approximated by a wormhole term plus a pair of linked half-wormholes. This resolves the factorization problem. We also propose the form of the pair of linked half-wormholes contribution in a matrix model with an arbitrary potential.