Half-wormholes in SYK with one time point

TL;DR

This paper analyzes the SYK model with a single time point, deriving explicit formulas for half-wormhole contributions, and shows how wormholes emerge as fluctuations around these half-wormholes, clarifying factorization.

Contribution

It provides an explicit hyperpfaffian formula for half-wormholes and demonstrates how wormholes arise from fluctuations around half-wormholes in the SYK model.

Findings

Explicit hyperpfaffian formula for half-wormholes

Finite-order perturbative expansion matches exact results

Wormholes viewed as large fluctuations around half-wormholes

Abstract

In this note we study the SYK model with one time point, recently considered by Saad, Shenker, Stanford, and Yao. Working in a collective field description, they derived a remarkable identity: the square of the partition function with fixed couplings is well approximated by a "wormhole" saddle plus a "pair of linked half-wormholes" saddle. It explains factorization of decoupled systems. Here, we derive an explicit formula for the half-wormhole contribution. It is expressed through a hyperpfaffian of the tensor of SYK couplings. We then develop a perturbative expansion around the half-wormhole saddle. This expansion truncates at a finite order and gives the exact answer. The last term in the perturbative expansion turns out to coincide with the wormhole contribution. In this sense the wormhole saddle in this model does not need to be added separately, but instead can be viewed as a large fluctuation around the linked half-wormholes.