Half-wormholes in SYK with one time point

TL;DR

This work analyzes a finite-dimensional SYK variant with a single time point to probe factorization and wormhole physics. By formulating $z^2$ in terms of a hyperpfaffian and a collective-field integral, the authors identify a non-self-averaging linked half-wormhole contribution $\Phi(0)$ that, together with the wormhole, reproduces the exact $z^2$ for fixed couplings. They show a finite-order perturbative expansion around the linked half-wormholes, whose last term matches the wormhole, implying the wormhole saddle is a large fluctuation around half-wormholes rather than an independent saddle. Extensions to two replicas with coupling $\mu$ reveal regimes where the wormhole dominates and half-wormholes are suppressed, connecting the factorization mechanism to coupling-induced enhancement. Overall, the paper provides explicit hyperpfaffian formulas, a controlled error analysis, and a clear semiclassical expansion framework for understanding factorization in a toy SYK model with fixed couplings.

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.