Half-wormholes in SYK with one time point

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 remarkable identity: square of partition function fixed couplings is well approximated "wormhole" saddle plus "pair linked half-wormholes" saddle. It explains factorization decoupled systems. Here, derive an explicit formula for half-wormhole contribution. expressed through hyperpfaffian tensor couplings. We then develop perturbative expansion around This truncates at finite order gives exact answer. The last term turns out to coincide wormhole sense does not need be added separately, but instead can viewed as large fluctuation half-wormholes.