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.
Highlights
It is becoming increasingly clear that wormholes play an important role in the physics of quantum black holes
They explain the long-time behavior of the spectral form factor [1], [2], correlation functions [3] and the entropy of Hawking radiation [4], [5]; can be traversable [6], [7], [8] and sometimes even by humans [9]
In some cases, there are wormhole solutions that dominate over the disconnected solutions and have no known instabilities, but do not have a known embedding in string theory
Summary
It is becoming increasingly clear that wormholes play an important role in the physics of quantum black holes. In some cases, there are wormhole solutions that dominate over the disconnected solutions and have no known instabilities, but do not have a known embedding in string theory (see [11] for a nice overview of these results and references therein) In the latter case, it is not clear whether we should expect to have two decoupled boundary systems with a factorized partition function. They considered a finite dimensional Grassman integral that can be thought of as an SYK model where the time direction is reduced to one point They computed a two-boundary observable zLzR and showed that even with fixed couplings one can introduce collective field variables GLR, ΣLR representing correlations between the two systems L and R. The wormhole saddle in this model need not be added separately, but can instead be described as a large fluctuation around linked half-wormholes
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