# Half-wormholes in SYK with one time point

### Submission summary

 Authors (as Contributors): Baurzhan Mukhametzhanov
Submission information
Date accepted: 2021-11-11
Date submitted: 2021-10-18 17:08
Submitted by: Mukhametzhanov, Baurzhan
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
Approach: Theoretical

### 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.

Published as SciPost Phys. 12, 029 (2022)

### List of changes

1) Change in the title from "Half-wormhole in ..." to "Half-wormholes in ...".
2) "Half-wormhole" is changed to "linked half-wormholes" in several places.
3) Definition of $sgn(A)$ added after (9)
4) Paragraphs at the beginning of section 4.2 and at the end of page 9 are added to clarify the meaning of the large $N$ expansion for typical couplings.

### Submission & Refereeing History

Resubmission 2105.08207v4 on 18 October 2021
Submission 2105.08207v3 on 21 July 2021

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### Report

The author has addressed my questions, and therefore I recommend publication.

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• originality: -
• clarity: -
• formatting: -
• grammar: -