Abstract
Haar integrals over the unitary group contain subleading terms that are needed for unitarity. We study analogous effects in the time evolution operators of JT gravity and Brownian SYK. In JT gravity with bulk matter we find an explanation for the first sub-leading terms, and in Brownian SYK we find configurations that can explain the full series. An important role is played by slightly off-shell modes that are exponentially amplified by chaos.
Highlights
The coefficients here are known as “Weingarten functions” [7, 8], depending on L, the dimension of the Hilbert space
We studied how the time evolution U (t) in two quantum chaotic systems (JT gravity with matter and Brownian SYK) can reproduce small but important effects present in the average over unitary matrices
In JT gravity, we matched the first subleading terms, and the most interesting effect came from the topology of a disk with a handle inserted
Summary
The coefficients here are known as “Weingarten functions” [7, 8], depending on L, the dimension of the Hilbert space. The first, second and fourth terms of (1.4) all contribute at order 1/L2, and the net effect is that the handle-disk gives a positive contribution in JT gravity, obscuring the origin of the interesting minus sign These different contributions can be distinguished in the bulk calculation as arising from different regions of the moduli space integral. They can be numerically separated from each other by including perturbative effects of bulk matter loops, which change V V relative to V V We conclude from this that at least some of the nontrivial unitarity-preserving subleading effects from the random unitary model are present in JT gravity with matter fields. We give further details on aspects of these two models
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