Charged eigenstate thermalization, Euclidean wormholes and global symmetries in quantum gravity

Abstract

We generalize the eigenstate thermalization hypothesis to systems with global symmetries. We present two versions, one with microscopic charge conservation and one with exponentially suppressed violations. They agree for correlation functions of simple operators, but differ in the variance of charged one-point functions at finite temperature. We then apply these ideas to holography and to gravitational low-energy effective theories with a global symmetry. We show that Euclidean wormholes predict a non-zero variance for charged one-point functions, which is incompatible with microscopic charge conservation. This implies that global symmetries in quantum gravity must either be gauged or explicitly broken by non-perturbative effects.

Interacting systems and wormholes

“…This can be shown to be possible in some simple two-dimensional models such as JT gravity or minimal models (and some rather exotic examples in 3d, see [21] and refs.). It has a merit in that it also conforms with expectations based on the eigenstate thermalization hypothesis (ETH), quantum ergodicity and the complicated/chaotic nature of gravity [22,23,24,25]. It brings however a certain tension with the extremely well-studied paradigms of AdS/CFT, involving single partner duals (such as N = 4 SYM), as well as with the principle of unitarity, and therefore seems to make sense in an approximate statistical sense (it is perhaps better to call such wormholes as statistical wormholes).…”

“…The next thing to notice is a consistency condition k 1 N 1 = k 2 N 2 for the C.S. levels and the gauge groups, due to the properties of the Kostka polynomials that are non-zero only for 24 . The model of common YM and MQM groups of section 2.1.3 is simply obtained by specialising N 1 = N 2 = n (and m 1,2 = 0) that leads to k 1 = k 2 = k. We also observe that had we picked k 1 = k 2 = 0, the only states that would contribute would be zero-weight states (inside a highest weight module) as was first noticed in [59,60,61] and described in section 2.1.3 and appendix H.…”

“…The function p (3) (n) is defined via χ Û (1) r = φ −r (q) = n∈Z p (r) (n)q n , |q| < 1 . (H. 24) with φ(q) the Euler function. This character is also the character of an r-fold module of a Heisenberg algebra.…”

Wormholes & Holography: An Introduction

“…In particular, [142] proposes a statistical hypothesis on the OPE coefficients of a dual two-dimensional CFT and relates this proposal to the Wormhole contribution in Euclidean AdS 3 . Furthermore, [143] combines a generalized notion of this OPE-statistics in the presence of a global symmetry with Wormholes to arrive at the desired conclusion that global symmetries cannot exist in quantum gravity. At the very least, it appears plausible that even though Wormholes may not be the complete story, they certainly encode some fine-grained quantum information of gravity.…”

“…. represent other components that need to be turned on to satisfy the equations of motion resulting from (143). The supergravity background here is the vacuum solution (AdS), which is dual to the ground state of the SU(N ) adjoint gauge theory.…”