Wormhole Renormalization: The gravitational path integral, holography, and a gauge group for topology change

TL;DR

This paper explores how wormholes in gravitational path integrals disrupt holographic factorization and proposes a renormalization-inspired method involving counter-wormholes and a gauge group to restore consistency.

Contribution

It introduces a gravitational analog of renormalization techniques, including a Hopf algebra framework and a gauge group, to systematically address wormhole contributions in holographic quantum gravity.

Findings

Counter-wormholes parametrize stringy effects restoring factorization.

A Hopf algebra structure underpins the symmetry organization of topology changes.

A gauge group acts to relate different topological contributions in the path integral.

Abstract

We study the Factorization Paradox from the bottom up by adapting methods from perturbative renormalization. Just as quantum field theories are plagued with loop divergences that need to be cancelled systematically by introducing counterterms, gravitational path integrals are plagued by wormhole contributions that spoil the factorization of the holographic dual. These wormholes must be cancelled by some stringy effects in a UV complete, holographic theory of quantum gravity. In a simple model of two-dimensional topological gravity, we outline a gravitational analog of the recursive BPHZ procedure in order to systematically introduce ``counter-wormholes" which parametrize the unknown stringy effects that lead to factorization. Underlying this procedure is a Hopf algebra of symmetries which is analogous to the Connes--Kreimer Hopf algebra underlying perturbative renormalization. The group dual to this Hopf algebra acts to reorganize contributions from spacetimes with distinct topology, and can be seen as a gauge group relating various equivalent ways of constructing a factorizing gravitational path integral.