Comments on Summing over bordisms in TQFT

Abstract

Recent works in quantum gravity, motivated by the "factorization problem" and "baby universes," have considered sums over bordisms with fixed boundaries in topological quantum field theory (TQFT). We discuss this construction and observe a curious splitting formula for the total amplitude.

A Panorama Of Physical Mathematics c. 2022

“…These involve the well-defined problem of summing over bordisms with fixed boundaries in a topological field theory. The model of [538] was interpreted as supporting the view that holographic duality involves ensemble averages, but other interpretations are possible [535]. As far as we know there is no previous mathematical work in this direction and it presents a potentially interesting opportunity for some interactions between physicists and mathematicians.…”

“…A curious related recent development, inspired by questions in quantum gravity, including the ensemble averaging question, are the "topological models of baby universes," investigated in [534][535][536][537][538]. These involve the well-defined problem of summing over bordisms with fixed boundaries in a topological field theory.…”

Combinatoric topological string theories and group theory algorithms

“…In the case when the group is a symmetric group S n , these theories admit defects, which have applications in describing counting and correlators in U(N) gauge theories [5] of interest in AdS/CFT [6][7][8]. Recent work on wormhole physics and baby universes [9][10][11][12][13][14][15], in the context of topology change in quantum gravity, considers sums over Riemann surfaces weighted by a string coupling g st , where each surface supports a Dijkgraaf-Witten theory. We will refer to these theories, summing over worldsheets, as combinatoric topological string theories or G-CTST.…”

“…It also has a related application in tensor model holography, relating the counting of connected and disconnected surfaces which are related to tensor model invariants [59]. Careful quantum gravitational discussions of the normalizations of partition functions relevant to combinatoric topological strings are in [9,10,13].…”