Bordisms between 9d type IIB supergravities and commutator widths of duality groups

TL;DR

This paper investigates the topological properties of bordisms between 9d supergravities from type IIB string theory, proposing a refined Swampland Conjecture related to duality groups and their commutator widths.

Contribution

It introduces a new perspective on the Swampland Cobordism Conjecture by linking bordism complexity to the commutator width of duality groups in quantum gravity.

Findings

Bordisms become more complex with larger monodromies, leading to suppression.

The refined conjecture relates unbounded commutator width to the necessity of infinite duality defects.

Expectations are confirmed across various duality groups, often in non-trivial ways.

Abstract

We study the topological properties of bordisms interpolating between different 9d gauged supergravities obtained from compactification of type IIB string theory on $\mathbb{S}^1$ with a non-trivial $\mathsf{SL}(2,\mathbb{Z})$ bundle. We describe how such bordisms implement the needed monodromies through stacks of $[p,q]$ 7-branes or gravitational solitons of non-trivial topology. For the later mechanism, we see that the topology of the bordism becomes increasingly complicated for large monodromies, which results in the associated bordisms being arbitrarily suppressed, against expectations on the breaking of global symmetries in Quantum Gravity. Motivated by this, we propose a refinement of the Swampland Cobordism Conjecture for the first bordism group $\Omega_1({\rm B}G)$ with a $G$ duality bundle. We argue that even if gravitational solitons can realize the monodromies associated with elements of the commutator subgroup of $G$, if the number of needed commutators is unbounded (in other words, the commutator width of $G$ diverges) then an infinite number of duality defects realizing elements in $G$ need to be included. We test this proposal for different duality groups $G$, and see that our expectations are realized, often in non-trivial ways.