On the Swampland Cobordism Conjecture and Non-Abelian Duality Groups

TL;DR

This paper explores the implications of the cobordism conjecture and non-Abelian duality groups in string theory, revealing how the structure of 7-branes and modular curves constrains F-theory vacua and Mordell-Weil torsion groups.

Contribution

It connects the cobordism conjecture with non-Abelian duality groups, providing a geometric description of 7-branes and predicting allowed Mordell-Weil torsion groups in 8D F-theory.

Findings

The space of closed paths on the moduli space captures non-Abelian braid statistics.

Only certain congruence subgroups of SL(2,Z) are compatible with 8D F-theory vacua.

Predicted Mordell-Weil torsion groups consistent with the conjectures.

Abstract

We study the cobordism conjecture of McNamara and Vafa which asserts that the bordism group of quantum gravity is trivial. In the context of type IIB string theory compactified on a circle, this predicts the presence of D7-branes. On the other hand, the non-Abelian structure of the IIB duality group $SL(2,\mathbb{Z})$ implies the existence of additional $[p,q]$ 7-branes. We find that this additional information is instead captured by the space of closed paths on the moduli space of elliptic curves parameterizing distinct values of the type IIB axio-dilaton. This description allows to recover the full structure of non-Abelian braid statistics for 7-branes. Combining the cobordism conjecture with an earlier Swampland conjecture by Ooguri and Vafa, we argue that only certain congruence subgroups $\Gamma \subset SL(2,\mathbb{Z})$ specifying genus zero modular curves can appear in 8D F-theory vacua. This leads to a successful prediction for the allowed Mordell-Weil torsion groups for 8D F-theory vacua.