On the Swampland Cobordism Conjecture and Non-Abelian Duality Groups

TL;DR

The paper tests the McNamara–Vafa cobordism conjecture in type IIB string theory compactified on a circle, finding that Abelian bordism alone misses the full non-Abelian duality data of $SL(2,\mathbb{Z})$ and its $[p,q]$ 7-branes. By tracking duality twists as closed paths on modular curves $\mathcal{M}_{\Gamma}$ and incorporating 7-branes around cusps and elliptic points, the authors recover the non-Abelian braid statistics and derive genus-zero constraints that select compatible congruence subgroups $\Gamma$. These constraints then predict the Mordell–Weil torsion groups for elliptically-fibered K3 manifolds in 8D F-theory vacua, matching known realizations and excluding outliers due to a 24-discriminant-zero bound. The work thus links Swampland cobordism ideas to arithmetic structures, suggesting a broader applicability to theories with non-Abelian dualities and potential connections to arithmetic geometry.

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 $Γ\subset SL(2,\mathbb{Z})$ 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.

Figures (3)

• Figure 1: Left: Duality twist in the circle compactification of type IIB string theory in the F-theory description. Right: Trivialization of the bordism group by the introduction of physical objects.
• Figure 2: Decomposition of elements $\gamma \in SL(2,\mathbb{Z})$ into factors $\gamma = \gamma_1 \gamma_2 \gamma_3$ and the corresponding brane picture.
• Figure 3: Two paths connecting $SL(2,\mathbb{Z})$ equivalent values of $\tau$, related by $S$ and $T$, are projected to closed paths around the cusp or one of the elliptic points in the fundamental domain.