On the Swampland Cobordism Conjecture and Non-Abelian Duality Groups

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.

25 pages, 3 figures; clarifications added

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• 8d Supergravity, Reconstruction of Internal Geometry and the Swampland
• Compactness of Brane Moduli and the String Lamppost Principle in $d>6$
• The anomaly that was not meant IIB
• Dynamical Cobordism of a Domain Wall and its Companion Defect 7-brane
• Dynamical Tadpoles, Stringy Cobordism, and the SM from Spontaneous Compactification
• Dynamical Cobordism Conjecture: Solutions for End-of-the-World Branes
• Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations
• IIB Explored: Reflection 7-Branes
• Taming the Landscape of Effective Theories
• Dynamical Cobordism and the Beginning of Time: Supercritical Strings and Tachyon Condensation
• One Loop to Rule Them All: Eight and Nine Dimensional String Vacua from Junctions
• Open-Closed Correspondence of K-theory and Cobordism
• New Supersymmetric String Theories from Discrete Theta Angles
• Looking for structure in the cobordism conjecture
• 4d strings at strong coupling
• Dimensional Reduction of Cobordism and K-theory
• Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges
• Gravitational Solitons and Completeness
• Modular Curves and the Refined Distance Conjecture
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• On genus one fibered Calabi-Yau threefolds with 5-sections
• Hitchhiker's Guide to the Swampland: The Cosmologist's Handbook to the string-theoretical Swampland Programme
• At the End of the World: Local Dynamical Cobordism
• Gravitational Background of Alice-Vortices and R7-Branes