The Chronicles of IIBordia: Dualities, Bordisms, and the Swampland

TL;DR

This paper explores the Swampland Cobordism Conjecture within type IIB string theory, linking bordism classes with known supersymmetric backgrounds and proposing new non-supersymmetric objects, using advanced algebraic topology tools.

Contribution

It explicitly computes bordism groups related to duality bundles in type IIB string theory and connects them to known string backgrounds, also predicting new non-supersymmetric objects.

Findings

Bordism classes correspond to known F-theory backgrounds.

Global symmetries are broken by string theoretic objects.

Introduction of reflection 7-branes predicts non-supersymmetric backgrounds.

Abstract

In this work we investigate the Swampland Cobordism Conjecture in the context of type IIB string theory geometries with non-trivial duality bundle. Quite remarkably, we find that many non-trivial bordism classes with duality bundles in Mp$(2,\mathbb{Z})$, a double cover of SL$(2,\mathbb{Z})$ related to fermions, correspond to asymptotic boundaries of well-known supersymmetric F-theory backgrounds. These include $[p,q]$-7-branes, non-Higgsable clusters, S-folds, as well as various lower-dimensional generalizations. These string theoretic objects break the global symmetries associated to the non-trivial bordism groups, providing a strong test of the Cobordism Conjecture. Further including worldsheet orientation reversal promotes the duality group to the Pin$^+$ cover of GL$(2,\mathbb{Z})$. The corresponding bordism groups require a new non-supersymmetric "reflection 7-brane" and its compactifications to ensure the absence of global symmetries, thus providing an interesting prediction of the Cobordism Conjecture for non-supersymmetric type IIB backgrounds. A major component of the present work is the explicit derivation of the involved bordism groups as well as their generators, which correspond to asymptotic boundaries of explicit string theory backgrounds. The main tool is the Adams spectral sequence, to which we provide a detailed introduction. We anticipate that the same techniques can be applied in a wide variety of settings.