Open-Closed Correspondence of K-theory and Cobordism

TL;DR

The authors propose a topological open-closed duality that links K-theory, which classifies D-brane charges, to cobordism groups that encode bulk geometric constraints. Grounded in the ABS orientation and the Hopkins–Hovey isomorphisms, they show how D-brane charges and cobordism invariants map to each other and how these relations reproduce tadpole cancellation conditions in type I and F-theory, providing a unified topological perspective on global symmetries in quantum gravity. This framework offers a bottom-up explanation for why certain backgrounds are allowed only when charges cancel via both open-string (K-theory) and closed-string (cobordism) data, and it suggests new avenues for extending to twisted and other generalized cobordism theories. The work strengthens the case that cobordism plays a fundamental role in consistent quantum gravity backgrounds and clarifies the gauging of global symmetries through explicit string-theoretic tadpole constraints.

Abstract

Non-trivial K-theory groups and non-trivial cobordism groups can lead to global symmetries which are conjectured to be absent in quantum gravity. Inspired by open-closed string duality, we propose a correspondence between the two groups, which can be considered as the physical manifestation of a generalisation of the classic Conner--Floyd isomorphism. The picture is exemplified by the relations between KO-groups and Spin-cobordisms and between K-groups and Spin$^c$-cobordisms. Global symmetries related by such isomorphism are eventually gauged. By combining K-theory and cobordism, we recover then tadpole cancellation conditions in type I string theory and F-theory from a bottom-up perspective.

Figures (1)

• Figure 1: Cobordism $(W,h)$ between pairs $(M,f)$ and $(N,g)$.