Open-Closed Correspondence of K-theory and Cobordism

TL;DR

This paper proposes a new correspondence between K-theory and cobordism groups inspired by open-closed string duality, with implications for understanding global symmetries and tadpole cancellation in string theory.

Contribution

It introduces a generalized open-closed correspondence between K-theory and cobordism, extending the Conner--Floyd isomorphism and applying it to string theory consistency conditions.

Findings

Relations between KO-groups and Spin-cobordisms demonstrated

Relations between K-groups and Spin$^c$-cobordisms demonstrated

Reproduction of tadpole cancellation conditions in string theory

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.