Dimensional Reduction of Cobordism and K-theory

TL;DR

This paper explores the relationship between cobordism and K-theory groups in string theory, showing they encode symmetry patterns and their breaking or gauging in quantum gravity via spectral sequence computations.

Contribution

It demonstrates that cobordism and K-theory groups of manifolds reproduce symmetry patterns from dimensional reduction, supporting their role in quantum gravity constraints.

Findings

Cobordism and K-theory groups match expected symmetry patterns.

These groups encode symmetry breaking and gauging in string compactifications.

Spectral sequence computations validate the theoretical predictions.

Abstract

It has been proposed that cobordism and K-theory groups, which can be mathematically related in certain cases, are physically associated to generalised higher-form symmetries. As a consequence, they should be broken or gauged in any consistent theory of quantum gravity, in accordance with swampland conjectures. We provide further support to this idea by showing that cobordism and K-theory groups of a general manifold $X$ reproduce the pattern of symmetries expected from the dimensional reduction of the theory on $X$, as well as their breaking and gauging. To this end, we employ the Atiyah-Hirzebruch spectral sequence to compute such groups for common choices of $X$ in string compactifications.