Generalised Symmetries and Swampland-Type Constraints from Charge Quantisation via Rational Homotopy Theory

TL;DR

This paper refines the understanding of charge quantisation in quantum field and string theories using rational homotopy theory, linking it to higher gauge theories and swampland constraints.

Contribution

It introduces a refined framework connecting charge quantisation, higher gauge theory, and homotopy types, providing new constraints on quantum field theories and insights into quantum gravity.

Findings

Homotopy groups of $\\mathcal A$ classify brane charges.

Homology groups of $\\mathcal A$ classify invertible higher-form symmetries.

Charge quantisation constrains gauge groups and field strengths.

Abstract

Sati and Schreiber [arXiv:2402.18473, arXiv:2512.12431] have proposed that charge quantisation in quantum field theory and string theory is governed by a homotopy type $\mathcal A$. We provide a refinement of this postulate, incorporating other currents including matter, connecting it to adjustments in higher gauge theory and providing a prescription for determining $\mathcal A$, and show that, while the homotopy groups of $\mathcal A$ classify the possible brane charges, the homology groups of $\mathcal A$ classify the invertible higher-form symmetries. Furthermore, we show that the charge-quantisation postulate implies a number of non-trivial constraints on quantum field theories similar to those implied by swampland conjectures; in particular, it rules out noncompact gauge groups and one-form field strengths that form a non-nilpotent Lie algebra. Finally, we argue that for theories of quantum gravity the space $\mathcal A$ must be contractible, in accordance with the swampland conjectures on the absence of global generalised symmetries and the completeness of the spectrum of charges, and explain how this explicitly arises in the case of Type I string theory.