Torsion in cohomology and dimensional reduction

TL;DR

This paper explores how torsion in the cohomology of compact manifolds can be understood through dimensional reduction in string theory, linking torsion cycles, gauge symmetries, and BPS objects in the effective field theory.

Contribution

It proposes a novel EFT perspective on torsion cohomology, connecting smeared delta forms, calibrated torsion cycles, and $ ext{Z}_N$ gauge symmetries in string compactifications.

Findings

Torsion cycles can be characterized by smeared delta forms in the EFT.

Calibrated torsion cycles relate to linking numbers via smeared delta forms.

Torsion factors correspond to $ ext{Z}_N$ gauge symmetries with BPS sources.

Abstract

Conventional wisdom dictates that $\mathbb{Z}_N$ factors in the integral cohomology group $H^p(X_n, \mathbb{Z})$ of a compact manifold $X_n$ cannot be computed via smooth $p$-forms. We revisit this lore in light of the dimensional reduction of string theory on $X_n$, endowed with a $G$-structure metric that leads to a supersymmetric EFT. If massive $p$-form eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to them will have a non-trivial smeared delta form, that is an EFT long-wavelength description of $p$-form currents of the $(n-p)$-cycles of $X_n$. We conjecture that, whenever torsion cycles are calibrated, their linking number can be computed via their smeared delta forms. From the EFT viewpoint, a torsion factor in cohomology corresponds to a $\mathbb{Z}_N$ gauge symmetry realised by a St\"uckelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.