Torsion in cohomology and dimensional reduction

Abstract

Conventional wisdom dictates that ℤN factors in the integral cohomology group Hp(Xn, ℤ) of a compact manifold Xn cannot be computed via smooth p-forms. We revisit this lore in light of the dimensional reduction of string theory on Xn, 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 Xn. 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 ℤN gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.