Torsion in cohomology and dimensional reduction

Abstract

Abstract Conventional wisdom dictates that ℤ N factors in the integral cohomology group H p (X n , 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 ℤ N gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.

Keywords

• D-Branes
• Differential and Algebraic Geometry
• Discrete Symmetries
• Flux Compactifications