R-torsion and linking numbers from simplicial abelian gauge theories

TL;DR

This paper develops a simplicial framework for abelian gauge theories that accurately reproduces continuum topological invariants like R-torsion and linking numbers, using dual triangulations and a novel doubling of fields.

Contribution

It introduces a new simplicial approach with dual triangulations and doubled fields to model topological invariants in abelian gauge theories.

Findings

Reproduces continuum partition functions and Wilson loop expectations.

Develops a simplicial analogue of Hodge-de Rham theory.

Identifies torsion pairings for homology torsion elements.

Abstract

Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R-torsion and linking numbers respectively. The new feature which makes this possible is the introduction of simplicial fields (cochains) associated with the dual triangulation of the background manifold, as well as with the triangulation itself. This doubling of fields, reminiscent of lattice fermion doubling, is required because the natural simplicial analogue of the Hodge star operator maps between cochains of a triangulation and cochains of the dual triangulation. The simplicial analogue of Hodge-de Rham theory is developed, along with a natural simplicial framework for considering linking numbers of framed loops. When the loops represent torsion elements of the homology of the manifold then Q/Z-valued torsion pairings appear in place of linking numbers for certain discrete values of the coupling parameter of the theory.