R-torsion and linking numbers from simplicial abelian gauge theories

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.

arXiv comment48 pages, latex, 9 figures (drawn in xfig)

Cited by in corpus (20)

• A geometric discretisation scheme applied to the Abelian Chern-Simons theory
• Laplacians on discrete and quantum geometries
• Geometric Structures on the Cochains of a Manifold
• Discretising geometry and preserving topology I: A discrete exterior calculus
• An analytic Approach to Turaev's Shadow Invariant
• Notes on simplicial BF theory
• Semi-Classical Holomorphic Transition Amplitudes in Covariant Loop Quantum Gravity
• Abelian Duality
• From simplicial Chern-Simons theory to the shadow invariant I
• From simplicial Chern-Simons theory to the shadow invariant II
• Equivalence of the Self-Dual Model and Maxwell-Chern-Simons Theory on Arbitrary Manifolds
• Discretising differential geometry via a new product on the space of chains
• Random Lattice Gauge Theories and Differential Forms
• Application of abelian holonomy formalism to the elementary theory of numbers
• Torus Knots and the Chern-Simons path integral: a rigorous treatment
• Matter on granular space-time
• Abelian Topological Order on Lattice Enriched with Electromagnetic Background
• Distributing the chiral and flavour components of Dirac-Kahler fermions across multiple lattices
• Discrete Connection Laplacians
• Trace Anomaly in Geometric Discretization