Discretising differential geometry via a new product on the space of chains

Details Discretising differential geometry via a new product on the space of chains Discretising differential geometry via a new product on the space of chains

2006-10-05

arxiv.org/abs/hep-th/0610065v1

Physics

High Energy Physics

High Energy Physics - Theory

39 pages, 2 figures, LOR2006

Scientific paper

A discretisation of differential geometry using the Whitney forms of algebraic topology is consistently extended via the introduction of a pairing on the space of chains. This pairing of chains enables us to give a definition of the discrete interior product and thus provides a solution to a notorious puzzle in discretisation techniques. Further prescriptions are made to introduce metric data, as a discrete substitute for the continuum vielbein, or Cartan formulation. The original topological data of the de Rham complex is then recovered as a discrete version of the Pontryagin class, a sketch of a few examples of the technique is also provided. A map of discrete differential geometry into the non-commutative geometry of graphs is constructed which shows in a precise way the difference between them.

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