Discrete Riemann Surfaces and the Ising model

Details Discrete Riemann Surfaces and the Ising model Discrete Riemann Surfaces and the Ising model

2009-09-19

arxiv.org/abs/0909.3600v1

Communications in Mathematical Physics 218, 1 (2001) 177-216

Mathematics

Differential Geometry

Scientific paper

10.1007/s002200000348

We de?ne a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is de?ned by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we de?ne a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.

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