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Discrete holomorphicity and Ising model transfer matrix formalism

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The transfer matrix formalism is a classical approach to the two-dimensional Ising model, and some of the most remarkable exact calculations about the model were obtained using this approach. Discrete holomorphicity, on the other hand, has recently lead to proofs of conformal invariance of energy and spin correlations in the scaling limit of the model taken at the critical point, and to convergence results of interfaces to Schramm-Loewner evolutions. In this talk we consider relations of discrete holomorphicity and transfer matrix formalism. In particular it will be shown that the transfer matrix can be reconstructed from discrete analytic continuation of s-holomorphic (or massive s-holomorphic) functions, and that fermion operators in the transfer matrix formalism are operator valued (massive) s-holomorphic functions. Finally we discuss rudiments of a fermionic conformal field theory description of the scaling limit of the Ising model based on these results.

This talk is part of the Probability series.

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