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Exactly solvable SOS models. Local height probabilities and theta function identities. (English) Zbl 0679.17010
Summary: The local height probabilities (LHPs) of a series of solvable solid-on- solid (SOS) models are obtained. The models have been constructed by a fusion procedure from the eight-vertex SOS model. The LHP results are expressed in terms of modular functions (which we call the branching coefficients) appearing in appropriate theta function identities. The critical behavior is studied by using their automorphic properties. Some of these identities result from the representation theory of affine Lie algebras. The branching coefficients are (not necessarily irreducible) characters of the Virasoro algebra constructed from the pairs $$(A_ 1^{(1)}+A_ 1^{(1)},A_ 1^{(1)})$$ or $$(A^{(1)}_{2\ell - 1},C_{\ell}^{(1)})$$. As the critical exponents the present lattice models realize the anomalous dimensions of known 2D conformal field theories (the minimal unitarizable theory and its supersymmetric extensions, the $$Z_ N$$-invariant theory, etc.).

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 82B05 Classical equilibrium statistical mechanics (general) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 11F03 Modular and automorphic functions 05A19 Combinatorial identities, bijective combinatorics
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