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Article

Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations

Journal of High Energy Physics. 2011. No. 11(097). P. 1-31.
Mironov A., Morozov A., Natanzon S. M.

Correlators in topological theories are given by the values of a linear form on the products of operators from a commutative associative algebra (CAA). As a corollary, partition functions of topological theory always satisfy the generalized WDVV equations of. We consider the Hurwitz partition functions, associated in this way with the CAA of cut-and-join operators. The ordinary Hurwitz numbers for a given number of sheets in the covering provide trivial (sums of exponentials) solutions to the WDVV equations, with finite number of time-variables. The generalized Hurwitz numbers from provide a non-trivial solution with infinite number of times. The simplest solution of this type is associated with a subring, generated by the dilatation operators ˆW1 = trD = trX@/@X.