7 research outputs found
Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations
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. 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 arXiv:0904.4227 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
tr X(d/dX).Comment: 24 page
Integrable Systems
Integrable systems which do not have an \u201cobvious\u201c group symmetry, beginning with the results of Poincar\ue9 and Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960\u2019s. Although a number of fundamental methods of mathematical physics were based essentially on the perturbation-theory analysis of the simplest integrable examples, ideas about the structure of nontrivial integrable systems did not exert any real influence on the development of physics