On Associativity Equations in Dispersionless Integrable Hierarchies

Details On Associativity Equations in Dispersionless Integrable Hierarchies On Associativity Equations in Dispersionless Integrable Hierarchies

2001-05-25

arxiv.org/abs/hep-th/0105260v2

Phys.Lett. B515 (2001) 483-492

Physics

High Energy Physics

High Energy Physics - Theory

16 pages, LaTeX

Scientific paper

10.1016/S0370-2693(01)00893-0

We discuss the origin of the associativity (WDVV) equations in the context of quasiclassical or Whitham hierarchies. The associativity equations are shown to be encoded in the dispersionless limit of the Hirota equations for KP and Toda hierarchies. We show, therefore, that any tau-function of dispersionless KP or Toda hierarchy provides a solution to associativity equations. In general, they depend on infinitely many variables. We also discuss the particular solution to the dispersionless Toda hierarchy that describes conformal mappings and construct a family of new solutions to the WDVV equations depending on finite number of variables.

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