Symmetries of WDVV equations

We say that a function F(tau) obeys WDVV equations, if for a given invertible symmetric matrix eta(alpha beta) and all tau is an element of T subset of R '', the expressions c(beta)(gamma)(alpha)(tau) = eta(alpha lambda)c(lambda beta gamma)(tau) = eta(alpha lambda)partial derivative(lambda)partial derivative(beta)partial derivative(gamma) F can be considered as structure constants of commutative associative algebra; the matrix eta(alpha beta) inverse to eta(alpha beta) determines an invariant scalar product on this algebra. A function x(alpha)(z, tau) obeying partial derivative(alpha)partial derivative(beta)x(gamma)(z, tau) = z(-1) c(alpha)(beta)(epsilon)partial derivative(epsilon)x(gamma)(z, tau) is called a calibration of a solution of WDVV equations. We show that there exists an infinite-dimensional group acting on the space of calibrated solutions of WDVV equations (in different form such a group was constructed in [A. Givental, math.AG/0305409]). We describe the action of Lie algebra of this group. (c) 2005 Elsevier B.V. All rights reserved.