Symmetries of WDVV equations

TL;DR

This paper explores the symmetries of WDVV equations, demonstrating the existence of an infinite-dimensional group acting on calibrated solutions and describing its Lie algebra, thus revealing deep structural properties of these equations.

Contribution

It introduces an infinite-dimensional symmetry group for solutions of WDVV equations and characterizes its Lie algebra, advancing understanding of their geometric and algebraic structure.

Findings

Existence of an infinite-dimensional symmetry group for WDVV solutions

Explicit description of the Lie algebra of this symmetry group

Connection to calibration functions and algebraic structures

Abstract

We say that a function F(tau) obeys WDVV equations, if for a given invertible symmetric matrix eta^{alpha beta} and all tau \in T \subset R^n, the expressions c^{alpha}_{beta gamma}(tau) = eta^{alpha lambda} c_{lambda beta gamma}(tau) = eta^{alpha lambda} \partial_{lambda} \partial_{beta} \partial_{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_{alpha} \partial_{beta} x^{gamma} (z, tau) = z^{-1} c^{varepsilon}_{alpha beta} \partial_{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 [2]). We describe the action of Lie algebra of this group.