Systems of correlators and solutions of the WDVV equations
In 1990, E. Witten, R. Dijkgraaf, E. Verlinde, H. Verlinde introduced a system of partial differential equations (the WDVV equations) which described deformations of two-dimensional topological field theories and consisted of three groups of equations: the associativity equations, the normalization equations and the quasihomogeneity equations. The aim of the paper is to construct solutions of these equations in a purely algebraic way in terms of Frobenius algebras. For an arbitrary Frobenius algebra \(A\) the authors build, by means of a bounded sequence of \(A\)-linear operators, an infinite family of holomorphic functions satisfying the associativity equations. If the Frobenius algebra is nilpotent then these solutions are polynomials. Constructing a normal extension of the Frobenius algebra the authors build a normalized holomorphic solution of the associativity equations. If the Frobenius algebra is graded then the functions, constructed above, appear to be homogeneous and thus satisfy all the WDVV equations. Application of this approach to cohomology algebra of closed oriented manifolds leads to polynomial solutions of the WDVV equations if the odd-dimensional cohomology is trivial. If the odd-dimensional cohomology is non-trivial one has to use the language of Frobenius super-algebras and super-WDVV equations. Application of this approach to the theory of quantum cohomology leads in general to new classes of solutions.
Witten-Dijkgraaf-Verlinde-Verlinde equations
cohomology algebra of closed oriented manifolds
scientific article; zbMATH DE number 1202576