We construct new solutions of the Witten–Dijkgraaf–E. Verlinde–H. Verlinde equations. Introduction In 1990 E. Witten [6] and R. Dijkgraaf, E. Verlinde, H. Verlinde [1] introduced a remarkable system of partial differential equations (the WDVV equations) determining deformations of 2-dimensional topological field theories. The WDVV equations consist of three groups of differential equations called the associativity equations, the normalisation equations, and the quasihomogeneity equations. B. Dubrovin [2] observed that solutions of these equations lead to new deep differential-geometrical structures on manifolds. He found that such structures naturally arise in various branches of mathematics including algebraic geometry, integrable systems, Coxeter groups, quantum cohomology, etc. The theory of the WDVV equations is closely related to the theory of Frobenius algebras. Namely, every solution of the associativity equations gives rise to a family of Frobenius algebras. In this paper we show that conversely every Frobenius algebra A over C gives rise in a purely algebraic way to a family of holomorphic solutions of the associativity equations. More precisely, we construct an infinite family of holomorphic functions A → C satisfying the associativity equations with respect to any choice of linear coordinates on A. If A has a unity 1 = 1A, then this family is parametrised by infinite sequences of elements of A. If A has no unity, then this family is parametrised by infinite sequences of A-linear homomorphisms A → A. The functions A → C constructed here in general do not satisfy the normalisation equations. We show how to modify them in order to satisfy these equations as well. We show finally that if the Frobenius algebra is graded, then the functions constructed in this paper are homogeneous and thus satisfy all the WDVV equations. Examples of 400 S. Natanzon, V. Turaev graded Frobenius algebras are provided by cohomology of closed oriented manifolds; in this case our methods yield polynomial solutions of the WDVV equations. Note that our solutions are in general different from those appearing in the theory of quantum cohomology, see [4]. It should be stressed that our aim here is not to recover the quantum cohomology but rather to study solutions of WDVV arising in an algebraic way via the theory of Frobenius algebras. The constructions of this paper can be easily generalised to produce solutions of the super-WDVV equations from Frobenius super-algebras [3]. We study solutions of WDVV in terms of the coefficients of their Taylor expansions, called correlators (cf. [2]). Section 1 is concerned with preliminaries on the associativity equations and correlators. In Sect. 2 we derive from a Frobenius algebra A a family of holomorphic solutions of the associativity equations. In Sect. 3 we show how to modify these solutions in order to satisfy the normalisation equations. In Sect. 4 we discuss the quasihomogeneity equations and consider examples of normalized quasihomogeneous systems of correlators related to the ordinary and quantum cohomology of manifolds. 1. Frobenius Algebras, Associativity Equations and Correlators Frobenius algebras. By a Frobenius algebra over C we mean a commutative associative finite dimensional algebra A over C provided with a non-degenerate symmetric bilinear form η : A × A → C such that η(ab, c) = η(a, bc), for any a, b, c ∈ A. The form η is called the inner product on A. If A has a unity 1 = 1A, then the inner product is determined by the linear functional a 7→ η(a, 1) : A → C because η(a, b) = η(ab, 1). The associativity equations. Let (η)λ,μ=1 be a non-degenerate symmetric N × N matrix over C. Let F = F (t1, ..., tN ) be a C-valued function on C where t1, ..., tN are the coordinates on C . The associativity equations (associated with (η)) are the following nonlinear partial differential equations: