Systems of correlators and solutions of the WDVV equations
type: Item
instance of: scholarly article
title: Systems of correlators and solutions of the WDVV equations
zbMATH Open document ID: 0906.35098
published in: Communications in Mathematical Physics
publication date: 1999-01-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q58
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 1202576
zbMATH Keywords: Witten-Dijkgraaf-Verlinde-Verlinde equations
zbMATH Keywords: WDVV equations
zbMATH Keywords: associativity equations
zbMATH Keywords: normalization equations
zbMATH Keywords: quasihomogeneity equations
zbMATH Keywords: Frobenius algebras
zbMATH Keywords: cohomology algebra of closed oriented manifolds
zbMATH Keywords: Frobenius super-algebras
zbMATH Keywords: super-WDVV equations
zbMATH Keywords: quantum cohomology
author: Sergey Natanzon
author: Vladimir Turaev
reviewed by: E. D. Belokolos
MaRDI profile type: Publication
full work available at URL: s002200050427
OpenAlex ID: W2065777339
DOI: 10.1007/S002200050427
description: scientific article; zbMATH DE number 1202576