Gromov-Witten classes, quantum cohomology, and enumerative geometry
type: Item
instance of: scholarly article
title: Gromov-Witten classes, quantum cohomology, and enumerative geometry
zbMATH Open document ID: 0853.14020
DOI: 10.1007/BF02101490
published in: Communications in Mathematical Physics
publication date: 1996-03-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/9402147
review text: The mathematical aspects of topological quantum field theory have recently led to the concept of quantum cohomology of a complex projective algebraic manifold. Basically, the quantum cohomology of such a manifold \(V\) is a formal deformation of its cohomology ring \(H^* (V, \mathbb{Q})\), whose parameters are the coordinates on the space \(H^* (V, \mathbb{Q})\) itself, and this object is used to construct what is called a cohomological field theory (CohFT). The physical concept of free energy (or potential) in such a CohFT corresponds mathematically to a formal series \(\Phi^V\) whose coefficients are given by the number of parametrized rational curves in \(V\) satisfying certain incidence conditions. Under the assumption that such a generating function (potential) \(\Phi^V\) really exists, i.e., that it can be meaningfully defined with respect to the numerical requirements concerning its coefficients, physicists have predicted its analytic behavior with such a precision that its uniqueness would be a consequence, just as the exact values of some numerical invariants of the space of parametrized pointed curves lying in \(V\). Some results on this conjectural, highly challenging subject have been obtained in the special case of a Calabi-Yau manifold \(V\), where \(\Phi^V\) would describe a variation of Hodge structures of the mirror dual manifold. In the present paper, the authors develop a formalism of defining (axiomatically) and investigating Gromov-Witten classes. Those appear as a collection of linear maps \[ I^V_{g,n, \beta} : H^* (V, \mathbb{Q})^{\otimes n} \to H^* (\overline M_{g,n}, \mathbb{Q}) \] with range in the cohomology ring of the moduli space \(\overline M_{g,n}\) of stable genus-\(g\) curves with \(n\) marked points and depending on integers \(g \geq 0\), \(n \geq 3 - 2g\), and homology classes \(\beta \in H_2 (V, \mathbb{Z}) \). The postulation of a series of formal and geometric properties (or axioms) for these Gromov-Witten classes, together wit…
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 58D30
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 851929
zbMATH Keywords: rational curves
zbMATH Keywords: stable algebraic curves
zbMATH Keywords: quantum cohomology of a complex projective algebraic manifold
zbMATH Keywords: cohomological field theory
zbMATH Keywords: variation of Hodge structures
zbMATH Keywords: mirror dual manifold
zbMATH Keywords: Gromov-Witten classes
zbMATH Keywords: operads for moduli spaces of stable curves
author: Maxim Kontsevich
author: Yuri I. Manin
reviewed by: Werner Kleinert
MaRDI profile type: Publication
arXiv ID: hep-th/9402147
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description: scientific article