Gromov-Witten classes, quantum cohomology, and enumerative geometry
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 with the geometric intuition behind them, is an elaboration of Witten's treatment [cf. \textit{E. Witten}, J. Differ. Geom. Suppl. 1, 243-310 (1991; Zbl 0808.32023)] and allows, in the sequel, to establish their existence formally, at least for some classes of Fano varieties \(V\) and \(g = 0\). This is then used to construct an appropriate potential function \(\Phi^V\), basically with the aid of zero-codimensional Gromov-Witten classes of genus zero. Two reconstruction theorems show that the Gromov-Witten classes can be calculated recursively, at least in certain particular situations, and that the codimension-zero classes regulate the corresponding compatibility conditions. It turns out that the potential function \(\Phi^V\), defined by the Gromov-Witten classes and the linear superspace structure of the cohomology ring \(H^*(V,\mathbb{Q})\), encodes an extremely rich geometric structure on its convergence domain in \(H^*(V,\mathbb{Q})\), including the Vafa quantum cohomology rings of the Fano variety \(V\). A special section is devoted to the discussion of concrete examples, always assuming that the relevant Gromov-Witten classes exist and the potential function can be calculated along the previously described procedure. This leads to precise statements on its coefficients, i.e., to enumerative results on the spaces of rational curves in \(V\), and -- via the reconstruction theorems -- to new (conjectural) number-theoretic identities. The concluding part of the paper provides two possible definitions of a cohomological field theory. One of them is directly based on the axiomatics of the authors' Gromov-Witten classes, and the other one is related to the so-called operads for moduli spaces of stable curves. The formalism developed here is applied to the calculation of the cohomology of the moduli space of curves of genus zero. Using a related result of \textit{S. Keel} [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)], the authors derive a complete system of linear relations between the homology classes of the boundary strata of these moduli spaces. Altogether, the present paper is a highly valuable contribution towards the understanding of correlation functions of topological sigma-models from a rigorous algebro-geometric point of view.
quantum cohomology of a complex projective algebraic manifold
operads for moduli spaces of stable curves
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