The Aspinwall-Morrison calculation and Gromov-Witten theory. (Q701277)

Let \(X\) be a Calabi-Yau threefold and \(H_1\), \(H_2\), \(H_3\) three cohomology classes in \(H_2(X,\mathbb{Z})\). In the framework of physical quantum field theories, the manifold \(X\) defines a so-called \(A\)-model, in which the 3-point correlator \(\langle H_1,H_2,H_3\rangle\) is expressed by the formula \[ \langle H_1,H_2,H_3\rangle= \int_X H_1H_2H_3+ \sum_{\beta\in H_2(X)} N_\beta(H_1,H_2,H_3) q^\beta, \] in which the path integral on the right is actually not a well-defined notion, and where the numbers \(N_\beta(H_1,H_2,H_3)\) for all homology classes of rational curves cannot be rigorously defined in the framework of topological field theories. The problem of giving this formula a precise meaning, on a firm mathematical basis, has been an issue for the past 15 years. Before the various approaches to establishing a mathematical theory of Gromov-Witten invariants were at hand, \textit{P. S. Aspinwall} and \textit{D. R. Morrison} [Commun. Math. Phys. 151, No. 2, 245--262 (1993; Zbl 0776.53043)] gave a calculation that determined the contribution of degree \(k\) multiple covers of a fixed smooth rational curve \(C\) in \(X\) to the numbers \(N_\beta\), thereby making the above formula precise for physicists. Based on ideas from topological field theory, the Aspinwall-Morrison calculation cannot be regarded as mathematically rigorous, since no direct comparison was made to any of the later definitions of Gromov-Witten invariants. Now, in the ``Gromov-Witten era'', the theory of the \(A\)-model is seen as a part of Gromov-Witten theory, where the spaces of maps that Aspinwall and Morrison dealt with appear as concrete moduli spaces. In this context, several geometers (after Aspinwall and Morrison) have found rigorous proofs of the crucial 3-point correlation formula, among them being \textit{C. Voisin} [Compos. Math. 104, No. 2, 135--151 (1996; Zbl 0951.14025)], \textit{M. Kontsevich} [in: The moduli space of curves, Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)], \textit{R. Pandharipande} [Commun. Math. Phys. 208, No. 2, 489--506 (1999; Zbl 0953.14036)], and others. In the paper under review, the author provides the lacking comparison between the original (mathematically incomplete) approach by Aspinwall and Morrison, on the one hand, and the modern approach via Gromov-Witten theory on the other. By showing that the Aspinwall-Morrison computations can be properly formalized in such a way that they turn out to be a push-forward version of Kontsevich's approach, he really justifies the former ones after nearly ten years of uncertainty. The author's work closes both a jawning gap and a historical chapter in this field of research, thereby demonstrating once more the always amazing mathematical truth in physical reasoning.