Associativity relations in quantum cohomology (Q1283576)

The author investigates quadratic relations for Gromov-Witten invariants, arising from the associativity of the quantum product (i.e., the famous WDVV equations). He concentrates to the case of rational curves (i.e. invariants associated to the moduli space of stable maps of genus zero). Here, the relations can be described by Feynman diagrams relating trees of rational curves. In several examples, the associativity has been used to compute all Gromov-Witten numbers recursively from some starting data. Considered as a purely algebraic problem, one may ask which invariants are needed in general as starting data such that the WDVV equations can be solved uniquely and consistently. The main result obtained by the author is the following. Strong reconstruction theorem. Let \(X\) be a complex projective manifold, and assume that \(H^{2*}(X,\mathbb{Q})\) is generated by divisors. Then it is sufficient to know all Gromov-Witten invariants \(N(\beta,d)\) with \(\sum_{i=1}^s d_s\leq 2\) (which have to fulfil a simple initial relation) to compute uniquely and consistently all Gromov-Witten invariants of \(X\) by means of the WDVV equations. The proof is constructive and gives in principle an algorithm (possibly quite complicated) for the computation. It is illustrated in detail with the examples of the product of a smooth quadric threefold with itself, \( \text{Sym}^2 {\mathbb{P}}^2 \), and the Grassmannians \(\text{Gr}(2,4)\) and \(\text{Gr}(2,5)\). Actually, only the first example satisfies the conditions of the reconstruction theorem, but similar techniques work well in the remaining cases. The author also points out that there are in general non-geometric solutions to a WDVV system, coming from other initial data than the geometric ones.