Localization Computations of Gromov-Witten Invariants

Introduction Gromov-Witten invariants are enumerative invariants of stable maps. Their definition in the context of mirror symmetry in physics allowed new approaches to old problems — for instance, counting the number of plane rational curves of degree d through 3d − 1 points — and solved all at once enumerative problems that had thwarted mathematicians for years. Any such innovation gives rise to a host of new questions in the search for structure, rigor, and generalizations. For rational domain curves C and convex target spaces X , Gromov-Witten invariants give a fairly straightforward count of rational curves in X by looking instead at stable maps to X . As an example, we’ll compute the number of lines through two points in P in section 5. We can also use Gromov-Witten invariants to give a “count” of rational curves in a quintic threefold, a question related to the Clemens Conjecture (given X ⊂ P a generic quintic threefold, and d a positive integer, there are finitely many rational curves of degree d in X [Kat06]). This question is not qualitatively so different from the first, but already the enumerative significance of the invariant becomes less obvious. We must take into account multiple covers — higher-degree reparametrizations of C which factor through lower-degree maps to X — which contribute to the Gromov-Witten invariant but overcount things enumeratively. We will compute this multiple cover contribution in section 5.1, but we’ll need the technique of localization to do so. As genus increases and non-homogeneous spaces are explored, it becomes impossible to claim that the Gromov-Witten invariant is a literal count of genus g curves in a space X . At the same time, more mathematical machinery is required to compute the invariants at all. For instance, Gromov-Witten invariants for g = 0 and X homogeneous are computed as an integral over the space of stable maps [M0,n(X,β)], which is a smooth Deligne-Mumford stack of the expected dimension. (We will look at this space and stable maps in section 1.) For g > 0 and X non-homogeneous, [Mg,n(X,β)] is a singular Deligne-Mumford stack with many components of a larger than expected dimension, and we can’t just do usual intersection theory. We must define a virtual fundamental class — defining virtual classes is our way of making spaces “act smooth.” This will occupy section 3.