Gromov-Witten Invariants of Symmetric Products of Projective Space

Abstract

Gromov-Witten invariants are numbers that roughly count curves of a fixed type on an algebraic variety X. For example, for 3 general points and 6 general lines in X=P^3, there are exactly 190 twisted cubics intersecting all of them, so 190 is a Gromov-Witten invariant of P^3. Gromov-Witten invariants appear in algebraic geometry and string theory. In the special case when X is a toric variety, Kontsevich found a method to compute any Gromov-Witten invariant of X. Givental and Lian-Liu-Yau used Kontsevich’s algorithm to prove a mirror theorem, which states that Gromov-Witten invariants of X have an interesting rigid structure predicted by physicists. The main result of this thesis is a mirror theorem for the nontoric orbifold X=Sym^d(P^r), the symmetric product of projective space, which parametrizes unordered d-tuples of points in P^r.