| The theory of quantum cohomology and Gromov-Witten invariants was first developed by Witten and has been the subject of active research in algebraic and symplectic geometry.;In this thesis, gauge theory techniques and the theory of flat connections are used to show that the small quantum product is a deformation of the cup product on a symplectic manifold M in a gauge theoretical sense, and to construct a moduli space of products on a complex vector space which are associative, commutative, Frobenius, and unital.;Both the quantum product and the cup product are commutative, Frobenius and associative, so they give rise to flat hermitian connections, the Dubrovin connections, on TH=THev M,C .;Specifically, to a constant product • : Hx H→H we associate the connection ∇YX( a) = dX(a)Y( a) + iX(a) • Y( a) where a∈H and X,Y are vector fields on H . The advantage of using a constant product (i.e. one in which • is independent of a) is that • is associative if and only if the Dubrovin connection ∇ is flat.;This approach is motivated in part by Givental's work on mirror symmetry, in which the flat sections for the Dubrovin connection on one smooth variety are related to the periods on the moduli space of complex structures on another "mirror" variety.;We first show that any flat connection ∇ on TH is gauge equivalent to the trivial connection d, through a gauge transformation g, and then we characterize g and the associated connection one-form in terms of the properties of the product.;We prove the following structure theorem: Theorem 0.1. (i) The space of constant products on a vector space V which are commutative, associative, Frobenius and unital modulo isomorphisms is isomorphic to the space of connection one-forms w with the properties: diXw =, wXY =wYX , w∧w=0, &angl0;wX,Y&angr0; +&angl0;wY ,X&angr0;=0 , and wX1 =-1X for all vector fields X,Y on TH , modulo an action of GL(V). (ii) Under only the assumption that the product • is associative and commutative, let a be the matrix valued function aij=- 1Gijktk where Gijk are the structure constants for • given by 6i•6j=Gk ij6k , for a basis 6i of TH . Then g=ea is the gauge transformation between d and the flat connection ∇. |
