Topological String Partition Functions as Equivariant Indices

In this work we exploit the relationship with certain equivariant genera of isntanton moduli spaces to study the string partition functions of some local Calabi-Yau geometries, in particular, the Gopakumar-Vafa conjecture for them [8]. Gromov-Witten invariants are in general rational numbers. However as conjectured by Gopakumar and Vafa [8] using M-theory, the generating series of GromovWitten invariants in all degrees and all genera has a particular form, determined by some integers. See [8] or Section 2 for precise formulation. There have been various proposals to the proof of this conjecture [14, 10, 19]. In this work we propose a new and geometric approach towards this conjecture for some interesting cases. The method relates the computations of Gromov-Witten invariants to equivariant index theory and 4 dimensional gauge theory. Recently there have been some progresses on the calculations of Gromov-Witten invariants, both in the physical approaches [2, 11, 1] and mathematical treatments [28, 17]. For toric local Calabi-Yau geometries, one can now express the string partition functions as sums over partitions. Furthermore, one can relate them to partition functions in gauge theory according to the idea of geometric engineering (see e.g. [21, 12, 13, 6, 29, 7, 9]). The latter are equivariant genera of framed moduli spaces and can be computed by localization formula [20, 9] (see also Section 4). The fixed points on moduli spaces are tuples of partitions hence one gets seemingly different sums over partitions. However as shown in some of the works mentioned above, one can use combinatorics to identify different expressions. To prove the GV conjecture, one needs to rewrite the sums over partitions as infinite products (see e.g. [9] or Section 2 for explanation). The purpose of this paper is to point out that in some cases if one pushes forward the calculations from the framed moduli spaces, which are the Gieseker partial compactification of the moduli space of genuine instantons, to the Uhlenbeck partial compactification, then one can achieve this. Let us briefly explain some relevant terminologies. For a compact complex d-manifold X , one is often interested in its Hirzebruch χy genus. It is defined by: