Two-sphere partition functions and Gromov-Witten invariants

Many $N=(2,2)$ two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as $N=(2,2)$ gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories – recently computed via localization by Benini et al. and Doroud et al. – yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. We compute these quantities for several examples giving us strong evidence for our conjecture.