Convergence of quantum cohomology by quantum Lefschetz
  Quantum Lefschetz theorem by Coates and Givental gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a byproduct, we prove the semi-simplicity and the Virasoro conjecture for the Gromov-Witten theories of (not necessarily Fano) projective toric manifolds. 
Hiroshi Iritani