Gromov-Witten Theory and Type II Extremal Transitions

Abstract

Extremal transitions are a contract-deform surgery, which play a significant role in spacetime topology change in string theory. From a physical point of view, we expect that mirror symmetry might be preserved under extremal transitions. Li- Ruan initiated the study of the change of Gromov-Witten theory under conifold transitions twenty years ago and some generalizations to other Type I transitions have been made over the years, but no work on Type II extremal transitions has been known before. In this thesis, we propose a very general conjectural framework which relates two quantum D-modules under a primitive extremal transition. We verify this framework for a number of examples of Type II extremal transitions. In the case of cubic extremal transitions, we find an interesting connection between the quantum D-module and the genus zero FJRW theory of the cubic singularity.