Gromov-Witten Theory of Deligne-Mumford Stacks

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Gromov-Witten theory concerns integrals over moduli spaces of stable maps from nodal Riemann surfaces to a target space. It had been a rapidly developing subject, with connections to many areas of mathematics and string theory. The crucial role of orbifolds/stacks in geometry and topology, for example the study of group actions and moduli problems, had been revealed in many works over the past decades. The two subjects were not considered together for a long time, until only very recently after Gromov-Witten theory was extended to stack/orbifold target spaces. The PI proposes to study several fundamental aspects of Gromov-Witten theory of orbifolds/stacks. The PI will pursue explicit calculations of orbifold Gromov-Witten invariants of toric targets and study structures of generating functions of Gromov-Witten invariants. Relations between Gromov-Witten theory and birational geometry, especially the so-called crepant resolution conjecture, will also be studied. Applications of Gromov-Witten theory of orbifolds/stacks to moduli spaces of curves, in particular properties and calculations of Hurwitz-Hodge integrals, will also be studied.The subject of Gromov-Witten theory lies on the boundary of several fields of mathematics and string theory. The proposed research will further advance the knowledge about these fields, in particular mirror symmetry for orbifolds/stacks, enumerative geometry of Deligne-Mumford stacks, integrable systems, the geometry of moduli spaces of curves. This will also further promote the existing interactions between algebraic geometry, symplectic geometry, combinatorics, mathematical physics, and string theory.