New symmetries for Gromov-Witten theory – ThéorieGW

This project is the natural follow-up of recent results by the applicants Boissière, Chaput, Chiodo, Mann, Perrin and Zvonkine, which shed new light on important symmetries in Gromov-Witten theory, a major branch in algebraic geometry inspired by theoretical physics. The method is to exploit the computational power of these correspondences to contribute to a major problem in the field: higher genus Gromov-Witten theory. We will do it in two tasks: Task 1. New symmetries from Landau-Ginzburg/Calabi-Yau correspondence. Task 2. New symmetries from quantum cohomology. As highlights for our tasks and long term objectives, we may point out two major conjectures both mainly due to Yongbin Ruan: Conjecture 1 (Task 1). The geometric LG/CY correspondence; Conjecture 2 (Task 2). The crepant resolution conjecture. This project involves the six algebraic geometers working on Gromov-Witten related problems most recently hired in French mathematics departments. They have recently contributed to many important results in the field: Witten's r-spin conjecture (open since 1992) has been solved by Faber, Shadrin and Zvonkine; Ruan's LG/CY correspondence has been proven in genus-zero for the quintic three-fold by Chiodo and Ruan; the crepant resolution conjecture has been established by Boissière, Mann, and Perroni in many cases at the level of quantum cohomology within Gromov-Witten theory; and a detailed study of the quantum cohomology of homogeneous spaces led by Chaput, Manivel and Perrin showed the appearance of new symmetry phenomena. The consequences of these new results are largely unexplored and already open the way to several computations in higher genera. It happened several times, however, that the implications in Gromov-Witten theory of some of the applicants' results were more rapidly exploited by other leading teams aboard. In a highly competitive field whose main experts work in teams based in the US and in the UK, the ANR project provides the ideal means allowing to coordinate research activity and to ensure that the expertise of the applicants in Gromov-Witten theory will not disperse. Finally, this project is an extraordinary chance to take advantage of the contacts established by the six participants with other leading mathematicians in the field (e.g. Faber, Ruan, Shadrin). In particular Yongbin Ruan ' a leading expert playing a crucial role in the foundation of Gromov-Witten theory and in the formulation of all the aforementioned conjectures ' has already given his full support to the project and is available to help in the organization of a workshop in 2010 in France (this would be the follow-up of the Clay workshop involving the main experts in the field: Carel Faber, Rahul Panharipande, Cumrun Vafa www.claymath.org/workshops/lgmodel/). In 2011, Gérard Besson ' responsible for the annual summer school taking place at the Institut Fourier in Grenoble ' offered to organize a summer school with the participation of Yongbin Ruan aiming to develop the interest in Gromov-Witten theory in Europe. The ANR funding would be vital to attract international experts and young researchers to this event.