New Recursion Formulae and Integrability for Calabi-Yau Spaces

University Of California-Davis, Davis CA

Investigators

Abstract

The organizing committee, consisting of Vincent Bouchard (University of Alberta), Tom Coates (Imperial College, London), Emma Previato (Boston University), Jian Zhou (Tsinghua University, Beijing), and Motohico Mulase (University of California, Davis) serving as chair, will organize a 5-day workshop titled "New Recursion Formulae and Integrablity for Calabi-Yau Spaces" at the Banff International Research Station during the week of October 16-21, 2011 (http://www.birs.ca/events/2011/5-day-workshops/11w5114). The planned workshop has a clear set of focused goals, and is unique among conferences in related subjects. The main objective is to establish a topological and geometric foundation of a newly discovered topological recursion formula of 2007 by physicists Eynard and Orantin in their work on random matrices, and its Gromov-Witten theoretic realization due to string theorists Marino and Bouchard-Klemm-Marino-Pasquetti.One of the major problems in this area is the Remodeling Conjecture due to them. It states that both open and closed Gromov-Witten invariants of an arbitrary toric Calabi-Yau 3-fold are, quite miraculously, calculated by the Eynard-Orantin topological recursion based on the complex analysis of the mirror curve. A good part of the workshop will be devoted to attacking this unsolved conjecture. Another emphasis of the workshop is placed on discovering yet unknown relation between the generating function of Gromov-Witten invariants of Calabi-Yau spaces and integrable nonlinear partial differential equations. The subject matter of the planned Workshop, which is the first full-scale international workshop specifically devoted to the topics mentioned above, does not fit in a single discipline of mathematics. An important aspect of the Workshop is its function of cross fertilization of different areas of mathematics and theoretical physics. The origin of the main topic is in the soil of statistical study of random matrices. It's geometric significance was discovered by string theorists. It's mathematical nature apparently lies in topology. The theory itself covers a large area of mathematics. So far rigorously established examples of the theory range from hyperbolic geometry to algebraic geometry and to combinatorics of topological graph theory. The mathematical apparatus of these rigorous theories is the Laplace transform and classical complex analysis. Any new understanding of the proposed topics is expected to enhance our current knowledge of mirror symmetry, Gromov-Witten theory, and certain combinatorial problems. The BIRS Workshop plans to bring a wide variety of researchers, to nurture international and interdisciplinary collaborations among young participants, and to generate a larger momentum in the discipline. Since the key players of the subjects are postdoctoral researchers and faculty members in their early careers, the Workshop is expected draw the attention of many graduate students and postdoctoral scholars.