Principal Investigator: Motohico Mulase
The project is aimed at identifying the geometric structure of the Eynard-Orantin recursion formula that was originally discovered in random matrix theory and statistical physics. A particular emphasis of the project is placed on developing a point of view of the Laplace transform as the mirror map. The goal of the project is to solve the Remodeling Conjecture in algebraic geometry due to string theory physicists Marino and Bouchard-Klemm-Marino-Pasquetti. The Remodeling Conjecture presents a concrete and universal recursion formula, based on the complex analysis of a Riemann surface, that computes both closed and open Gromov-Witten invariants of an arbitrary toric Calabi-Yau 3-fold. Bouchard and Marino proposed a conjecture on Hurwitz numbers as a limit case of the Remodeling Cojecture. They defined a sequence of generating functions of simple Hurwitz numbers, and conjectured that these quantities satisfy the Eynard-Orantin recursion formula. The Bouchard-Marino conjecture was solved by the PI and his collaborators in 2009-10. The current understanding that has emerged since the publication of the PI's papers is the following. The Gromov-Witten theory of a toric Calabi-Yau 3-fold is a combinatorial counting problem on the A-model side of a topological string theory. The generating function of the solution to this counting problem satisfies a combinatorial equation, called the cut- and-join equation by Zhou. Now take the Laplace transform of this function. The result is a symmetric meromorphic function defined on the product of a Riemann surface. This Riemann surface is identified as the mirror curve of the toric Calabi-Yau 3-fold. The Laplace transform of the combinatorial equation becomes (conjecturally) the Eynard-Orantin recursion on the B-model side. The PI's contribution to this general picture is the identification of the role of the Laplace transform as the mirror map. This type of the Laplace transform was further investigated in the PI's recent papers, and has been utilized by other researchers in solving many related problems.
Pure mathematical research is about a sharp excitement of discovery. This excitement energizes young students in science and engineering. The PI has been collaborating with both undergraduate and graduate students, engaging them into the heart of the research excitement. Instead of peeking into research experience, these students have participated in the real excitement of mathematical discovery. These lasting impacts have inspired the students to become research mathematicians, and they are continuing to produce new results. The relation between algebraic geometry, symplectic geometry, combinatorics, random matrix theory, topology, integrable systems, and string theory in the Gromov-Witten theory was apparent in the early 1990s. After a long time of extremely fruitful developments in each individual area, we are back to a new level of interaction once again. Instead of pushing an existing problem or conjecture, the proposed project is aimed at understanding a new point of view. It is expected to bring a further cross-fertilization of these quite different ideas/areas in mathematical sciences. The PI has organized several workshops specifically aimed at this subject in the last two and a half years, which were attended by many young researchers. He will continue to do so for further dissemination.