Topological recursion, the Laplace transform, and integrable systems

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AbstractAward: DMS-1104734Principal Investigator: Motohico MulaseThe project is aimed at identifying the geometric structure of theEynard-Orantin recursion formula that was originally discovered inrandom matrix theory and statistical physics. A particular emphasis ofthe project is placed on developing a point of view of the Laplacetransform as the mirror map. The goal of the project is to solve theRemodeling Conjecture in algebraic geometry due to string theoryphysicists Marino and Bouchard-Klemm-Marino-Pasquetti. The RemodelingConjecture presents a concrete and universal recursion formula, basedon the complex analysis of a Riemann surface, that computes bothclosed and open Gromov-Witten invariants of an arbitrary toricCalabi-Yau 3-fold. Bouchard and Marino proposed a conjecture onHurwitz numbers as a limit case of the Remodeling Cojecture. Theydefined a sequence of generating functions of simple Hurwitz numbers,and conjectured that these quantities satisfy the Eynard-Orantinrecursion formula. The Bouchard-Marino conjecture was solved by the PIand his collaborators in 2009-10. The current understanding that hasemerged since the publication of the PI's papers is thefollowing. The Gromov-Witten theory of a toric Calabi-Yau 3-fold is acombinatorial counting problem on the A-model side of a topologicalstring theory. The generating function of the solution to thiscounting problem satisfies a combinatorial equation, called the cut-and-join equation by Zhou. Now take the Laplace transform of thisfunction. The result is a symmetric meromorphic function defined onthe product of a Riemann surface. This Riemann surface is identifiedas the mirror curve of the toric Calabi-Yau 3-fold. The Laplacetransform of the combinatorial equation becomes (conjecturally) theEynard-Orantin recursion on the B-model side. The PI'scontribution to this general picture is the identification of the roleof the Laplace transform as the mirror map. This type of the Laplacetransform was further investigated in the PI's recent papers, andhas been utilized by other researchers in solving many relatedproblems.Pure mathematical research is about a sharp excitement ofdiscovery. This excitement energizes young students in scienceand engineering. The PI has been collaborating with bothundergraduate and graduate students, engaging them into the heartof the research excitement. Instead of peeking into researchexperience, these students have participated in the realexcitement of mathematical discovery. These lasting impacts haveinspired the students to become research mathematicians, and theyare continuing to produce new results. The relation betweenalgebraic geometry, symplectic geometry, combinatorics, randommatrix theory, topology, integrable systems, and string theory inthe Gromov-Witten theory was apparent in the early 1990s. After along time of extremely fruitful developments in each individualarea, we are back to a new level of interaction onceagain. Instead of pushing an existing problem or conjecture, theproposed project is aimed at understanding a new point ofview. It is expected to bring a further cross-fertilization ofthese quite different ideas/areas in mathematical sciences. ThePI has organized several workshops specifically aimed at thissubject in the last two and a half years, which were attended bymany young researchers. He will continue to do so for furtherdissemination.