Conference on Number Theory, Geometry, Moonshine & Strings

Organized by
Jeffrey Harvey, Ph.D.Professor, University of Chicago

September 6 - 8, 2017

The recognition that Calabi-Yau manifolds are fundamental for string compactification initiated a synthesis of geometry and string theory in the 1980s. This reinvigorated enumerative geometry, reshaped Gromov-Witten theory, and precipitated the development of new fields such as Donaldson-Thomas theory, quantum cohomology and homological mirror symmetry. Contemporaneously, efforts to unravel the mysteries of monstrous moonshine produced new algebraic theories which elucidated the role of automorphic forms in string theory and facilitated further developments in these and other fields.

In this century, the connections between number theory, geometry and string theory have deepened, with developments in the theory of harmonic Maass forms, the appearance of mock modular forms in enumerative geometry and string theory, and new relationships between automorphic forms and arithmetic geometry. New forms of moonshine have illuminated new paths for exploration.

The Conference on Number Theory, Geometry, Moonshine & Strings is the first in a series of meetings that will bring together mathematicians and physicists to elucidate, synthesize, develop and disseminate these emerging relationships between number theory, geometry, moonshine and string theory. The first meeting will focus on exploring number theoretic and string theoretic aspects of some recently discovered half-integral weight forms of moonshine which exhibit connections to modularity of varieties and the Birch–Swinnerton-Dyer conjecture.

The first conference was held on Sep. 6-8 and consisted of a series of seminars, a discussion section, and a public lecture “From Moonshine to Black Holes: Number Theory in Math- ematics and Physics” by Jeff Harvey. The schedule of seminars with speakers and titles is included below. The conference was organized by Jeff Harvey, John Duncan, Shamit Kachru and Ken Ono. There were 25 supported participants and a smaller number of people who came with their own support. Several mathematicians from local universities attended parts of the conference including Alberto Baider (CUNY), Herv ́e Jacquet (Columbia), Steve Miller (Rutgers), and Karen Taylor (NYU).

The conference brought together number theorists and physicists and many people com- mented on the unusual opportunities this presented. A number of new collaborations were started at the conference and others benefited from new points of view. For example, Daniel Persson wrote “I interacted quite a lot with Minhyong Kim who gave me completely new per- spectives on what cross-fertilizations between number theory and physics might entail…Steve Miller also showed up for one day, and that was enough for us to start a joint project together with Boris Pioline.” Larry Rolen said “ I learned a huge amount about the problems that people are studying in moonshine and physics, and it was especially useful for me to learn about what some of their goals are and what things they are interested in knowing about, as well as learning about cutting-edge results in moonshine. Motivation from physics has inspired many important applications and ideas in mock theta functions, as you know, and I am always interested to learn about this perspective as physicists have a different set of motivations.”

The discussion section was organized around several topics that helped to break down the lan- guage barrier between mathematicians and physicists. David Morrison gave several concrete examples of elliptic curves with interesting arithmetic structure, Greg Moore addressed the question ”What are BPS states and how do you count them?” and Jan Brunier and Yingkun Li discussed Generalized Jacobians and their role in moonshine and number theory. Several people commented that they would like to see more of this kind of presentation in future conferences.

The second conference in March will include some number theory but it is my intention to include more material on geometry and connections to black hole physics. I believe these two conferences will lead to new collaborations and cross-fertilization of ideas, but given the pace of mathematical research it may take a year or two to fully see the impact of these meetings. Because of this, if the Simons Foundation decides to fund additional conferences in this area I believe it would be best if the conferences were spaced a year apart rather than six months apart as the first two conference were in order to provide sufficient time for new developments.

Overall I think the conference was a great success and I look forward to the next one in March, 2018.