Conference on Number Theory, Geometry, Moonshine & Strings II

The second conference was held March 14-16, 2018, and consisted of a series of seminars and a public lecture, “The Riemann Hypothesis,” by Ken Ono. 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 including some postdocs and graduate students. Several mathematicians from local universities attended parts of the conference including Michael Harris (Columbia), Jim Lepowsky (Rutgers), Steve Miller (Rutgers), and Karen Taylor (NYU).

The conference brought together number theorists, geometers and physicists and many people commented on the unusual opportunities this presented. A number of new collaborations were started at the conference and others benefited from new points of view. Some examples: a new collaboration between Ken Ono and Sergei Gukov involving relations between ongoing work of Gukov and collaborators on generating functions for mock modular forms and three-manifold invariants and earlier work of Ono on quantum modular forms; a new collaboration between Shamit Kachru and Ken Ono connecting number theoretic aspects of quadratic forms to counting of black holes on K3 _ T 2; discussions between Kachru, Jim Bryan, John Duncan, Sheldon Katz, Albrecht Klemm and Georg Oberdieck on the definition of and predictions for twined Gromov-Witten invariants; fruitful discussions between Chris Beem and Jeff Harvey on the modularity of Schur indices in N = 2 superconformal field theories in four dimensions; a new collaboration between Duncan, Harvey, Michael Griffin and Brandon Rayhaun on moonshine for “Pariah” sporadic groups connected to weight 3=2 modular forms. These examples are just those I know of or have been made aware of. I am sure there are many other such examples.

In an email following the conference, Shamit Kachru said “This meeting helps show them [graduate students] that within the sphere of mathematical string theory, there are other directions, related to or quite distinct from what they have seen before, being pursued by other people who are fun to talk to. Interactions at such meetings have helped lead to projects and new collaborators for Sarah Harrison, Natalie Paquette, and Nathan Benjamin in the past, and continue to help people (e.g. Brandon Rayhaun). It would be hard for me to imagine raising mathematical physicists without such meetings.”

Sergei Gukov said in email to me that “many of the talks were directly or indirectly related to my current projects. For example, Martin’s talk about mock modular forms was directly related to above mentioned work in progress with Miranda (Cheng) et.al. (And, from Martin’s talk, I learned about a different perspective on mock modular forms relative to that I used so far.) Even your own talk on characters was extremely useful to my line of work on Vertex Operator Algebras labeled by 4-manifolds (that arise in compactification of 6d (0,2) theory on 4-manifold). I can go on with this list, but overall, it is hard to point to another two to three days in this academic year that would be equally helpful and, at the same time, relatively relaxed and enjoyable!”

Minhyong Kim said, “The second conference on number theory, geometry, moonshine and strings was a great success from my point of view. The first conference gave a clear sense of the kind of topics to read about in the past months, so I was able to follow the physics lectures much better this time around. I found most of the lectures greatly illuminating and tantalizing, especially those of Jeff Harvey, Shamit Kachru, Greg Moore and Sameer Murthy. All these people were very responsive to questions as well. In both number theory and high- energy physics, the ubiquity of modularity is a phenomenon at the exciting boundary of understanding and mystery, and there is much to be gained by continued interaction between the two communities.”

Although there were a number of notable presentations, one that impressed me the most regarding the potential for new relations between string theory and number theory was the talk by Albrecht Klemm on relations between masses of D-brane states in Calabi-Yau geometry, arithmetic geometry and values of L-functions. The work he presented is forging new connections between some very deep aspects of string theory and number theory.

The third conference is scheduled for February 27-March 1, 2019. It is too soon to set the agenda for this meeting, but I hope that some of the collaborations and new avenues of research that were promoted by the first two meetings will reach fruition and that part of the meeting will be devoted to hearing new results that were discovered as a result of the first two meetings. The first two meetings have strengthened my belief (and that of others) that these new interactions between mathematicians and physicists involving the topics of number theory, geometry, moonshine and string theory central to this series of conferences will lead to important, new results and connections between these areas. The support provided by the Simons Foundation has been crucial for the development of these new research directions.

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