- Organized by
Jeffrey Harvey, Ph.D.Professor, University of Chicago
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.
This third conference on the topic brought together mathematicians and physicists to elucidate, synthesize, develop and disseminate emerging relationships between number theory, geometry, moonshine and string theory. In addition, the conference revisited topics from the first two conferences to survey the progress that has been made in these new directions and to facilitate discussions that will lead to future research topics that can benefit from combining the insights and techniques of physicists and mathematicians.
The third and final conference in this series was held February 27–March 1, 2019, and consisted of a series of seminars and a public lecture, “How to Count in String Theory,” by Shamit Kachru. 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, including some postdocs and graduate students. Several mathematicians from local universities attended parts of the conference, including Dorian Goldfeld (Columbia), Jim Lepowsky (Rutgers) and Steve Miller (Rutgers).
The conference brought together number theorists, algebraic geometers and physicists. This led to many opportunities for people to learn about new points of view and new problems that they could attack using the ideas and tools that grew out of these meetings. There are so many examples of this that I can’t list them all, but I will highlight some of the new collaborations and work that grew directly out of these meetings.
At the second meeting, Jim Bryan and Georg Oberdieck had discussions with physicists on what are known as ‘CHL models’ and, based on this, developed a method to define and compute Donaldson-Thomas invariants for these models. Oberdieck reported on these results at the third meeting.
Shamit Kachru, Arnav Tripathy and collaborators have written a series of interesting papers on aspects of number theory that are visible in the physics of the attractor mechanism for supersymmetric black holes. These papers directly grew out of conversations that started during these meetings. Sergei Gukov reported on work involving modularity arising from 3D physics and geometry that is joint with Cheng, Chun, Ferrari and Harrison. Gukov’s interest in number theory developed largely due to this series of meetings, and he continues to build on these ideas.
On the moonshine side of things, work on moonshine for the O’Nan sporadic group with connections to the Mordell-Weil groups of elliptic curves by John Duncan, Michael Mertens and Ken Ono was presented at these meetings, and a new collaboration by Cheng, Dunan and Mertens grew out of related discussions at this meeting. New directions are also emerging from talks at the last meeting. Ono, Greg Moore and Minhyong Kim highlighted the talk by Wendland on the geometry of K3 surfaces and Mathieu Moonshine as an important development, Jeff Harvey and Minhyong Kim were inspired by discussions with Sameer Murthy on D-branes on elliptic curves and connections to L-functions, and there are many other examples of lively discussions that will probably lead to new collaborations.
Another testament to the influence of these meetings is that a number of other meetings have been held or are being planned that build on ideas and themes that started in these meetings. Examples include a thematic program on arithmetic geometry and quantum field theory at the Korea Institute for Advanced Study from 2017 to 2019 and a planned workshop on modularity in physics and mathematics at the KITP in Santa Barbara in the fall of 2020.
Overall, these meetings have been a huge success. They have led to new collaborations, ideas and results that likely would not have happened without the interactions between mathematicians and physicists that these meetings made possible. More generally, these developments support the view that physics and number theory have a great deal to teach each other and that pursuing these connections will lead to far-ranging and important, new insights. The support provided by the Simons Foundation has been crucial for the development of these new research directions.
University of Chicago
For further information on Prof. Kachru’s Lecture, see its event page.