Mathematics

Moonshine in Spacetime: Automorphy, Algebras, and String Compactifications

Natalie Paquette (Caltech)

Abstract: New examples of moonshine--- relationships between finite groups and special classes of (mock) modular forms--- have been proliferating in recent years, starting with the discovery of a Mathieu group moonshine apparently connected to conformal field theory (CFT) on the K3 surface. While many aspects of these new moonshines remain mysterious, in this talk we will stress the power of spacetime string theory---as opposed to worldsheet string theory or CFT---to shine light on some of moonshine's mysteries. We will exhibit this in two vignettes. In part 1, we give a conceptual, physical explanation of the genus zero property of Monstrous moonshine using properties of a heterotic string compactification, concomitantly placing algebraic aspects of Borcherds' proof, such as the Monstrous Lie algebra, in a physical context. This gives a precise instantiation of the role of Generalized Kac-Moody algebras organizing BPS states in string theory, as first suggested by Harvey and Moore. In part 2, we completely determine a class of elliptic genera encoding the possible symmetries acting on BPS states in K3 CFT using wall-crossing properties of spacetime BPS states on K3 x T2 and orbifolds thereof. These in turn produce a class of 1/4-BPS counting functions in spacetime. The latter are Siegel automorphic forms that constitute predictions for the reduced Gromov-Witten theory of orbifolds of K3 x T2 and account for the entropy of supersymmetric black holes.

Generalized Weyl modules and nonsymmetric q-Whittaker functions

Daniel Orr   [email] (Virginia Tech)

Abstract: For the current algebra of a finite-dimensional simple Lie algebra, there are universal highest weight modules known as Weyl modules. By works of Cherednik and Braverman-Finkelberg, it is known that the graded characters of these modules form a joint eigenfunction of the q-Toda difference operators (and such functions are known as q-Whittaker functions). These works reveal connections to semi-infinite flag manifolds, double affine Hecke algebras (DAHAs), and Macdonald polynomials. E. Feigin and Makedonskyi introduced "generalized" Weyl modules for the Iwahori subalgebra of the current algebra, in order to give a representation-theoretic interpretation of positivity exhibited by certain specialized nonsymmetric Macdonald polynomials. I will explain some results from our joint work which show that the graded characters of generalized Weyl modules form an eigenfunction of a nonsymmetric variant of the q-Toda system arising from a representation of the nil-DAHA.

Lecture III: Descriptive graph combinatorics

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. Alexander Kechris will present "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.