Umbral Moonshine and Monstrous Moonshine
In 2010 Eguchi, Ooguri and Tachikawa observed a remarkable connection between the elliptic genus of K3 surfaces and the representation theory of the Mathieu group M24. This has been dubbed Mathieu Moonshine because of the analogy to Monstrous Moonshine, a connection between the modular j function and the representation theory of the monster group. Mathieu Moonshine turns out to be the first of 23 examples of a new Umbral Moonshine phenomenon linking special sets of mock modular forms to the representation theory of a series of finite groups appearing in the automorphism groups of the 23 Niemeier lattices. There is also evidence for M24 moonshine in threshold corrections in N=2 heterotic string compactifications and the related counting problems in the Type IIA dual related to Gromov-Witten invariants. I am currently interested in extending these connections to some of the other groups and mock modular forms appearing in umbral moonshine and more generally in finding some explicit construction of the infinite dimensional modules suggested by these new moonshine phenomenon, hopefully connected to aspects of BPS states in string theory. Lance Dixon, Paul Ginsparg and I wrote a paper in 1988 with my all-time favorite title: Beauty and the Beast: Superconformal Symmetry in a Monster Module. In this paper we provided a translation into CFT and string theory language of the work of Frenkel, Lepowsky and Meurman on the construction of a natural infinite dimensional module for the Monster. We pointed out the existence of a previously overlooked superconformal symmetry in their construction, and we provided a physical interpretation of the generalized moonshine of Norton, namely that generalized moonshine is a statement about the modular properties of the partition function weighted by an element g of the monster in a Hilbert space constructed by twisting by h for every commuting pair of element (g,h) in the monster group. In later work, Ben Craps, Matthias Gaberdiel and I constructed and classified D-brane states in the monster CFT, leading to "Monstrous Branes."
Research Papers:
Umbral Moonshine, with M. Cheng and J. Duncan, pdf
Mathieu Moonshine and N=2 String Compactifications, with M. Cheng, X. Dong, J. Duncan, S. Kachru and T. Wrase,
pdf
Monstrous Branes, with B. Craps and M.R.Gaberdiel,
pdf
Beauty and the Beast:Superconformal Symmetry in a Monster Module, with L. Dixon and P. Ginsparg,
pdf
Umbral Moonshine and the Niemeier Lattices, with M. Cheng and J. Duncan, in preparation.
Contact Information
Email: j-harvey at uchicago.edu
Phone: +1 773-702-9897
Work Address: Enrico Fermi Institute, 5620 Ellis Ave., Chicago IL 60637 USA
