(Mock) Modularity, Moonshine and String Theory

Subfactors and VOAs should both describe CFT, but what is relatively easy in one formulation can be very difficult in the other. In my talk I'll describe lessons the VOA world can learn from the subfactor one.

I will talk about computation of the Witten index of 1d N=4 gauged linear sigma model which describes wall-crossing of BPS states in 4d N=2 theories. In the phase where the gauge group is broken to a finite group, the index is expressed as the JK-residue integral. Using this result, I am going to examine large-rank behaviour of the Kronecker quivers which describes the most simplest wall-crossing phenomena. I will also talk about how the refined Witten indices of quivers are preserved under the mutation process.

Recently, 23 cases of umbral moonshine, relating mock modular forms and finite groups, have been discovered in the context of the 23 even unimodular Niemeier lattices. One of the 23 cases in fact coincides with the so-called Mathieu moonshine, discovered in the context of K3 non-linear sigma models. Here we establish a uniform relation between all 23 cases of umbral moonshine and K3 sigma models, and thereby take a first step in placing umbral moonshine into a geometric and physical context. This is achieved by relating the ADE root systems of the Niemeier lattices to the ADE du Val singularities that a K3 surface can develop, and the configuration of smooth rational curves in their resolutions. A geometric interpretation of our results is given in terms of the marking of K3 surfaces by Niemeier lattices.