Abstract
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. In this paper 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.
Highlights
Introduction and SummaryMock modular forms are interesting functions playing an increasingly important role in various areas of mathematics and theoretical physics
To discuss the relation between the mock modular form H X and the K 3 elliptic genus, we first take a closer look at the construction of H X from the root system X
For any of the 23 Niemeier lattices, the root system is a union of -laced root systems with an ADE classification with the same Coxeter number m
Summary
Mock modular forms are interesting functions playing an increasingly important role in various areas of mathematics and theoretical physics. While the discovery of these 23 cases of moonshine perhaps adds to the beauty of the Mathieu moonshine relation, it adds more mystery. It was previously entirely unclear what the physical or geometrical context for these other instances of umbral moonshine could be. In this paper we establish a relation between K 3 sigma models and all cases of umbral moonshine, and thereby take a first step in incorporating umbral moonshine into the realm of geometry and theoretical physics
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