Umbral Moonshine and K3 Surfaces

          @misc{ pirsa_PIRSA:15040135,
            doi = {10.48660/15040135},
            url = {https://pirsa.org/15040135},
            author = {Harrison, Sarah},
            keywords = {Quantum Fields and Strings},
            language = {en},
            title = {Umbral Moonshine and K3 Surfaces},
            publisher = {Perimeter Institute},
            year = {2015},
            month = {apr},
            note = {PIRSA:15040135 see, \url{https://pirsa.org}}
          }
          

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. 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.