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Title: Umbral moonshine and K3 surfaces

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. In addition, 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.
Authors:
 [1] ;  [2]
  1. Univ. of Amsterdam, Amsterdam (The Netherlands)
  2. Stanford Univ., Stanford, CA (United States)
Publication Date:
OSTI Identifier:
1228046
Grant/Contract Number:
AC02-76SF00515
Type:
Accepted Manuscript
Journal Name:
Communications in Mathematical Physics
Additional Journal Information:
Journal Volume: 339; Journal Issue: 1; Journal ID: ISSN 0010-3616
Publisher:
Springer
Research Org:
SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)
Sponsoring Org:
USDOE Office of Science (SC)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING