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 socalled Mathieu moonshine, discovered in the context of K3 nonlinear 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]}
 Univ. of Amsterdam, Amsterdam (The Netherlands)
 Stanford Univ., Stanford, CA (United States)
 Publication Date:
 Report Number(s):
 SLACPUB16469
Journal ID: ISSN 00103616; PII: 2398
 Grant/Contract Number:
 AC0276SF00515
 Type:
 Accepted Manuscript
 Journal Name:
 Communications in Mathematical Physics
 Additional Journal Information:
 Journal Volume: 339; Journal Issue: 1; Journal ID: ISSN 00103616
 Publisher:
 Springer
 Research Org:
 SLAC National Accelerator Lab., Menlo Park, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
 OSTI Identifier:
 1228046
 Alternate Identifier(s):
 OSTI ID: 1243080
Cheng, Miranda C. N., and Harrison, Sarah. Umbral moonshine and K3 surfaces. United States: N. p.,
Web. doi:10.1007/s0022001523985.
Cheng, Miranda C. N., & Harrison, Sarah. Umbral moonshine and K3 surfaces. United States. doi:10.1007/s0022001523985.
Cheng, Miranda C. N., and Harrison, Sarah. 2015.
"Umbral moonshine and K3 surfaces". United States.
doi:10.1007/s0022001523985. https://www.osti.gov/servlets/purl/1228046.
@article{osti_1228046,
title = {Umbral moonshine and K3 surfaces},
author = {Cheng, Miranda C. N. and Harrison, Sarah},
abstractNote = {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 socalled Mathieu moonshine, discovered in the context of K3 nonlinear 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.},
doi = {10.1007/s0022001523985},
journal = {Communications in Mathematical Physics},
number = 1,
volume = 339,
place = {United States},
year = {2015},
month = {6}
}