Umbral moonshine and K3 surfaces

Cheng, Miranda C. N.; Harrison, Sarah

doi:10.1007/s00220-015-2398-5

Title: Umbral moonshine and K3 surfaces

Journal Article · Thu Jun 25 00:00:00 EDT 2015 · Communications in Mathematical Physics

DOI:https://doi.org/10.1007/s00220-015-2398-5· OSTI ID:1228046

Cheng, Miranda C. N. ^[1]; Harrison, Sarah ^[2]

Univ. of Amsterdam, Amsterdam (The Netherlands)
Stanford Univ., Stanford, CA (United States)

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.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC02-76SF00515

OSTI ID:: 1228046

Alternate ID(s):: OSTI ID: 1243080

Report Number(s):: SLAC-PUB-16469; PII: 2398

Journal Information:: Communications in Mathematical Physics, Vol. 339, Issue 1; ISSN 0010-3616

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 27 works

Citation information provided by
Web of Science

References (80)

Catastrophes and the classification of conformal theories Vafa, Cumrun; Warner, Nicholas Physics Letters B, Vol. 218, Issue 1 https://doi.org/10.1016/0370-2693(89)90473-5	journal	February 1989
Little string theory in a double scaling limit Giveon, Amit; Kutasov, David Journal of High Energy Physics, Vol. 1999, Issue 10 https://doi.org/10.1088/1126-6708/1999/10/034	journal	October 1999
Modular invariant partition functions in two dimensions Cappelli, A.; Otzykson, C.; Zuber, J. -B. Nuclear Physics B, Vol. 280 https://doi.org/10.1016/0550-3213(87)90155-6	journal	January 1987
Algebraic geometry and effective lagrangians Martinec, Emil J. Physics Letters B, Vol. 217, Issue 4 https://doi.org/10.1016/0370-2693(89)90074-9	journal	February 1989
The Partition Function of the Two-Dimensional Black Hole Conformal Field Theory Hanany, Amihay; Prezas, Nikolaos; Troost, Jan Journal of High Energy Physics, Vol. 2002, Issue 04 https://doi.org/10.1088/1126-6708/2002/04/014	journal	April 2002
Modular invariant partition functions for N = 2 superconformal field theories Qiu, Zongan Physics Letters B, Vol. 198, Issue 4 https://doi.org/10.1016/0370-2693(87)90906-3	journal	December 1987
New N=2 superconformal field theories and superstring compactification Kazama, Yoichi; Suzuki, Hisao Nuclear Physics B, Vol. 321, Issue 1 https://doi.org/10.1016/0550-3213(89)90250-2	journal	July 1989
Character formulas for the N = 4 superconformal algebra Eguchi, Tohru; Taormina, Anne Physics Letters B, Vol. 200, Issue 3 https://doi.org/10.1016/0370-2693(88)90778-2	journal	January 1988
Unravelling Mathieu moonshine Govindarajan, Suresh Nuclear Physics B, Vol. 864, Issue 3 https://doi.org/10.1016/j.nuclphysb.2012.07.005	journal	November 2012
Two-dimensional black hole and singularities of CY manifolds Ooguri, Hirosi; Vafa, Cumrun Nuclear Physics B, Vol. 463, Issue 1 https://doi.org/10.1016/0550-3213(96)00008-9	journal	March 1996
N=2 superconformal symmetry and SO(2,1) current algebra Dixon, Lance J.; Peskin, Michael E.; Lykken, Joseph Nuclear Physics B, Vol. 325, Issue 2 https://doi.org/10.1016/0550-3213(89)90459-8	journal	October 1989
Derived equivalences of K3 surfaces and twined elliptic genera Duncan, John F. R.; Mack-Crane, Sander Research in the Mathematical Sciences, Vol. 3, Issue 1 https://doi.org/10.1186/s40687-015-0050-9	journal	February 2016
Finite automorphism groups of complex tori of dimension two Fujiki, Akira Publications of the Research Institute for Mathematical Sciences, Vol. 24, Issue 1 https://doi.org/10.2977/prims/1195175326	journal	January 1988
On the Complete Classification of Unitary N = 2 Minimal Superconformal Field Theories Gray, Oliver Communications in Mathematical Physics, Vol. 312, Issue 3 https://doi.org/10.1007/s00220-012-1478-z	journal	May 2012
Note on twisted elliptic genus of K3 surface Eguchi, Tohru; Hikami, Kazuhiro Physics Letters B, Vol. 694, Issue 4-5 https://doi.org/10.1016/j.physletb.2010.10.017	journal	January 2011
A strong coupling test of S-duality Vafa, Cumrun; Witten, Edward Nuclear Physics B, Vol. 431, Issue 1-2 https://doi.org/10.1016/0550-3213(94)90097-3	journal	December 1994
Unitary representations of the N=4 superconformal algebra Eguchi, Tohru; Taormina, Anne Physics Letters B, Vol. 196, Issue 1 https://doi.org/10.1016/0370-2693(87)91679-0	journal	September 1987
Elliptic genera and N = 2 superconformal field theory Kawai, Toshiya; Yamada, Yasuhiko; Yang, Sung-Kil Nuclear Physics B, Vol. 414, Issue 1-2 https://doi.org/10.1016/0550-3213(94)90428-6	journal	February 1994
Quotient d'un Espace Analytique par un Groupe d'Automorphismes Cartan, Henri Algebraic Geometry and Topology https://doi.org/10.1515/9781400879915-007	book	December 1957
String propagation in black hole geometry Dijgraaf, Robbert; Verlinde, Herman; Verlinde, Erik Nuclear Physics B, Vol. 371, Issue 1-2 https://doi.org/10.1016/0550-3213(92)90237-6	journal	March 1992
Space-time supersymmetry in compactified string theory and superconformal models Gepner, Doron Nuclear Physics B, Vol. 296, Issue 4 https://doi.org/10.1016/0550-3213(88)90397-5	journal	January 1988
Elliptic Curves and Modular Forms in Algebraic Topology Landweber, Peter S. Lecture Notes in Mathematics https://doi.org/10.1007/bfb0078035	book	January 1988
On the unitary representations of N=2 and N=4 superconformal algebras Eguchi, Tohru; Taormina, Anne Physics Letters B, Vol. 210, Issue 1-2 https://doi.org/10.1016/0370-2693(88)90360-7	journal	August 1988
Superconformal algebras and string compactification on manifolds with SU(n) holonomy Eguchi, Tohru; Ooguri, Hirosi; Taormina, Anne Nuclear Physics B, Vol. 315, Issue 1 https://doi.org/10.1016/0550-3213(89)90454-9	journal	March 1989
The partition function of the supersymmetric two-dimensional black hole and little string theory Israel, Dan; Kounnas, Costas; Pakman, Ari Journal of High Energy Physics, Vol. 2004, Issue 06 https://doi.org/10.1088/1126-6708/2004/06/033	journal	June 2004
Perspectives in Representation Theory Etingof, Pavel; Khovanov, Mikhail; Savage, Alistair https://doi.org/10.1090/conm/610/12195	book	January 2014
A natural representation of the Fischer-Griess Monster with the modular function J as character Frenkel, I. B.; Lepowsky, J.; Meurman, A. Proceedings of the National Academy of Sciences, Vol. 81, Issue 10 https://doi.org/10.1073/pnas.81.10.3256	journal	May 1984
Duality of the fermionic 2d black hole and N = 2 Liouville theory as mirror symmetry Hori, Kentaro; Kapustin, Anton Journal of High Energy Physics, Vol. 2001, Issue 08 https://doi.org/10.1088/1126-6708/2001/08/045	journal	August 2001
Monstrous Moonshine Conway, J. H.; Norton, S. P. Bulletin of the London Mathematical Society, Vol. 11, Issue 3 https://doi.org/10.1112/blms/11.3.308	journal	October 1979
Monstrous moonshine and monstrous Lie superalgebras Borcherds, Richard E. Inventiones Mathematicae, Vol. 109, Issue 1 https://doi.org/10.1007/BF01232032	journal	December 1992
The non-compact elliptic genus: mock or modular Troost, Jan Journal of High Energy Physics, Vol. 2010, Issue 6 https://doi.org/10.1007/JHEP06(2010)104	journal	June 2010
K3 surfaces, $\mathcal{N}=4$ dyons and the Mathieu group $M_{24}$ Cheng, Miranda C. N. Communications in Number Theory and Physics, Vol. 4, Issue 4 https://doi.org/10.4310/CNTP.2010.v4.n4.a2	journal	January 2010
Mathieu twining characters for K3 Gaberdiel, Matthias R.; Hohenegger, Stefan; Volpato, Roberto Journal of High Energy Physics, Vol. 2010, Issue 9 https://doi.org/10.1007/JHEP09(2010)058	journal	September 2010
Mathieu Moonshine in the elliptic genus of K3 Gaberdiel, Matthias R.; Hohenegger, Stefan; Volpato, Roberto Journal of High Energy Physics, Vol. 2010, Issue 10 https://doi.org/10.1007/JHEP10(2010)062	journal	October 2010
On Rademacher sums, the largest Mathieu group and the holographic modularity of moonshine Cheng, Miranda C. N.; Duncan, John F. R. Communications in Number Theory and Physics, Vol. 6, Issue 3 https://doi.org/10.4310/CNTP.2012.v6.n3.a4	journal	January 2012
Generalized Mathieu Moonshine Gaberdiel, Matthias R.; Persson, Daniel; Ronellenfitsch, Henrik Communications in Number Theory and Physics, Vol. 7, Issue 1 https://doi.org/10.4310/CNTP.2013.v7.n1.a5	journal	January 2013
The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24 Taormina, A.; Wendland, K. Journal of High Energy Physics, Vol. 2013, Issue 8 https://doi.org/10.1007/JHEP08(2013)125	journal	August 2013
Symmetries of K3 sigma models Gaberdiel, Matthias R.; Hohenegger, Stefan; Volpato, Roberto Communications in Number Theory and Physics, Vol. 6, Issue 1 https://doi.org/10.4310/CNTP.2012.v6.n1.a1	journal	January 2012
Mathieu moonshine and $ \mathcal{N}=2 $ string compactifications Cheng, Miranda C. N.; Dong, Xi; Duncan, John F. R. Journal of High Energy Physics, Vol. 2013, Issue 9 https://doi.org/10.1007/JHEP09(2013)030	journal	September 2013
Moonshine in fivebrane spacetimes Harvey, Jeffrey A.; Murthy, Sameer Journal of High Energy Physics, Vol. 2014, Issue 1 https://doi.org/10.1007/JHEP01(2014)146	journal	January 2014
Mathieu moonshine in four dimensional $ \mathcal{N} $ = 1 theories Wrase, Timm Journal of High Energy Physics, Vol. 2014, Issue 4 https://doi.org/10.1007/JHEP04(2014)069	journal	April 2014
The A-D-E classification of minimal andA 1 (1) conformal invariant theories Cappelli, A.; Itzykson, C.; Zuber, J. B. Communications in Mathematical Physics, Vol. 113, Issue 1 https://doi.org/10.1007/BF01221394	journal	March 1987
Kählerian K3 surfaces and Niemeier lattices. I Nikulin, V. V. Izvestiya: Mathematics, Vol. 77, Issue 5 https://doi.org/10.1070/IM2013v077n05ABEH002666	journal	October 2013
String theory dynamics in various dimensions Witten, Edward Nuclear Physics B, Vol. 443, Issue 1-2 https://doi.org/10.1016/0550-3213(95)00158-O	journal	June 1995
Enhanced gauge symmetries and K3 surfaces Aspinwall, Paul S. Physics Letters B, Vol. 357, Issue 3 https://doi.org/10.1016/0370-2693(95)00957-M	journal	September 1995
Geometric engineering of quantum field theories Katz, Sheldon; Klemm, Albrecht; Vafa, Cumrun Nuclear Physics B, Vol. 497, Issue 1-2 https://doi.org/10.1016/S0550-3213(97)00282-4	journal	July 1997
U(1)m modular invariants, N = 2 minimal models, and the quantum Hall effect Gannon, Terry Nuclear Physics B, Vol. 491, Issue 3 https://doi.org/10.1016/S0550-3213(97)00032-1	journal	May 1997
Phases of N = 2 theories in two dimensions Witten, Edward Nuclear Physics B, Vol. 403, Issue 1-2 https://doi.org/10.1016/0550-3213(93)90033-L	journal	August 1993
On the Landau-Ginzburg Description of n=2 Minimal Models Witten, Edward International Journal of Modern Physics A, Vol. 09, Issue 27 https://doi.org/10.1142/S0217751X9400193X	journal	October 1994
Ramond sector characters and N = 2 Landau-Ginzburg models Di Francesco, P.; Yankielowicz, S. Nuclear Physics B, Vol. 409, Issue 1 https://doi.org/10.1016/0550-3213(93)90452-U	journal	November 1993
A twisted non-compact elliptic genus Ashok, Sujay K.; Troost, Jan Journal of High Energy Physics, Vol. 2011, Issue 3 https://doi.org/10.1007/JHEP03(2011)067	journal	March 2011
Localization and real Jacobi forms Ashok, Sujay K.; Doroud, Nima; Troost, Jan Journal of High Energy Physics, Vol. 2014, Issue 4 https://doi.org/10.1007/JHEP04(2014)119	journal	April 2014
A holomorphic anomaly in the elliptic genus Murthy, Sameer Journal of High Energy Physics, Vol. 2014, Issue 6 https://doi.org/10.1007/JHEP06(2014)165	journal	June 2014
Elliptic genera and quantum field theory Witten, Edward Communications in Mathematical Physics, Vol. 109, Issue 4 https://doi.org/10.1007/BF01208956	journal	December 1987
Definite quadratische formen der dimension 24 und diskriminante 1 Niemeier, Hans-Volker Journal of Number Theory, Vol. 5, Issue 2 https://doi.org/10.1016/0022-314X(73)90068-1	journal	April 1973
A characterisation of Leech's lattice Conway, J. H. Inventiones Mathematicae, Vol. 7, Issue 2 https://doi.org/10.1007/BF01389796	journal	June 1969
Notes on Sphere Packings Leech, John Canadian Journal of Mathematics, Vol. 19 https://doi.org/10.4153/CJM-1967-017-0	journal	January 1967
Some Sphere Packings in Higher Space Leech, John Canadian Journal of Mathematics, Vol. 16 https://doi.org/10.4153/CJM-1964-065-1	journal	January 1964
Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces Kond?, Shigeyuki Duke Mathematical Journal, Vol. 92, Issue 3 https://doi.org/10.1215/S0012-7094-98-09217-1	journal	April 1998
Integral Symmetric Bilinear Forms and some of Their Applications Nikulin, V. V. Mathematics of the USSR-Izvestiya, Vol. 14, Issue 1 https://doi.org/10.1070/IM1980v014n01ABEH001060	journal	February 1980
Finite groups of automorphisms of K3 surfaces and the Mathieur group Mukai, Shigeru Inventiones Mathematicae, Vol. 94, Issue 1 https://doi.org/10.1007/BF01394352	journal	February 1988
Space-Time Supersymmetry in Compactified String Theory and Superconformal Models Gepner, Doron Current Physics–Sources and Comments https://doi.org/10.1016/b978-0-444-87492-4.50027-6	book	January 1989
Mathieu twining characters for K3 Gaberdiel, Matthias R.; Hohenegger, Stefan; Volpato, Roberto ETH Zurich https://doi.org/10.3929/ethz-b-000025519	text	January 2010
Mathieu Moonshine in the elliptic genus of K3 R., Gaberdiel, Matthias; Stefan, Hohenegger,; Roberto, Volpato, ETH Zurich https://doi.org/10.3929/ethz-b-000158838	text	January 2010
On the complete classification of the unitary N=2 minimal superconformal field theories Gray, Oliver arXiv https://doi.org/10.48550/arxiv.0812.1318	text	January 2008
Symmetries of K3 sigma models Gaberdiel, Matthias R.; Hohenegger, Stefan; Volpato, Roberto arXiv https://doi.org/10.48550/arxiv.1106.4315	text	January 2011
Kahlerian K3 surfaces and Niemeier lattices Nikulin, Viacheslav V. arXiv https://doi.org/10.48550/arxiv.1109.2879	text	January 2011
Generalised Mathieu Moonshine Gaberdiel, Matthias R.; Persson, Daniel; Ronellenfitsch, Henrik arXiv https://doi.org/10.48550/arxiv.1211.7074	text	January 2012
Mathieu moonshine in four dimensional $\mathcal{N}=1$ theories Wrase, Timm arXiv https://doi.org/10.48550/arxiv.1402.2973	text	January 2014
Derived Equivalences of K3 Surfaces and Twined Elliptic Genera Duncan, John F. R.; Mack-Crane, Sander arXiv https://doi.org/10.48550/arxiv.1506.06198	preprint	January 2015
The partition function of the supersymmetric two-dimensional black hole and little string theory Israel, Dan; Kounnas, Costas; Pakman, Ari arXiv https://doi.org/10.48550/arxiv.hep-th/0403237	text	January 2004
Phases of $N=2$ Theories In Two Dimensions Witten, Edward arXiv https://doi.org/10.48550/arxiv.hep-th/9301042	text	January 1993
A Strong Coupling Test of S-Duality Vafa, Cumrun; Witten, Edward arXiv https://doi.org/10.48550/arxiv.hep-th/9408074	text	January 1994
Little String Theory in a Double Scaling Limit Giveon, A.; Kutasov, D. arXiv https://doi.org/10.48550/arxiv.hep-th/9909110	text	January 1999
Notes on the K3 Surface and the Mathieu group M_24 Eguchi, Tohru; Ooguri, Hirosi; Tachikawa, Yuji arXiv https://doi.org/10.48550/arxiv.1004.0956	text	January 2010
A Twisted Non-compact Elliptic Genus Ashok, Sujay K.; Troost, Jan arXiv https://doi.org/10.48550/arxiv.1101.1059	text	January 2011
The overarching finite symmetry group of Kummer surfaces in the Mathieu group M_24 Taormina, Anne; Wendland, Katrin arXiv https://doi.org/10.48550/arxiv.1107.3834	text	January 2011
D3-instantons, Mock Theta Series and Twistors Alexandrov, Sergei; Manschot, Jan; Pioline, Boris arXiv https://doi.org/10.48550/arxiv.1207.1109	text	January 2012
Ramond Sector Characters and N=2 Landau-Ginzburg Models Di Francesco, P.; Yankielowicz, S. arXiv https://doi.org/10.48550/arxiv.hep-th/9305037	text	January 1993
Fifteen Characterizations of Rational Double Points and Simple Critical Points H., Durfee, Alan Fondation L'Enseignement Mathématique https://doi.org/10.5169/seals-50375	text	January 1979

Cited By (16)

From Moonshine to Mock Moonshine Marathe, Kishore Vietnam Journal of Mathematics, Vol. 47, Issue 1 https://doi.org/10.1007/s10013-018-0310-2	journal	September 2018
K3 Elliptic Genus and an Umbral Moonshine Module Anagiannis, Vassilis; Cheng, Miranda C. N.; Harrison, Sarah M. Communications in Mathematical Physics, Vol. 366, Issue 2 https://doi.org/10.1007/s00220-019-03314-w	journal	March 2019
Squashed Toric Sigma Models and Mock Modular Forms Gupta, Rajesh Kumar; Murthy, Sameer Communications in Mathematical Physics, Vol. 360, Issue 1 https://doi.org/10.1007/s00220-017-3069-5	journal	January 2018
A Borcherds–Kac–Moody Superalgebra with Conway Symmetry Harrison, Sarah M.; Paquette, Natalie M.; Volpato, Roberto Communications in Mathematical Physics, Vol. 370, Issue 2 https://doi.org/10.1007/s00220-019-03518-0	journal	July 2019
K3 string theory, lattices and moonshine Cheng, Miranda C. N.; Harrison, Sarah M.; Volpato, Roberto Research in the Mathematical Sciences, Vol. 5, Issue 3 https://doi.org/10.1007/s40687-018-0150-4	journal	July 2018
Some comments on symmetric orbifolds of K3 Volpato, Roberto Journal of High Energy Physics, Vol. 2019, Issue 10 https://doi.org/10.1007/jhep10(2019)082	journal	October 2019
Derived equivalences of K3 surfaces and twined elliptic genera Duncan, John F. R.; Mack-Crane, Sander Research in the Mathematical Sciences, Vol. 3, Issue 1 https://doi.org/10.1186/s40687-015-0050-9	journal	February 2016
Derived Equivalences of K3 Surfaces and Twined Elliptic Genera Duncan, John F. R.; Mack-Crane, Sander arXiv https://doi.org/10.48550/arxiv.1506.06198	preprint	January 2015
Landau-Ginzburg Orbifolds and Symmetries of K3 CFTs Cheng, Miranda C. N.; Ferrari, Francesca; Harrison, Sarah M. arXiv https://doi.org/10.48550/arxiv.1512.04942	preprint	January 2015
K3 String Theory, Lattices and Moonshine Cheng, Miranda C. N.; Harrison, Sarah M.; Volpato, Roberto arXiv https://doi.org/10.48550/arxiv.1612.04404	preprint	January 2016
K3 Elliptic Genus and an Umbral Moonshine Module Anagiannis, Vassilis; Cheng, Miranda C. N.; Harrison, Sarah M. arXiv https://doi.org/10.48550/arxiv.1709.01952	preprint	January 2017
No More Walls! A Tale of Modularity, Symmetry, and Wall Crossing for 1/4 BPS Dyons Paquette, Natalie M.; Volpato, Roberto; Zimet, Max arXiv https://doi.org/10.48550/arxiv.1702.05095	text	January 2017
Umbral Moonshine and String Duality Zimet, Max arXiv https://doi.org/10.48550/arxiv.1803.07567	preprint	January 2018
TASI Lectures on Moonshine Anagiannis, Vassilis; Cheng, Miranda C. N. arXiv https://doi.org/10.48550/arxiv.1807.00723	preprint	January 2018
Some comments on symmetric orbifolds of K3 Volpato, Roberto arXiv https://doi.org/10.48550/arxiv.1902.11093	text	January 2019
Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the $\mathbb{Z}_2$ Model Fischbach, Fabian; Klemm, Albrecht; Nega, Christoph arXiv https://doi.org/10.48550/arxiv.2005.07712	text	January 2020

Similar Records

3D string theory and Umbral moonshine

Journal Article · Tue Sep 05 00:00:00 EDT 2017 · Journal of Physics. A, Mathematical and Theoretical · OSTI ID:1228046

Kachru, Shamit; Paquette, Natalie M.; Volpato, Roberto

Generalised Umbral Moonshine

Journal Article · Sat Mar 02 00:00:00 EST 2019 · Symmetry, Integrability and Geometry: Methods and Applications · OSTI ID:1228046

Cheng, Miranda C. N.; de Lange, Paul; Whalen, Daniel P. Z.

Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine

Journal Article · Thu Jul 19 00:00:00 EDT 2018 · Journal of High Energy Physics (Online) · OSTI ID:1228046

Cheng, Miranda C. N.; Duncan, John F. R.; Harrison, Sarah M.; +3 more

Related Subjects

97 MATHEMATICS AND COMPUTING

Title: Umbral moonshine and K3 surfaces

Citation Formats

References (80)

Cited By (16)

Similar Records

Related Subjects