Abstract
Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3ellipticgeneratwinedbydiscretesymmetriesthataremanifestintheUVdescription, though often inaccessible in the IR. We obtain various twining functions coinciding with moonshine predictions that have not been observed in physical theories before. These include twining functions arising from Mathieu moonshine, other cases of umbral moonshine, and Conway moonshine. For instance, all functions arising from M 11 ⊂ 2.M 12 moonshine appear as explicit twining genera in the LG models, which moreover admit a uniform description in terms of its natural 12-dimensional representation. Our results provide strong evidence for the relevance of umbral moonshine for K3 symmetries, as well as new hints for its eventual explanation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 Surface and the Mathieu group M 24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu Moonshine in the elliptic genus of K3, JHEP 10 (2010) 062 [arXiv:1008.3778] [INSPIRE].
A. Taormina and K. Wendland, The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24, JHEP 08 (2013) 125 [arXiv:1107.3834] [INSPIRE].
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine, Commun. Num. Theor. Phys. 08 (2014) 101 [arXiv:1204.2779] [INSPIRE].
M.R. Gaberdiel and R. Volpato, Mathieu Moonshine and orbifold K3s, Contrib. Math. Comput. Sci. 8 (2014) 109 [arXiv:1206.5143] [INSPIRE].
T. Creutzig, G. Hoehn and T. Miezaki, The McKay-Thompson series of Mathieu Moonshine modulo two, arXiv:1211.3703.
M.R. Gaberdiel, D. Persson, H. Ronellenfitsch and R. Volpato, Generalized Mathieu Moonshine, Commun. Num. Theor Phys. 07 (2013) 145 [arXiv:1211.7074] [INSPIRE].
M.R. Gaberdiel, D. Persson and R. Volpato, Generalised Moonshine and holomorphic orbifolds, Proc. Symp. Pure Math. 90 (2015) 73 [arXiv:1302.5425] [INSPIRE].
A. Taormina and K. Wendland, Symmetry-surfing the moduli space of Kummer K3s, Proc. Symp. Pure Math. 90 (2015) 129 [arXiv:1303.2931] [INSPIRE].
A. Taormina and K. Wendland, A twist in the M 24 moonshine story, arXiv:1303.3221 [INSPIRE].
M.C.N. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru and T. Wrase, Mathieu moonshine and N = 2 string compactifications, JHEP 09 (2013) 030 [arXiv:1306.4981] [INSPIRE].
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine and the Niemeier lattices, arXiv:1307.5793 [INSPIRE].
J.A. Harvey and S. Murthy, Moonshine in fivebrane spacetimes, JHEP 01 (2014) 146 [arXiv:1307.7717] [INSPIRE].
S. Harrison, S. Kachru and N.M. Paquette, Twining genera of (0, 4) supersymmetric σ-models on K3, JHEP 04 (2014) 048 [arXiv:1309.0510] [INSPIRE].
T. Creutzig and G. Höhn, Mathieu Moonshine and the geometry of K3 surfaces, Commun. Num. Theor. Phys. 08 (2014) 295 [arXiv:1309.2671] [INSPIRE].
M.R. Gaberdiel, A. Taormina, R. Volpato and K. Wendland, A K3 σ-model with \( {\mathbb{Z}}_2^8:{\mathbb{M}}_{20} \) symmetry, JHEP 02 (2014) 022 [arXiv:1309.4127] [INSPIRE].
D. Persson and R. Volpato, Second quantized Mathieu Moonshine, Commun. Num. Theor. Phys. 08 (2014) 403 [arXiv:1312.0622] [INSPIRE].
T. Wrase, Mathieu moonshine in four dimensional \( \mathcal{N}=1 \) theories, JHEP 04 (2014) 069 [arXiv:1402.2973] [INSPIRE].
K. Ono, L. Rolen and S. Trebat-Leder, Classical and umbral Moonshine: connections and p-adic properties, J. Ramanujan Math. Soc. 30 (2015) 135 [arXiv:1403.3712] [INSPIRE].
M.C.N. Cheng, X. Dong, J.F.R. Duncan, S. Harrison, S. Kachru and T. Wrase, Mock modular Mathieu Moonshine modules, arXiv:1406.5502 [INSPIRE].
N.M. Paquette and T. Wrase, Comments on M 24 representations and CY 3 geometries, JHEP 11 (2014) 155 [arXiv:1409.1540] [INSPIRE].
J.A. Harvey, S. Murthy and C. Nazaroglu, ADE double scaled little string theories, Mock modular forms and umbral Moonshine, JHEP 05 (2015) 126 [arXiv:1410.6174] [INSPIRE].
J.F.R. Duncan, M.J. Griffin and K. Ono, Moonshine, arXiv:1411.6571 [INSPIRE].
N. Benjamin, S.M. Harrison, S. Kachru, N.M. Paquette and D. Whalen, On the elliptic genera of manifolds of Spin(7) holonomy, Annales Henri Poincaré 17 (2016) 2663 [arXiv:1412.2804] [INSPIRE].
M.C.N. Cheng, S.M. Harrison, S. Kachru and D. Whalen, Exceptional algebra and sporadic groups at c = 12, arXiv:1503.07219 [INSPIRE].
J.F.R. Duncan and S. Mack-Crane, Derived equivalences of K3 surfaces and twined elliptic genera, arXiv:1506.06198 [INSPIRE].
T. Gannon, Much ado about Mathieu, Adv. Math. 301 (2016) 322 [arXiv:1211.5531] [INSPIRE].
J.F.R. Duncan, M.J. Griffin and K. Ono, Proof of the umbral Moonshine conjecture, arXiv:1503.01472 [INSPIRE].
M.C.N. Cheng, K3 surfaces, N = 4 dyons and the Mathieu group M 24, Commun. Num. Theor. Phys. 4 (2010) 623 [arXiv:1005.5415] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu twining characters for K3, JHEP 09 (2010) 058 [arXiv:1006.0221] [INSPIRE].
T. Eguchi and K. Hikami, Note on twisted elliptic genus of K3 surface, Phys. Lett. B 694 (2011) 446 [arXiv:1008.4924] [INSPIRE].
M.C.N. Cheng and J.F.R. Duncan, On Rademacher sums, the largest Mathieu group and the holographic modularity of Moonshine, Commun. Num. Theor. Phys. 6 (2012) 697 [arXiv:1110.3859] [INSPIRE].
M.C.N. Cheng and J.F.R. Duncan, The largest Mathieu group and (Mock) automorphic forms, arXiv:1201.4140 [INSPIRE].
V.V. Nikulin, Kählerian K3 Surfaces and Niemeier Lattices, Izv. Math. 77 (2013) 954 [arXiv:1109.2879].
M.C.N. Cheng and S. Harrison, Umbral Moonshine and K3 surfaces, Commun. Math. Phys. 339 (2015) 221 [arXiv:1406.0619] [INSPIRE].
S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Inv. Math, 94 (1988) 1.
S. Kondo, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces, Duke Math. J. 92 (1998) 593.
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Symmetries of K3 σ-models, Commun. Num. Theor. Phys. 6 (2012) 1 [arXiv:1106.4315] [INSPIRE].
D. Huybrechts, On derived categories of K3 surfaces and Mathieu groups, arXiv:1309.6528 [INSPIRE].
M.C.N. Cheng, J.F.R. Duncan, S.M. Harrison and S. Kachru, Equivariant K3 invariants, arXiv:1508.02047 [INSPIRE].
M.C.N. Cheng, S.M. Harrison, R. Volpato and M. Zimet, K3 string theory, lattices and moonshine, arXiv:1612.04404 [INSPIRE].
I.B. Frenkel, J. Lepowsky and A. Meurman, A moonshine module for the Monster, in Vertex operators in mathematics and physics, J. Lepowski et al., Springer, Germany (1985).
I. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, volume 134, Academic Press Inc., Boston U.S.A. (1988).
J.F. Duncan, Super-moonshine for Conway’s largest sporadic group, Duke Math. J. 139 (2007) 255 [math/0502267].
J.F.R. Duncan and S. Mack-Crane, The Moonshine module for Conway’s group, SIGMA 3 (2015) e10 [arXiv:1409.3829] [INSPIRE].
E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
D.A. Kastor, E.J. Martinec and S.H. Shenker, RG flow in N = 1 discrete series, Nucl. Phys. B 316 (1989) 590 [INSPIRE].
S. Cecotti, N = 2 Landau-Ginzburg versus Calabi-Yau σ-models: nonperturbative aspects, Int. J. Mod. Phys. A 6 (1991) 1749 [INSPIRE].
P. Fendley and K.A. Intriligator, Scattering and thermodynamics in integrable N = 2 theories, Nucl. Phys. B 380 (1992) 265 [hep-th/9202011] [INSPIRE].
O. Gray, On the complete classification of the unitary N = 2 minimal superconformal field theories, Commun. Math. Phys. 312 (2012) 611 [arXiv:0812.1318] [INSPIRE].
A. Cappelli, C. Itzykson and J.B. Zuber, Modular invariant partition functions in two-dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].
D. Gepner and Z.-a. Qiu, Modular invariant partition functions for parafermionic field theories, Nucl. Phys. B 285 (1987) 423 [INSPIRE].
E.J. Martinec, Algebraic geometry and effective lagrangians, Phys. Lett. B 217 (1989) 431 [INSPIRE].
C. Vafa and N.P. Warner, Catastrophes and the classification of conformal theories, Phys. Lett. B 218 (1989) 51 [INSPIRE].
A.N. Schellekens and N.P. Warner, Anomalies and modular invariance in string theory, Phys. Lett. B 177 (1986) 317 [INSPIRE].
A.N. Schellekens and N.P. Warner, Anomaly cancellation and selfdual lattices, Phys. Lett. B 181 (1986) 339 [INSPIRE].
A.N. Schellekens and N.P. Warner, Anomalies, characters and strings, Nucl. Phys. B 287 (1987) 317 [INSPIRE].
T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].
V. Gritsenko, Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms, math/9906190 [INSPIRE].
V.K. Dobrev, Characters of the unitarizable highest weight modules over the N = 2 superconformal algebras, Phys. Lett. B 186 (1987) 43 [INSPIRE].
E. Kiritsis, Character formulae and the structure of the representations of the N = 1, N = 2 superconformal algebras, Int. J. Mod. Phys. A 3 (1988) 1871 [INSPIRE].
P. Di Francesco and S. Yankielowicz, Ramond sector characters and N = 2 Landau-Ginzburg models, Nucl. Phys. B 409 (1993) 186 [hep-th/9305037] [INSPIRE].
D. Gepner, Space-time supersymmetry in compactified string theory and superconformal models, Nucl. Phys. B 296 (1988) 757 [INSPIRE].
B.R. Greene, C. Vafa and N.P. Warner, Calabi-Yau manifolds and renormalization group flows, Nucl. Phys. B 324 (1989) 371 [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
V. Gorbounov and F. Malikov, Vertex algebras and the Landau-Ginzburg/Calabi-Yau correspondence, Mosc. Math. J. 4 (2004) 729.
L.A. Borisov, Vertex algebras and mirror symmetry, Commun. Math. Phys. 215 (2001) 517 [math/9809094] [INSPIRE].
T. Eguchi, H. Ooguri, A. Taormina and S.-K. Yang, Superconformal algebras and string compactification on manifolds with SU(N) holonomy, Nucl. Phys. B 315 (1989) 193 [INSPIRE].
P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, in Mirror symmetry II, B. Greene and S.T. Yau eds., American Mathematical Society, U.S.A. (2001), hep-th/9404151 [INSPIRE].
P.S. Aspinwall, Enhanced gauge symmetries and K3 surfaces, Phys. Lett. B 357 (1995) 329 [hep-th/9507012] [INSPIRE].
P.S. Aspinwall, K3 surfaces and string duality, in Differential geometry inspired by string theory, S.T. Yau ed., International Press of Boston, U.S.A. (1999), hep-th/9611137 [INSPIRE].
W. Nahm and K. Wendland, A Hiker’s guide to K3: aspects of N = (4, 4) superconformal field theory with central charge c = 6, Commun. Math. Phys. 216 (2001) 85 [hep-th/9912067] [INSPIRE].
J.H. Conway, Atlas of finite groups: maximal subgroups and ordinary characters for simple groups, Oxford University Press, Oxford U.K. (1985).
R.T. Curtis, On subgroups of ·0. I. Lattice stabilizers, J. Algebra 27 (1973) 549.
J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Springer, Germany (2013).
G. Hoehn and G. Mason, Finite groups of symplectic automorphisms of hyper-Kähler manifolds of type K3[2], arXiv:1409.6055 [INSPIRE].
B.R. Greene, M.R. Plesser and S.S. Roan, New constructions of mirror manifolds: Probing moduli space far from Fermat points, in Mirror Symmetry I, S.T. Yau ed., American Mathematical Society, U.S.A. (1993).
S. Levy, The eightfold way: the beauty of Klein’s quartic curve, Cambridge University Press, Cambridge U.K. (2001).
K. Frantzen, K3-surfaces with special symmetry, Ph.D. thesis, Ruhr-Universität Bochum, Bochum, Germany (2008), arXiv:0902.3761.
M.C.N. Cheng, F. Ferrari, S.M. Harrison, D. Israël and N.M. Paquette, work in progress.
J. Distler and S. Kachru, (0, 2) Landau-Ginzburg theory, Nucl. Phys. B 413 (1994) 213 [hep-th/9309110] [INSPIRE].
A. Beauville and R. Donagi, La variété des droites d’une hypersurface cubique de dimension 4, CR Acad. Sci. Paris Sér. I 301 (1985) 703.
L. Fu, Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds, arXiv:1303.2241.
G. Mongardi, Automorphisms of Hyperkähler manifolds, Ph.D. thesis, University of Roma Tre, Rome, Italy (2013), arXiv:1303.4670.
D. Huybrechts, The K3 category of a cubic fourfold, arXiv:1505.01775.
R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE].
A. Adler, On the automorphism group of a certain cubic threefold, Amer. J. Math. 100 (1978) 1275.
J.P. Smith, Picard-Fuchs differential equations for families of K3 surfaces, Ph.D. thesis, University of Warwick, Warwick, U.K. (2006), arXiv:0705.3658.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1512.04942
On leave from CNRS, France (Miranda C.N. Cheng).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Cheng, M.C., Ferrari, F., Harrison, S.M. et al. Landau-Ginzburg orbifolds and symmetries of K3 CFTs. J. High Energ. Phys. 2017, 46 (2017). https://doi.org/10.1007/JHEP01(2017)046
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2017)046