A twist in the M 24 moonshine story
Confluentes Mathematici, Volume 7 (2015) no. 1, p. 83-113

Prompted by the Mathieu Moonshine observation, we identify a pair of 45-dimensional vector spaces of states that account for the first order term in the massive sector of the elliptic genus of K3 in every 2 -orbifold CFT on K3. These generic states are uniquely characterized by the fact that the action of every geometric symmetry group of a 2 -orbifold CFT yields a well-defined faithful representation on them. Moreover, each such representation is obtained by restriction of the 45-dimensional irreducible representation of the Mathieu group M 24 constructed by Margolin. Thus we provide a piece of evidence for Mathieu Moonshine explicitly from SCFTs on K3.

The 45-dimensional irreducible representation of M 24 exhibits a twist, which we prove can be undone in the case of 2 -orbifold CFTs on K3 for all geometric symmetry groups. This twist however cannot be undone for the combined symmetry group ( 2 ) 4 A 8 that emerges from surfing the moduli space of Kummer K3s. We conjecture that in general, the untwisted representations are exclusively those of geometric symmetry groups in some geometric interpretation of a CFT on K3. In that light, the twist appears as a representation theoretic manifestation of the maximality constraints in Mukai’s classification of geometric symmetry groups of K3.

DOI : https://doi.org/10.5802/cml.19
Classification:  81T40,  81T60,  14J28
@article{CML_2015__7_1_83_0,
     author = {Taormina, Anne and Wendland, Katrin},
     title = {A twist in the $M\_{24}$ moonshine story},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {7},
     number = {1},
     year = {2015},
     pages = {83-113},
     doi = {10.5802/cml.19},
     language = {en},
     url = {http://www.numdam.org/item/CML_2015__7_1_83_0}
}
Taormina, Anne; Wendland, Katrin. A twist in the $M_{24}$ moonshine story. Confluentes Mathematici, Volume 7 (2015) no. 1, pp. 83-113. doi : 10.5802/cml.19. http://www.numdam.org/item/CML_2015__7_1_83_0/

[1] O. Alvarez, T.P. Killingback, M. Mangano, and P. Windey. String theory and loop space index theorems, Comm. Math. Phys. 111:1–12, 1987. | MR 896755 | Zbl 0627.58033

[2] P.S. Aspinwall and D.R. Morrison. String theory on K3 surfaces, in: Mirror symmetry II, B. Greene and S.T. Yau, eds., AMS, 1994, pp. 703–716; hep-th/9404151. | MR 1416354 | Zbl 0931.14020

[3] J.H. Conway, R.T. Curtis, S.P. Norton, R.W. Parker, and R.A. Wilson. Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, with computational assistance from J. G. Thackray. | MR 827219 | Zbl 0568.20001

[4] M.C.N. Cheng. K3 surfaces, N=4 dyons, and the Mathieu group M 24 , Commun. Number Theory Phys. 4:623, 2010; arXiv:1005.5415[hep-th]. | MR 2793423 | Zbl 1231.14031

[5] A. M. Cohen. Finite complex reflection groups, Ann. Sci. École Norm. Sup., 4:379, 1976. | Numdam | MR 422448 | Zbl 0359.20029

[6] A. Dabholkar, S. Murthy and D. Zagier. Quantum black holes, wall crossing and mock modular forms; arXiv:1208.4074 [hep-th].

[7] T. Eguchi and K. Hikami. Note on Twisted Elliptic Genus of K3 Surface, Phys. Lett. B694:446–455, 2011; arXiv1008.4924 [hep-th]. | MR 2748168

[8] Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa. Notes on K3 surface and the Mathieu group M 24 , Exper. Math. 20:91–96, 2011; arXiv:1004.0956 [hep-th]. | MR 2802725 | Zbl 1266.58008

[9] T. Eguchi, H. Ooguri, A. Taormina, and S.-K. Yang. Superconformal algebras and string compactification on manifolds with SU(n) holonomy, Nucl. Phys. B315:193–221, 1989. | MR 985505

[10] T. Eguchi and A. Taormina. Extended superconformal algebras and string compactifications, in: Trieste School 1988: Superstrings, pp. 167–188. | MR 1091555

[11] J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert. Defect lines, dualities, and generalised orbifolds, in: Proc. XVIth ICMP (2010), World Sci. Publ., Hackensack NJ, pp. 608–613. | MR 2730830

[12] T. Gannon. Much ado about Mathieu; arXiv:1211.5531 [math.RT].

[13] M.R. Gaberdiel, S. Hohenegger, and R. Volpato. Mathieu moonshine in the elliptic genus of K3, J. High Ener. Phys. 1010:062, 2010; arXiv:1008.3778 [hep-th]. | MR 2780524 | Zbl 1291.81312

[14] —. Mathieu twining characters for K3, J. High Ener. Phys. 1009:058, 2010; arXiv:1006.0221 [hep-th]. | MR 2776956

[15] M.R. Gaberdiel, S. Hohenegger and R. Volpato. Symmetries of K3 and sigma models, Comm. Numb. Th. Phys. 6:1–50, 2012. | MR 2955931 | Zbl 1275.81085

[16] M.R. Gaberdiel, D. Persson, H. Ronellenfitsch, and R. Volpato. Generalised Mathieu Moonshine; arXiv:1211.7074 [hep-th].

[17] M.R. Gaberdiel, D. Persson, and R. Volpato. Generalised Moonshine and Holomorphic Orbifolds; arXiv:1302.5425 [hep-th].

[18] V. Kac. Vertex algebras for beginners, University Lecture Series 10, AMS 1998. | MR 1651389 | Zbl 0861.17017

[19] A. Kapustin and D. Orlov. Vertex algebras, mirror symmetry, and D-branes: the case of complex tori, Comm. Math. Phys. 233:79–136, 2003. | MR 1957733 | Zbl 1051.17017

[20] F. Klein. Über die Transformation siebenter Ordnung der elliptische Funktionen. in: Gesammelte Math. Abhandlungen III, 1878, p. 90.

[21] S. Kondo. Niemeier lattices, Mathieu groups and finite groups of symplectic automorphisms of K3 surfaces, Duke Math. J. 92:593–603, 1998, Appendix by S. Mukai. | MR 1620514 | Zbl 0958.14025

[22] W. Lerche, C. Vafa, and N.P. Warner. Chiral rings in N=2 superconformal theories, Nucl. Phys. B324:427–474, 1989. | MR 1025424

[23] R. S. Margolin. A geometry for M 24 , J. Algebra 156:370, 1993. | MR 1216474 | Zbl 0791.20010

[24] S. Mukai. Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94:183–221, 1988. | MR 958597 | Zbl 0705.14045

[25] 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. 216:85–138, 2001; hep-th/9912067. | MR 1810775 | Zbl 1018.81048

[26] H. Ooguri. Superconformal symmetry and geometry of Ricci flat Kähler manifolds, Int. J. Mod. Phys. A4:4303, 1989. | MR 1018479

[27] A. Taormina and K. Wendland. The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24 , J. High Ener. Phys. 08:125, 2013; arXiv:1107.3834 [hep-th]. | MR 3106313

[28] —. Symmetry-surfing the moduli space of Kummer K3s; arXiv:1303.2931 [hep-th].

[29] K. Wendland. Moduli spaces of unitary conformal field theories. Ph.D. thesis, University of Bonn, 2000.

[30] K. Wendland. On the geometry of singularities in quantum field theory, in: Proc. of the ICM 2010, Hyderabad, Hindustan Book Agency, 2010, pp. 2144-2170. | MR 2827880 | Zbl 1226.14020

[31] K. Wendland. Snapshots of conformal field theory; in: Mathematical Aspects of Quantum Field Theories, D. Calaque and Th. Strobl, eds., Mathematical Physics Studies, Springer 2015, pp. 89–129; arXiv:1404.3108 [hep-th]. | MR 3330240

[32] R. A. Wilson. The geometry of the Hall-Janko group as quaternionic reflection group, Geom. Dedic. 20:157, 1986. | MR 833844 | Zbl 0589.51023

[33] E. Witten. Elliptic genera and quantum field theory, Commun. Math. Phys. 109:525–536, 1987. | MR 885560 | Zbl 0625.57008