Abstract
A few years ago a connection between the elliptic genus of the K3 manifold and the largest Mathieu group M24 was proposed. We study the elliptic genera for Calabi-Yau manifolds of larger dimensions and discuss potential connections between the expansion coefficients of these elliptic genera and sporadic groups. While the Calabi-Yau 3-fold case is rather uninteresting, the elliptic genera of certain Calabi-Yau d-folds for d > 3 have expansions that could potentially arise from underlying sporadic symmetry groups. We explore such potential connections by calculating twined elliptic genera for a large number of Calabi-Yau 5-folds that are hypersurfaces in weighted projected spaces, for a toroidal orbifold and two Gepner models.
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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.C.N. Cheng, K3 Surfaces, N = 4 Dyons and the Mathieu Group M24, 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].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu Moonshine in the elliptic genus of K3, JHEP 10 (2010) 062 [arXiv:1008.3778] [INSPIRE].
T. Eguchi and K. Hikami, Note on twisted elliptic genus of K3 surface, Phys. Lett. B 694 (2011) 446 [arXiv:1008.4924] [INSPIRE].
T. Gannon, Much ado about Mathieu, Adv. Math. 301 (2016) 322 [arXiv:1211.5531] [INSPIRE].
S. Govindarajan and K. Gopala Krishna, BKM Lie superalgebras from dyon spectra in Z(N) CHL orbifolds for composite N, JHEP 05 (2010) 014 [arXiv:0907.1410] [INSPIRE].
S. Govindarajan, BKM Lie superalgebras from counting twisted CHL dyons, JHEP 05 (2011) 089 [arXiv:1006.3472] [INSPIRE].
J.F.R. Duncan, M.J. Griffin and K. Ono, Moonshine, arXiv:1411.6571 [INSPIRE].
S. Kachru, Elementary introduction to Moonshine, arXiv:1605.00697 [INSPIRE].
S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988) 183.
S. Kondo, Niemeier lattices, mathieu groups, and finite groups of symplectic automorphisms of k3 surfaces, Duke Math. J. 92 (1998) 593.
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].
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 M24 moonshine story, arXiv:1303.3221 [INSPIRE].
M.R. Gaberdiel, C.A. Keller and H. Paul, Mathieu Moonshine and Symmetry Surfing, J. Phys. A 50 (2017) 474002 [arXiv:1609.09302] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Symmetries of K3 σ-models, Commun. Num. Theor. Phys. 6 (2012) 1 [arXiv:1106.4315] [INSPIRE].
M.C.N. Cheng, S.M. Harrison, R. Volpato and M. Zimet, K3 String Theory, Lattices and Moonshine, arXiv:1612.04404 [INSPIRE].
N.M. Paquette, R. Volpato and M. Zimet, No More Walls! A Tale of Modularity, Symmetry and Wall Crossing for 1/4 BPS Dyons, JHEP 05 (2017) 047 [arXiv:1702.05095] [INSPIRE].
S. Kachru, N.M. Paquette and R. Volpato, 3D String Theory and Umbral Moonshine, J. Phys. A 50 (2017) 404003 [arXiv:1603.07330] [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.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine and the Niemeier Lattices, arXiv:1307.5793 [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].
M.M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston U.S.A. (1985).
V. Gritsenko, Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms, math/9906190 [INSPIRE].
T. Eguchi and K. Hikami, N = 2 Superconformal Algebra and the Entropy of Calabi-Yau Manifolds, Lett. Math. Phys. 92 (2010) 269 [arXiv:1003.1555] [INSPIRE].
E. Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525.
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].
T. Eguchi and K. Hikami, Enriques moonshine, J. Phys. A 46 (2013) 312001 [arXiv:1301.5043] [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].
T. Wrase, Mathieu moonshine in four dimensional \( \mathcal{N} \) = 1 theories, JHEP 04 (2014) 069 [arXiv:1402.2973] [INSPIRE].
N.M. Paquette and T. Wrase, Comments on M 24 representations and CY 3 geometries, JHEP 11 (2014) 155 [arXiv:1409.1540] [INSPIRE].
S. Datta, J.R. David and D. Lüst, Heterotic string on the CHL orbifold of K3, JHEP 02 (2016) 056 [arXiv:1510.05425] [INSPIRE].
A. Chattopadhyaya and J.R. David, \( \mathcal{N} \) = 2 heterotic string compactifications on orbifolds of K3×T 2, JHEP 01 (2017) 037 [arXiv:1611.01893] [INSPIRE].
A. Chattopadhyaya and J.R. David, Dyon degeneracies from Mathieu moonshine symmetry, Phys. Rev. D 96 (2017) 086020 [arXiv:1704.00434] [INSPIRE].
T. Eguchi and K. Hikami, N=2 Moonshine, Phys. Lett. B 717 (2012) 266 [arXiv:1209.0610] [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].
M.C.N. Cheng, S.M. Harrison, S. Kachru and D. Whalen, Exceptional Algebra and Sporadic Groups at c = 12, arXiv:1503.07219 [INSPIRE].
I.B. Frenkel, J. Lepowsky and A. Meurman, A Moonshine Module for the Monster, in Vertex Operators in Mathematics and Physics , Springer, New York U.S.A. (1985), pg. 231.
J.F. Duncan, Super-moonshine for Conway’s largest sporadic group, math/0502267.
J.F.R. Duncan, M.J. Griffin and K. Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [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].
M. Kreuzer and H. Skarke, Calabi-Yau data, http://hep.itp.tuwien.ac.at/∼kreuzer/CY/.
A.P. Braun, J. Knapp, E. Scheidegger, H. Skarke and N.-O. Walliser, PALP — a User Manual, in Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer , A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger eds., World Scientific, Singapore (2012), pg. 461 [arXiv:1205.4147].
M. Kreuzer and H. Skarke, No mirror symmetry in Landau-Ginzburg spectra!, Nucl. Phys. B 388 (1992) 113 [hep-th/9205004] [INSPIRE].
C. Vafa, Quantum Symmetries of String Vacua, Mod. Phys. Lett. A 4 (1989) 1615 [INSPIRE].
S. Kachru and A. Tripathy, The Hodge-elliptic genus, spinning BPS states and black holes, Commun. Math. Phys. 355 (2017) 245 [arXiv:1609.02158] [INSPIRE].
K. Wendland, Hodge-elliptic genera and how they govern K3 theories, arXiv:1705.09904 [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].
M.C.N. Cheng, F. Ferrari, S.M. Harrison and N.M. Paquette, Landau-Ginzburg Orbifolds and Symmetries of K3 CFTs, JHEP 01 (2017) 046 [arXiv:1512.04942] [INSPIRE].
S. Odake, Extension of N = 2 Superconformal Algebra and Calabi-Yau Compactification, Mod. Phys. Lett. A 4 (1989) 557 [INSPIRE].
S. Odake, Character Formulas of an Extended Superconformal Algebra Relevant to String Compactification, Int. J. Mod. Phys. A 5 (1990) 897 [INSPIRE].
S. Odake, C = 3-d Conformal Algebra With Extended Supersymmetry, Mod. Phys. Lett. A 5 (1990)561 [INSPIRE].
T. Eguchi and A. Taormina, Character Formulas for the N = 4 Superconformal Algebra, Phys. Lett. B 200 (1988) 315 [INSPIRE].
T. Eguchi and K. Hikami, N = 4 Superconformal Algebra and the Entropy of HyperKähler Manifolds, JHEP 02 (2010) 019 [arXiv:0909.0410] [INSPIRE].
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Banlaki, A., Chowdhury, A., Kidambi, A. et al. Calabi-Yau manifolds and sporadic groups. J. High Energ. Phys. 2018, 129 (2018). https://doi.org/10.1007/JHEP02(2018)129
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DOI: https://doi.org/10.1007/JHEP02(2018)129