Abstract
We show using string dualities that Mathieu moonshine controls Gromov-Witten invariants and periods of the holomorphic 3-form Ω for certain CY 3 manifolds. We also discuss how the period vectors appear in flux compactifications on these CY 3 manifolds and work out the connection between the sporadic group M24 and the Yukawa couplings in four dimensional theories that arise from heterotic string theory compactifications on these CY 3 manifolds.
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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].
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine, Commun. Num. TheorPhys. 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].
M.C.N. Cheng and S. Harrison, Umbral Moonshine and K3 Surfaces, arXiv:1406.0619 [INSPIRE].
D. Persson and R. Volpato, Second Quantized Mathieu Moonshine, arXiv:1312.0622 [INSPIRE].
T. Gannon, Much ado about Mathieu, arXiv:1211.5531 [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].
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, arXiv:1303.2931 [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].
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].
S. Hohenegger and S. Stieberger, BPS Saturated String Amplitudes: K3 Elliptic Genus and Igusa Cusp Form, Nucl. Phys. B 856 (2012) 413 [arXiv:1108.0323] [INSPIRE].
J.A. Harvey and S. Murthy, Moonshine in Fivebrane Spacetimes, JHEP 01 (2014) 146 [arXiv:1307.7717] [INSPIRE].
T. Wrase, Mathieu moonshine in four dimensional \( \mathcal{N}=1 \) theories, JHEP 04 (2014) 069 [arXiv:1402.2973] [INSPIRE].
B.R. Greene and M.R. Plesser, Duality in Calabi-Yau Moduli Space, Nucl. Phys. B 338 (1990) 15 [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].
D. Lüst, String vacua with N = 2 supersymmetry in four-dimensions, hep-th/9803072 [INSPIRE].
A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
M. Alim and E. Scheidegger, Topological Strings on Elliptic Fibrations, arXiv:1205.1784 [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [INSPIRE].
A. Giryavets, S. Kachru, P.K. Tripathy and S.P. Trivedi, Flux compactifications on Calabi-Yau threefolds, JHEP 04 (2004) 003 [hep-th/0312104] [INSPIRE].
M. Graña, Flux compactifications in string theory: A Comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].
T.W. Grimm and J. Louis, The effective action of N = 1 Calabi-Yau orientifolds, Nucl. Phys. B 699 (2004) 387 [hep-th/0403067] [INSPIRE].
D. Robbins and T. Wrase, D-terms from generalized NS-NS fluxes in type-II, JHEP 12 (2007) 058 [arXiv:0709.2186] [INSPIRE].
S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].
J.H. Conway and S.P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979) 308.
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].
J. Polchinski, String theory. Vol. 2: Superstring theory and beyond, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2005).
B. de Wit, V. Kaplunovsky, J. Louis and D. Lüst, Perturbative couplings of vector multiplets in N = 2 heterotic string vacua, Nucl. Phys. B 451 (1995) 53 [hep-th/9504006] [INSPIRE].
I. Antoniadis, S. Ferrara, E. Gava, K.S. Narain and T.R. Taylor, Perturbative prepotential and monodromies in N = 2 heterotic superstring, Nucl. Phys. B 447 (1995) 35 [hep-th/9504034] [INSPIRE].
K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [INSPIRE].
M. Alim, M. Hecht, P. Mayr and A. Mertens, Mirror Symmetry for Toric Branes on Compact Hypersurfaces, JHEP 09 (2009) 126 [arXiv:0901.2937] [INSPIRE].
S. Katz, A. Klemm and R. Pandharipande, On the motivic stable pairs invariants of K3 surfaces, arXiv:1407.3181 [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995) 501 [hep-th/9406055] [INSPIRE].
S. Hosono, A. Klemm and S. Theisen, Lectures on mirror symmetry, in Integrable models and strings, Springer, Berlin Germany (1994), pg. 235.
K. Hori, Mirror symmetry. Vol. 1, American Mathematical Society Press, Providence U.S.A. (2003).
P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [INSPIRE].
B.R. Greene, M. Plesser and S. Roan, New constructions of mirror manifolds: Probing moduli space far from fermat points, AMS/IP Stud. Adv. Math. 9 (1998) 347.
V.V. Batyrev, Dual polyhedra and mirror symmetry for calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003].
D.R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, alg-geom/9202026.
A. Klemm, Instanton, http://www.th.physik.uni-bonn.de/th/People/netah/cy/codes/inst.m.
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Paquette, N.M., Wrase, T. Comments on M24 representations and CY 3 geometries. J. High Energ. Phys. 2014, 155 (2014). https://doi.org/10.1007/JHEP11(2014)155
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DOI: https://doi.org/10.1007/JHEP11(2014)155