Abstract
We show that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi-Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skew-holomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This suggests a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.
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References
A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 Surface and the Mathieu Group M24, Exper. Math. 20 (2011) 91.
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Umbral Moonshine and the Niemeier Lattices, Res. Math. Sci. 1 (2014) 3 [arXiv:1307.5793] [INSPIRE].
M.C.N. Cheng, J.F.R. Duncan and J.A. Harvey, Weight One Jacobi Forms and Umbral Moonshine, J. Phys. A 51 (2018) 104002 [arXiv:1703.03968] [INSPIRE].
N.-P. Skoruppa, Developments in the theory of Jacobi forms, in Automorphic functions and their applications, Khabarovsk (1988), pp. 167-185.
N.-P. Skoruppa, Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms, Invent. Math. 102 (1990) 501.
J.A. Harvey and B.C. Rayhaun, Traces of Singular Moduli and Moonshine for the Thompson Group, Commun. Num. Theor. Phys. 10 (2016) 23 [arXiv:1504.08179] [INSPIRE].
J.F.R. Duncan, J.A. Harvey and B.C. Rayhaun, Skew-Holomorphic Jacobi Forms and Moonshine, in preparation.
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].
D. Gaiotto, A. Strominger and X. Yin, The M5-Brane Elliptic Genus: Modularity and BPS States, JHEP 08 (2007) 070 [hep-th/0607010] [INSPIRE].
J. de Boer, M.C.N. Cheng, R. Dijkgraaf, J. Manschot and E. Verlinde, A Farey Tail for Attractor Black Holes, JHEP 11 (2006) 024 [hep-th/0608059] [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
J. Manschot and G.W. Moore, A Modern Farey Tail, Commun. Num. Theor. Phys. 4 (2010) 103 [arXiv:0712.0573] [INSPIRE].
M. Alim, B. Haghighat, M. Hecht, A. Klemm, M. Rauch and T. Wotschke, Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes, Commun. Math. Phys. 339 (2015) 773 [arXiv:1012.1608] [INSPIRE].
M.C.N. Cheng and J.F.R. Duncan, Optimal Mock Jacobi Theta Functions, arXiv:1605.04480 [INSPIRE].
M.C.N. Cheng and J.F.R. Duncan, Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules, arXiv:1707.01336 [INSPIRE].
N.-P. Skoruppa and D. Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988) 113.
N.-P. Skoruppa, Heegner cycles, modular forms and Jacobi forms, J. Théor. Nombres Bordeaux 3 (1991) 93.
J. Duncan, M. Mertens and K. Ono, O’Nan moonshine and arithmetic, arXiv:1702.03516.
G.W. Moore, Arithmetic and attractors, hep-th/9807087 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997).
G.W. Moore and N. Seiberg, Naturality in Conformal Field Theory, Nucl. Phys. B 313 (1989) 16 [INSPIRE].
A. Dabholkar and J.A. Harvey, Nonrenormalization of the Superstring Tension, Phys. Rev. Lett. 63 (1989) 478 [INSPIRE].
V.V. Nikulin, Integral Symmetric Bilinear Forms and Some of their Applications, Math. USSR Izv. 14 (1980) 103 [Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 111].
J.H. Conway and N.J.A. Sloane. Sphere packings, lattices and groups, vol. 290, Springer Science & Business Media (2013).
W. Ebeling, Lattices and Codes. A course partially based on lectures by Friedrich Hirzebruch, Advanced Lectures in Mathematics, Springer Spektrum (2013).
R. Minasian, G.W. Moore and D. Tsimpis, Calabi-Yau black holes and (0, 4) σ-models, Commun. Math. Phys. 209 (2000) 325 [hep-th/9904217] [INSPIRE].
L. Göttsche, The Betti numbers of the Hilbert schemes of points on a smooth projective surface, Math. Ann. 286 (1990) 193.
D. Zagier, Nombres de classes et formes modulaires de poids 3/2, Sém. Théor. Nombres Bordeaux 4 (1975) 1.
M. Cheng, J. Duncan and M. Mertens, Class Number Moonshine, in preparation.
K. Yoshioka, The Betti numbers of the moduli space of stable sheaves of rank 2 on ℙ2, J. Reine Angew. Math. 453 (1994) 193.
K. Yoshioka, The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface, Math. Ann. 302 (1995) 519.
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
R. Griess and A. Ryba, Finite simple groups which projectively embed in an exceptional Lie group are classified!, Bull. Am. Math. Soc. 36 (1999) 75.
S. Kachru and A. Tripathy, Black Holes and Hurwitz Class Numbers, Int. J. Mod. Phys. D 26 (2017) 1742003 [arXiv:1705.06295] [INSPIRE].
J.A. Minahan, D. Nemeschansky, C. Vafa and N.P. Warner, E strings and N = 4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581 [hep-th/9802168] [INSPIRE].
M.C.N. Cheng and J.F.R. Duncan, Rademacher Sums and Rademacher Series, Contrib. Math. Comput. Sci. 8 (2014) 143 [arXiv:1210.3066] [INSPIRE].
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Cheng, M.C.N., Duncan, J.F.R., Harrison, S.M. et al. Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine. J. High Energ. Phys. 2018, 130 (2018). https://doi.org/10.1007/JHEP07(2018)130
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DOI: https://doi.org/10.1007/JHEP07(2018)130