Abstract
Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The \({\mathcal{N} = 1}\) superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group \({\mathcal{S}W(3/2,2)}\). In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper.
Article PDF
Similar content being viewed by others
References
Berger M.: Sur les groupes d’holonomie homogène des variétés à connexion affines et des variétés riemanniennes. Bull. Soc. Math. Fr. 83, 279–330 (1955)
Joyce D.: Compact 8-manifolds with holonomy spin (7). Invent. Math. 123(3), 507–552 (1996)
Joyce D.: Compact Manifolds with Special Holonomy. Oxford University Press, New York (2000)
Cheng, M.C.N., Harrison, S.M., Kachru, S., Whalen, D.: Exceptional algebra and sporadic groups at c = 12. arXiv:1503.0721
Shatashvili, S.L., Vafa, C.: Superstrings and manifold of exceptional holonomy. Selecta Math. 1, 347 (1995). hep-th/9407025
Gepner, D., Noyvert, B.: Unitary representations of SW(3/2,2) superconformal algebra. Nucl. Phys. B610, 545–577 (2001). hep-th/0101116
Eguchi, T., Sugawara, Y., Yamaguchi, S.: Supercoset CFT’s for string theories on noncompact special holonomy manifolds. Nucl. Phys. B657, 3–52 (2003). hep-th/0301164
Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the K3 surface and the Mathieu group M 24. Exp. Math. 20, 91–96 (2011). arXiv:1004.0956
Cheng, M.C.: K3 surfaces, N = 4 dyons, and the Mathieu group M24. Commun. Number Theor. Phys. 4, 623–658 (2010). arXiv:1005.5415
Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu twining characters for K3. JHEP 1009, 058 (2010). arXiv:1006.0221
Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu moonshine in the elliptic genus of K3. JHEP 1010, 062 (2010). arXiv:1008.3778
Eguchi, T., Hikami, K.: Note on twisted elliptic genus of K3 surface. Phys. Lett. B694, 446–455 (2011). arXiv:1008.4924
Gannon, T.: Much ado about Mathieu. arXiv:1211.5531
Mukai S.: Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math. 94(1), 183–221 (1988)
Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Symmetries of K3 sigma models. Commun. Number Theor. Phys. 6, 1–50 (2012). arXiv:1106.4315
Cheng, M.C., Duncan, J.F.: The largest Mathieu group and (Mock) automorphic forms. arXiv:1201.4140
Cheng, M.C., Duncan, J.F., Harvey, J.A.: Umbral moonshine. arXiv:1204.2779
Cheng, M.C., Dong, X., Duncan, J., Harvey, J., Kachru, S., Wrase, T.: Mathieu moonshine and N = 2 string compactifications. JHEP 1309, 030 (2013). arXiv:1306.4981
Cheng, M.C.N., Duncan, J.F.R., Harvey, J.A.: Umbral moonshine and the Niemeier lattices. arXiv:1307.5793
Cheng, M.C.N., Dong, X., Duncan, J.F.R., Harrison, S., Kachru, S., Wrase, T.: Mock modular Mathieu moonshine modules. arXiv:1406.5502
Harrison, S., Kachru, S., Paquette, N.M.: Twining genera of (0,4) supersymmetric sigma models on K3. JHEP 1404, 048 (2014). arXiv:1309.0510
Cheng, M.C.N., Harrison, S.: Umbral moonshine and K3 surfaces. arXiv:1406.0619
Taormina, A., Wendland, K.: The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24. JHEP 1308, 125 (2013). arXiv:1107.3834
Taormina, A., Wendland, K.: Symmetry-surfing the moduli space of Kummer K3s. arXiv:1303.2931
Taormina, A., Wendland, K.: A twist in the M24 moonshine story. arXiv:1303.3221
Gaberdiel, M.R., Taormina, A., Volpato, R., Wendland, K.: A K3 sigma model with \({\mathbb{Z}^8_2}\): \({\mathbb{M}_{20}}\) symmetry. JHEP 1402, 022 (2014). arXiv:1309.4127
Harvey, J.A., Murthy, S.: Moonshine in fivebrane spacetimes. JHEP 1401, 146 (2014). arXiv:1307.7717
Paquette, N.M., Wrase, T.: Comments on \({{\rm M}_{24}}\) representations and CY 3 geometries. JHEP 1411, 155 (2014). arXiv:1409.1540
Harvey, J.A., Murthy, S., Nazaroglu, C.: ADE double scaled little string theories, mock modular forms and umbral moonshine. arXiv:1410.6174
Duncan, J.F., Griffin, M.J., Ono, K.: Moonshine (2014). arXiv:1411.6571
Duncan, J.F.: Super-moonshine for conway’s largest sporadic group. arXiv:math/0502267
Duncan, J.F., Mack-Crane, S.: The moonshine module for conway’s group (2014). arXiv:1409.3829
Hirzebruch F., Berger T., Jung R., Landweber P.S.: Manifolds and Modular Forms, vol. 20. Springer, Berlin (1992)
Witten, E.: Three-dimensional gravity revisited. arXiv:0706.3359
Witten, E.: The index of the dirac operator in loop space. In: Elliptic curves and modular forms in algebraic topology, pp. 161–181. Springer, Berlin (1988)
Salamon S.: Quaternionic kahler manifolds. Invent. Math. 67, 143–171 (1982)
Neumann, C.D.D.: The Elliptic genus of Calabi–Yau 3-folds and 4-folds: product formulae and generalized Kac–Moody algebras. J. Geom. Phys. 29, 5–12 (1999). hep-th/9607029
Kawai, T., Yamada, Y., Yang, S.-K.: Elliptic genera and \({{\rm N} = 2}\) superconformal field theory. Nucl. Phys. B414, 191–212 (1994). hep-th/9306096
David, J.R., Jatkar, D.P., Sen, A.: Product representation of Dyon partition function in CHL models. JHEP 0606, 064 (2006). hep-th/0602254
Vafa C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys. B273, 592 (1986)
Gaberdiel, M.R., Kaste, P.: Generalized discrete torsion and mirror symmetry for g(2) manifolds. JHEP 0408, 001 (2004). hep-th/0401125
Kiritsis E.B.: Character formulae and the structure of the representations of the n = 1, n = 2 superconformal algebras. Int. J. Modern Phys. A 3(08), 1871–1906 (1988)
Bouwknegt, P., McCarthy, J.G., Pilch, K.: The W(3) algebra: modules, semiinfinite cohomology and BV algebras. Lect. Notes Phys. M42, 1–204 (1996). hep-th/9509119
de Boer, J., Harmsze, F., Tjin, T.: Nonlinear finite W symmetries and applications in elementary systems. Phys. Rep. 272, 139–214 (1996). hep-th/9503161
Whalen, D.: An algorithm for evaluating Gram matrices in Verma modules of W-algebras. arXiv:1412.0759
Bowcock, P., Taormina, A.: Representation theory of the affine Lie superalgebra sl(2/1:C) at fractional level. Commun. Math. Phys. 185, 467–493 (1997). hep-th/9605220
Bowcock, P., Hayes, M., Taormina, A.: Characters of admissible representations of the affine superalgebra sl(2–1:C)-k. Nucl. Phys. B 510, 739–764 (1998). hep-th/9705234
Dorrzapf, M.: The embedding structure of unitary N = 2 minimal models. Nucl. Phys. B 529, 639–655 (1998). hep-th/9712165
Zwegers, S.: Mock theta functions. arXiv:0807.4834
Cheng, M.C.N., Duncan, J.F.R.: On Rademacher sums, the largest Mathieu group, and the holographic modularity of moonshine (2011). arXiv:1110.3859
Whalen, D.: Vector-valued Rademacher sums and automorphic integrals (2014). arXiv:1406.0571
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Boris Pioline.
Rights and permissions
About this article
Cite this article
Benjamin, N., Harrison, S.M., Kachru, S. et al. On the Elliptic Genera of Manifolds of Spin(7) Holonomy. Ann. Henri Poincaré 17, 2663–2697 (2016). https://doi.org/10.1007/s00023-015-0454-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-015-0454-5