Abstract
The (complex) Hodge-elliptic genus and its conformal field theoretic counterpart were recently introduced by Kachru and Tripathy, refining the traditional complex elliptic genus. We construct a different, so-called chiral Hodge-elliptic genus, which is expected to agree with the generic conformal field theoretic Hodge-elliptic genus, in contrast to the complex Hodge-elliptic genus as originally defined. For K3 surfaces X, the chiral Hodge-elliptic genus is shown to be independent of all moduli. Moreover, employing Kapustin’s results on infinite volume limits it is shown that it agrees with the generic conformal field theoretic Hodge-elliptic genus of K3 theories, while the complex Hodge-elliptic genus does not. This new invariant governs part of the field content of K3 theories, supporting the idea that all their spaces of states have a common subspace which underlies the generic conformal field theoretic Hodge-elliptic genus, and thereby the complex elliptic genus. Mathematically, this space is modelled by the sheaf cohomology of the chiral de Rham complex of X. It decomposes into irreducible representations of the \(N=4\) superconformal algebra such that the multiplicity spaces of all massive representations have precisely the dimensions required in order to furnish the representation of the Mathieu group \(M_{24}\) that is predicted by Mathieu Moonshine. This is interpreted as evidence in favour of the ideas of symmetry surfing, which have been proposed by Taormina and the author, along with the claim that the sheaf cohomology of the chiral de Rham complex is a natural home for Mathieu Moonshine. These investigations also imply that the generic chiral algebra of K3 theories is precisely the \(N=4\) superconformal algebra at central charge \(c=6\), if the usual predictions on infinite volume limits from string theory hold true.
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Ademollo, M., Brink, L., D’Adda, A., D’Auria, R., Napolitano, E., Sciuto, S., Del Giudice, E., Di Vecchia, P., Ferrara, S., Gliozzi, F., Musto, R., Pettorino, R.: Supersymmetric strings and color confinement. Phys. Lett. B 62, 105–110 (1976)
Atiyah, M., Bott, R., Patodi, V.K.: On the heat equation and the index theorem. Invent. Math 19, 279–330 (1973). Errata: Invent. Math. 28, 277–280 (1975)
Alvarez, O., Killingback, T.P., Mangano, M., Windey, P.: String theory and loop space index theorems. Commun. Math. Phys. 111, 1–12 (1987)
Aspinwall, P.S., Morrison, D.R.: String theory on K3 surfaces. In: Greene, B., Yau, S.T. (eds.) Mirror Symmetry II, pp. 703–716. AMS, Providence (1994). arXiv:hep-th/9404151
Aspinwall, P.S., Melnikov, I.V., Plesser, M.R.: (0, 2) Elephants. JHEP 01, 060 (2012). arXiv:1008.2156 [hep-th]
Ben-Zvi, D., Heluani, R., Szczesny, M.: Supersymmetry of the chiral de Rham complex. Compos. Math. 144(2), 503–521 (2008). arXiv:math/0601532 [math.QA]
Borisov, L.A., Libgober, A.: Elliptic genera of toric varieties and applications to mirror symmetry. Invent. Math. 140(2), 453–485 (2000). arXiv:math/9904126 [math.AG]
Borisov, L.A., Libgober, A.: Elliptic genera of singular varieties. Duke Math. J. 116(2), 319–351 (2003). arXiv:math/0007108 [math.AG]
Borisov, L.A.: Vertex algebras and mirror symmetry. Commun. Math. Phys. 215(2), 517–557 (2001). arXiv:math/9809094 [math.AG]
Berglund, P., Parkes, L., Hubsch, T.: The complete matter sector in a three generation compactification. Commun. Math. Phys. 148, 57–100 (1992)
Candelas, P., de la Ossa, X.C., Green, P.S., Parkes, L.: A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B 359, 21–74 (1991)
Cecotti, S.: \({N}=2\) Landau–Ginzburg vs. Calabi–Yau \(\sigma \)-models: non-perturbative aspects. Int. J. Mod. Phys. A6, 1749–1813 (1991)
Casher, A., Englert, F., Nicolai, H., Taormina, A.: Consistent superstrings as solutions of the \(D=26\) bosonic string theory. Phys. Lett. B 162, 121–126 (1985)
Creutzig, Th, Höhn, G.: Mathieu moonshine and the geometry of K3 surfaces. Commun. Number Theory Phys. 8(2), 295–328 (2014). arXiv:1309.2671 [math.QA]
Cheng, M.C.N.: K3 surfaces, \(N=4\) dyons, and the Mathieu group \(M_{24}\). Commun. Number Theory Phys. 4, 623–657 (2010). arXiv:1005.5415 [hep-th]
Candelas, P., Lynker, M., Schimmrigk, R.: Calabi–Yau manifolds in weighted P(4). Nucl. Phys. B341, 383–402 (1990)
Distler, J., Greene, B.: Some exact results on the superpotential from Calabi–Yau compactifications. Nucl. Phys. B 309, 295–316 (1988)
Dixon, L.J., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B 261, 678–686 (1985)
Dixon, L.J., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds II. Nucl. Phys. B 274, 285–314 (1986)
Dixon, L.J.: Some world-sheet properties of superstring compactifications, on orbifolds and otherwise. In: Superstrings, Unified Theories and Cosmology 1987 (Trieste, 1987). ICTP Series in Theoretical Physics, vol. 4, pp. 67–126. World Sci. Publ., Teaneck (1988)
Di Francesco, P., Yankielowicz, S.: Ramond sector characters and \({N}=2\) Landau–Ginzburg models. Nucl. Phys. B 409, 186–210 (1993). arXiv:hep-th/9305037
Eguchi, T., Hikami, K.: Superconformal algebras and mock theta functions 2. Rademacher expansion for K3 surface. Commun. Number Theory Phys. 3, 531–554 (2009). arXiv:0904.0911 [math-ph]
Eguchi, T., Hikami, K.: Note on twisted elliptic genus of K3 surface. Phys. Lett. B 694, 446–455 (2011). arXiv:1008.4924 [hep-th]
Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the \(K3\) surface and the Mathieu group \(M_{24}\). Exp. Math. 20(1), 91–96 (2011). arXiv:1004.0956 [hep-th]
Eguchi, T., Ooguri, H., Taormina, A., Yang, S.-K.: Superconformal algebras and string compactification on manifolds with \({SU}(n)\) holonomy. Nucl. Phys. B 315, 193–221 (1989)
Eguchi, T., Taormina, A.: Unitary representations of the \({N}=4\) superconformal algebra. Phys. Lett. B 196, 75–81 (1987)
Eguchi, T., Taormina, A.: Character formulas for the \({N}=4\) superconformal algebra. Phys. Lett. B 200, 315–322 (1988)
Eguchi, T., Taormina, A.: Extended superconformal algebras and string compactifications, pp. 167–188. Trieste School, Superstrings (1988)
Eguchi, T., Taormina, A.: On the unitary representations of \({N}=2\) and \({N}=4\) superconformal algebras. Phys. Lett. 210, 125–132 (1988)
Eguchi, T., Yang, S.-K.: N = 2 superconformal models as topological field theories. Mod. Phys. Lett. A 5, 1693–1701 (1990)
Frenkel, E., Szczesny, M.: Chiral de Rham complex and orbifolds. J. Algebr. Geom. 16(4), 599–624 (2007). arXiv:math/0307181 [math.AG]
Gannon, T.: Much ado about Mathieu. Adv. Math. 301, 322–358 (2016). arXiv:1211.5531 [math.RT]
Getzler, E.: Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem. Commun. Math. Phys. 92(2), 163–178 (1983)
Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu moonshine in the elliptic genus of K3. JHEP 1010, 062 (2010). arXiv:1008.3778 [hep-th]
Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Mathieu twining characters for K3. JHEP 1009, 058 (2010). arXiv:1006.0221 [hep-th]
Gaberdiel, M.R., Hohenegger, S., Volpato, R.: Symmetries of K3 sigma models. Commun. Number Theory Phys. 6, 1–50 (2012). arXiv:1106.4315 [hep-th]
Gilkey, P.B.: Curvature and the eigenvalues of the Dolbeault complex for Kaehler manifolds. Adv. Math. 11, 311–325 (1973)
Gaberdiel, M.R., Keller, Ch., Paul, H.: Mathieu moonshine and symmetry surfing. J. Phys. A50(47), 474002 (2017). arXiv:1609.09302 [hep-th]
Gorbounov, V., Malikov, F.: Vertex algebras and the Landau–Ginzburg/Calabi–Yau correspondence. Mosc. Math. J. 4(3), 729–779 (2004). arXiv:math.AG/0308114
Greene, B.R., Plesser, M.R.: Duality in Calabi–Yau moduli space. Nucl. Phys. B 338, 15–37 (1990)
Greene, B.R.: String theory on Calabi–Yau manifolds. In: Fields, Strings and Duality (Boulder, CO, 1996), pp. 543–726. World Sci. Publishing, River Edge (1997). arXiv:hep-th/9702155
Grimm, F.: The chiral de Rham complex of tori and orbifolds. Ph.D. thesis, Albert-Ludwigs-Universität Freiburg (2016). Supervisor: K. Wendland. http://home.mathematik.uni-freiburg.de/mathphys/mitarbeiter/wendland/GrimmPhD.pdf
Haefliger, A.: Orbi-espaces. In: Ghys, Etienne, de la Harpe, Pierre (eds.) Sur les groupes hyperboliques d’après Mikhael Gromov, pp. 203–213. Birkhäuser, Boston (1990)
Heluani, R.: Supersymmetry of the chiral de Rham complex. II. Commuting sectors. Int. Math. Res. Not. 6, 953–987 (2009). arXiv:0806.1021 [math.QA]
Hirzebruch, F.: Elliptic genera of level \(N\) for complex manifolds. In: Bleuler, K., Werner, M. (eds.) Differential Geometric Methods in Theoretical Physics, pp. 37–63. Kluwer Acad. Publ., Dordrecht (1988)
Huybrechts, D.: The tangent bundle of a Calabi–Yau manifold. Deformations and restriction to rational curves. Commun. Math. Phys. 171(1), 139–158 (1995)
Kapustin, A.: Chiral de Rham complex and the half-twisted sigma-model. arXiv:hep-th/0504074
Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2, pp. 120–139. Birkhäuser, Basel (1994). arXiv:alg-geom/9411018
Krichever, I.: Generalized elliptic genera and Baker–Akhiezer functions. Math. Notes 47, 132–142 (1990)
Kachru, S., Tripathy, A.: The Hodge-elliptic genus, spinning BPS states, and black holes. Commun. Math. Phys. 355, 245–259 (2017). arXiv:1609.02158 [hep-th]
Kachru, S., Tripathy, A.: BPS jumping loci and special cycles. arXiv:1703.00455 [hep-th]
Lian, B.H., Linshaw, A.R.: Chiral equivariant cohomology. I. Adv. Math. 209(1), 99–161 (2007). arXiv:math/0501084 [math.DG]
Lerche, W., Vafa, C., Warner, N.P.: Chiral rings in \({N}=2\) superconformal theories. Nucl. Phys. B 324, 427–474 (1989)
Mordell, L.J.: The definite integral \(\int _{-\infty }^\infty {e^{ax^2+b}\over e^{cx}+d}dx\) and analytic theory of numbers. Acta Math. 61, 323–360 (1933)
Malikov, F., Schechtman, V.: Chiral Poincaré duality. Math. Res. Lett. 6(5–6), 533–546 (1999). arXiv:math/9905008 [math.AG]
Malikov, F., Schechtman, V., Vaintrob, A.: Chiral de Rham complex. Commun. Math. Phys. 204(2), 439–473 (1999). arXiv:math/9803041 [math.AG]
Mukai, S.: Finite groups of automorphisms of K3 surfaces and the Mathieu group. Invent. Math. 94, 183–221 (1988)
Narain, K.S.: New heterotic string theories in uncompactified dimensions \(<10\). Phys. Lett. 169B, 41–46 (1986)
Nahm, W., Wendland, K.: 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). arXiv:hep-th/9912067
Ooguri, H.: Superconformal symmetry and geometry of Ricci flat Kahler manifolds. Int. J. Mod. Phys. A 4, 4303–4324 (1989)
Patodi, V.K.: An analytic proof of Riemann–Roch–Hirzebruch theorem for Kaehler manifolds. J. Differ. Geom. 5, 251–283 (1971)
Seiberg, N.: Observations on the moduli space of superconformal field theories. Nucl. Phys. B 303, 286–304 (1988)
Sen, A.: \((2,0)\) supersymmetry and space–time supersymmetry in the heterotic string theory. Nucl. Phys. B 278, 289–308 (1986)
Sen, A.: Heterotic string theory on Calabi–Yau manifolds in the Green–Schwarz formalism. Nucl. Phys. B 284, 423–448 (1987)
Song, B.: Vector bundles induced from jet schemes. arXiv:1609.03688 [math.DG]
Song, B.: Chiral Hodge cohomology and Mathieu moonshine. arXiv:1705.04060 [math.QA]
Taormina, A.: The \({N}=2\) and \({N}=4\) superconformal algebras and string compactifications. Mathematical Physics, Islamabad, pp. 349–370. World Sci. Publishing (1989)
Thurston, W.P.: Three-Dimensional Geometry and Topology, vol. 1, Princeton University Press, Princeton, ed. Silvio Levy (1997)
Taormina, A., Wendland, K.: The overarching finite symmetry group of Kummer surfaces in the Mathieu group \(M_{24}\). JHEP 08, 125 (2013). arXiv:1107.3834 [hep-th]
Taormina, A., Wendland, K.: Symmetry-surfing the moduli space of Kummer K3s. In: Proceedings of the Conference String-Math 2012, Proceedings of Symposia in Pure Mathematics, no. 90, pp. 129–153. arXiv:1303.2931 [hep-th]
Taormina, A., Wendland, K.: A twist in the \(M_{24}\) moonshine story. Confluentes Mathematici 7, 83–113 (2015). arXiv:1303.3221 [hep-th]
Taormina, A., Wendland, K.: The Conway moonshine module is a reflected K3 theory. arXiv:1704.03813 [hep-th]
Wendland, K.: Moduli spaces of unitary conformal field theories. Ph.D. thesis, University of Bonn (2000). Supervisor: W. Nahm; available on request
Wendland, K.: Snapshots of conformal field theory. In: Calaque, D., Strobl, Th (eds.) Mathematical Aspects of Quantum Field Theories, pp. 89–129. Springer, Berlin (2015). arXiv:1404.3108 [hep-th]
Wendland, K.: K3 en route from geometry to conformal field theory. In: Proceedings of the 2013 Summer School “Geometric, Algebraic and Topological Methods for Quantum Field Theory” in Villa de Leyva, Colombia, pp. 75–110. World Scientific. arXiv:1503.08426 [math.DG]
Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys. B 202, 253–316 (1982)
Witten, E.: Elliptic genera and quantum field theory. Commun. Math. Phys. 109, 525–536 (1987)
Witten, E.: The index of the Dirac operator in loop space. In: Landweber, P. (ed.) Elliptic Curves and Modular Forms in Algebraic Geometry, pp. 161–181. Springer, Berlin (1988)
Witten, E.: Mirror manifolds and topological field theory. In: Essays on Mirror Manifolds, pp. 120–158. Internat. Press, Hong Kong (1992). arXiv:hep-th/9112056
Witten, E.: On the Landau–Ginzburg description of \({N}=2\) minimal models. Int. J. Mod. Phys. A 9, 4783–4800 (1994). arXiv:hep-th/9304026
Acknowledgements
It is my great pleasure to thank Thomas Creutzig, Shamit Kachru, Stefan Kebekus, Anatoly Libgober, Emanuel Scheidegger, and Anne Taormina for very helpful communications and discussions. I particularly thank Bailin Song for his comments on an earlier version of this note, which lead to the formulation of Prop. 4.4 and to a considerable extension of the interpretation of my calculations. I am grateful to an anonymous referee for their diligent reading of the manuscript and constructive criticism. I am also grateful to the organisers of the program on Automorphic forms, mock modular forms and string theory in September 2016 and to the Simons Center for Geometry and Physics at Stony Brook for the support and hospitality, and for the inspiring environment during the initial steps of this work.
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Wendland, K. Hodge-Elliptic Genera and How They Govern K3 Theories. Commun. Math. Phys. 368, 187–221 (2019). https://doi.org/10.1007/s00220-019-03425-4
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DOI: https://doi.org/10.1007/s00220-019-03425-4