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Hodge-Elliptic Genera and How They Govern K3 Theories

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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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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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    Katrin Wendland

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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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