  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
M24 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.
