The elliptic genus and its relatives

My book contribution

gives a summary of the special role that the elliptic genus plays in Mathieu moonshine, and of its more general properties. Indeed, independently of conformal field theory, the elliptic genus is a topological invariant of complex manifolds, which for any (compact) Calabi-Yau manifold X allows an interpretation as regularized U(1)-equivariant index of a Dirac operator on the loop space of X. The U(1)-equivariance together with the Calabi-Yau condition entail a dependence on two complex parameters on this invariant, which is modular with respect to a natural action of SL(2,Z). The same modular covariant is expected to arise as part of the partition function in every conformal field theory which describes superstring theory on X. In general, a mathematical proof for the reappearance of this modular covariant is lacking. With my student Yuhang Hou, we have now proved this claim for the minimal resolutions of ADE singularities, that is, for the simplest nontrivial examples of non-compact Calabi-Yau speces. Moreover, the above book contribution contains a proof of the folklore classification theorem for superconformal field theories with space-time supersymmetry at central charges c=6 by means of the elliptic genus. I had already sketched the ideas of proof in my Ph.D. thesis and could not find the argument elsewhere in the literature.

Some of the relevant background from geometry is summarized in a companion paper,

Topics include the complex and Kähler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss toroidal superconformal field theories and their Z₂ orbifolds, yielding a discussion of K3 surfaces as the simplest class of Calabi-Yau manifolds where non-linear sigma model constructions bear mysteries to the very day. The elliptic genus in CFT and in geometry is presented as an instructional piece of evidence in favor of a deep connection between geometry and conformal field theory.

Following ideas that underlie the definition of the chiral de Rham complex, for every Calabi-Yau manifold one finds a virtual bundle, which possesses an associated sheaf of vertex operator algebras, and from which one recovers the elliptic genus as a graded Euler characteristic. In the above-mentioned book contribution, I also discover and formulate a conjectural global decomposition of this virtual bundle, which is governed by the extended supersymmetry of the vertex operator algebras in question. With Dr. Thomas Creutzig I have now proved my conjecture. Our technique of proof generalizes to other settings with extended supersymmetry. We are currently exploring to which generality this is the case and whether or not there are implications on Mathieu Moonshine.

In the publication I confirm the predictions of the above book contribution that the cohomology of the chiral de Rham complex on K3 surfaces should be key to explaining Mathieu Moonshine. Indeed, I introduce the notion of a generic space of states, shared by all K3 theories, which is modeled by the cohomology of the chiral de Rham complex of K3 surfaces. I introduce the chiral Hodge-elliptic genus, a refinement of the (complex) elliptic genus which yields a bigraded dimension of the generic space of states. I use it to show that under a mild assumption on infinite volume limits, string theory correctly predicts the generic chiral algebra of K3 theories. As predicted in my book contribution above and independently discovered by Bailin Song in his publication Chiral Hodge cohomology and Mathieu moonshine, the generic space of states carries an M₂₄ action which induces Mathieu Moonshine. This action is compatible with symmetry surfing Kummer surfaces.

Given the results summarized above, to confirm symmetry surfing, it remains to prove compatibility with the action of M₂₄ in general and with the structure of a vertex operator algebra. In the publication

with Prof. Anne Taormina we therefore further study the generic space of states. Keller and Zadeh had already confirmed for K3 theories which are Z₂ orbifolds that under arbitrary deformations away from the orbifolds, up to second order deformation theory, the space of massive ground states is isomorphic to the generic space of states. However the selection of states that contribute to the generic space of states severely depends on the chosen deformation. In the above work, we find an SU(2)-action which channels the selection of generic states in K3 theories that are Z₂ orbifolds under the most symmetric deformation away from the orbifolds. Along the way, we introduce a number of novel counting functions and conjecture as yet unknown positivity results for them.