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
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