Hecke operators in equivariant elliptic cohomology and generalized moonshine

Abstract

Contents 1. Introduction 1.1. Background on generalized Moonshine 1.2. Background on equivariant elliptic cohomology and statement of results 1.3. Acknowledgements 2. Principal bundles over complex elliptic curves 2.1. The moduli space of principal G-bundles 2.2. Line bundles over M G 2.3. Construction and properties of the Freed-Quinn line bundle L α 2.4. Sections of L α 3. Abelian groups and level structures 3.1. Isogenies 3.2. Duals 3.3. Cyclic groups 4. Symmetric groups and coverings 5. Hopkins-Kuhn-Ravenel character theory 5.1. Products 5.2. Change of groups 6. Hecke operators and power operations 6.1. The Hecke correspondence 6.2. Power operations in K-theory 6.3. Symmetric and exterior powers on Ell α G 6.4. Replicability 7. The Witten genus 7.1. Twisted Thom isomorphisms 7.2. A few words about physics 7.3. The Witten genus and replicability? Appendix A. Generalities on principal bundles References

Second-quantized Mathieu moonshine

“…equivariant, with respect to the M 24 -action on the pair (g, h), and it should incorporate the α-twist in the modular transformation. Equivariant versions of Hecke operators acting on modular forms was proposed by Ganter [55] in the context of generalized Monstrous moonshine, and we shall see that this result also applies here after some minor modifications.…”

“…Before we close this section we shall offer a more geometric perspective on the twisted twining genera which will be useful later on. In [55], Ganter showed that the natural home for the Norton series f (g, h; τ ) [19] is the equivariant elliptic cohomology developed in [56]. We shall here give a short review of Ganter's perspective, adapted to the case of Jacobi forms on (2.26) ¶ This should really be a moduli stack but we ignore this technical point; see [55] for a more precise description.…”

“…which is twisted by the 3-cocycle α ∈ H 3 (M 24 , U (1)) [55]. Thus, for fixed (g, h) we can think of the twisted twining genus φ g,h as a section of L α g,h .…”

The Moonshine Anomaly

“…The description of A G as a conformal boundary condition for Z(Vect ω [G]) explains the relationship between anomalies and the multipliers for twisted-twining genera [GPRV13, Section 3]. Indeed, following [Gan09], let M = M 1 denote the moduli stack of elliptic curves and let M G denote the moduli stack of elliptic curves equipped with a principal G-bundle, and let L ω denote the line bundle thereon constructed from ω. The Hilbert space that the Dijkgraaf-Witten theory Z(Vect ω [G]) assigns to an elliptic curve E is the fiber over E ∈ M of the pushforward of L ω along M G → M. The "twisted-twining genera" are the (genus-one) conformal blocks of A G ; abstract nonsense of boundary field theories says that they are sections of L ω .…”

Third Homology of some Sporadic Finite Groups

“…We have also been inspired by the idea that 3-cocycles G × G × G → U(1) (when G is finite, these represent classes in H 4 (G; Z)) can explain and predict some features of moonshine [4,16,17,18]. Such a cocycle can arise as the gauge anomaly of a G-action on a conformal field theory.…”