# Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more concrete, then indicate some motivation for it.

## The guess

Recall the following description of modular forms. Let $X$ be (a suitable compactification of) the moduli stack of elliptic curves. On $X$ there is a canonical line bundle $\omega$ whose fiber at an elliptic curve $E$ is the space $\Omega^1(E)$ of $1$-forms on $E$. Then a modular form of weight $k$ is a section of $\omega^{\otimes k}$.

Now let $G$ be a finite group (for simplicity). Let $X_G$ be (a suitable compactification of) the moduli stack of pairs (an elliptic curve $E$, a principal $G$-bundle on $E$). On $X_G$ there is again a canonical line bundle $\omega_G$, the pullback of $\omega$ from above along the forgetful map $X_G \to X$ which forgets the $G$-bundle. Then a $G$-equivariant modular form of weight $k$ should be a section of $\omega^{\otimes k}_G$. ($\omega_G$ might be the wrong bundle here.)

Finally, let $\alpha$ be a cohomology class in $H^3(BG, U(1)) \cong H^4(BG, \mathbb{Z})$. I believe that $\alpha$ determines a line bundle on $X_G$ which I will also call $\alpha$. Then an $\alpha$-twisted $G$-equivariant modular form of weight $k$ should be a section of $\alpha \otimes \omega^{\otimes k}_G$.

## Making the guess more concrete

Starting from an elliptic curve $E$ over $\mathbb{C}$ and a principal $G$-bundle on $E$, pick an oriented basis for $H_1(E)$. Then on the one hand a standard construction lets us associate to this data a complex number $z \in \mathbb{H}$ describing the elliptic curve $E$ together with its oriented basis for $H_1(E)$, and on the other hand we can consider the monodromy of the principal $G$-bundle with respect to the oriented basis, obtaining a pair of commuting elements $(g, h) \in G^2$, or equivalently a homomorphism

$$H_1(E) \cong \pi_1(E) \cong \mathbb{Z}^2 \to G.$$

The space of such triples $(g, h, z)$ has a natural action of $\text{SL}_2(\mathbb{Z})$ corresponding to change of oriented basis, and an $\alpha$-twisted $G$-equivariant modular form of weight $k$ should admit a more concrete description as a function on such triples, holomorphic in $z$, satisfying a suitable cocycle condition under the action of $\text{SL}_2(\mathbb{Z})$ determined by $\alpha$ and $k$, and maybe also satisfying some growth conditions.

## Motivation, etc.

One motivation for writing down this definition is that the tmf of a point admits a natural map to modular forms which is an isomorphism after tensoring with $\mathbb{Q}$. The data of a finite group $G$ and a class $\alpha \in H^3(BG, U(1)) \cong H^4(BG, \mathbb{Z})$ should determine a notion of $\alpha$-twisted $G$-equivariant tmf, which should admit a description analogous to the above but more derived, and one might hope that on a point this admits an analogous map to some $\alpha$-twisted $G$-equivariant version of modular forms which is again an isomorphism after tensoring with $\mathbb{Q}$.

A closely related one is that modular-looking functions of triples $(g, h, z)$ satisfying a suitable cocycle condition under the action of $\text{SL}_2(\mathbb{Z})$ appear explicitly in the generalized moonshine conjecture. These functions should arise as something like characters for actions of $G$ on vertex algebras. The connection to tmf comes from the relationship between vertex algebras and conformal field theories, and then from the relationship between conformal field theories and the Segal-Stolz-Teichner perspective on elliptic cohomology.

A version of this story appears in a paper by Berwick-Evans and another one appears in a paper by Morava describing work by Ganter. But as far as I can tell, these papers are about elliptic cohomology itself and not about the simpler underived story above, which should approximate it. For more general $G$ there is also clearly some connection to conformal blocks and nonabelian theta functions but I don't know enough about these to write down a precise statement.

• (Probably you are aware of it anyway) - the space $X_G$ has been considered by Ginzburg - Kapranov - Vasserot in "Elliptic algebras and equivariant elliptic cohomology": their version of equivariant elliptic cohomology takes values in coherent sheaves on $X_G$ (more precisely, on the fibre of $X_G\to X$ over a point of $X$, but this point is arbitrary, so the generic point may be taken); in particular, values on the spheres are tensor powers of $\omega_G$. Also work of Devoto and of Tanabe must be relevant. – მამუკა ჯიბლაძე Jan 30 '15 at 6:35
• You should search for the term "twisted sector", if you have not already done so. – Neil Strickland Jan 30 '15 at 10:30
• Ganter has a fairly detailed discussion of the Freed-Quinn line bundle $\mathcal{L}^\alpha$ in section 2 of arxiv.org/abs/0706.2898 but I have yet to translate the construction into a language that makes sense to me. – S. Carnahan Jan 30 '15 at 11:25