# What is known about the connection of positive energy representations of loop groups and modular forms

At the end of Section 14.1 in Pressley, Segal "Loop Groups" there is the remark that the partition function is a modular function in the sense that the Dedekind $\eta$ function is a modular form. I am interested about the following remark:

It follows from the Kac character formula that the characters of all positive energy representations of loop groups are constructed from modular functions in an appropriate sense. No explanation of this phenomenon seems to be known at present.

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The idea is as follows, to each vertex operator algebra $V$ (the one associated to an affine Kac-Moody Lie algebra in the case of your question) and an algebraic curve $X$ you can associate a vector space of coinvariants of $V$, $C(X,V)$. As the curve $X$ moves in the moduli space of curves of a given genus, the vector spaces $C(X,V)$ arrange to form a vector bundle on this space (it is generally a quasi-coherent sheaf, but under certain finiteness condition which holds in your case it is a bundle). This can be generalized by looking at pointed curves and modules of $V$ over the marked points of $X$. In the particular case that you look at the moduli space of elliptic curves, the sections of this bundle are the characters of the representations (where $q = e^{2 \pi i \tau}$ is the modular parameter of the corresponding curve). This is a "reason" for those characters to have modular properties, however, the fact that we expected already these spaces to form a bundle on $M_g$ comes from CFTs considerations.