Trace-function frames and modularity of genus-one conformal blocks

Abstract

The recursive relations and modular transformation laws of trace functions have a common geometric source: conformal blocks on pointed elliptic curves. For a simple, rational, $C_2$-cofinite, CFT-type vertex operator algebra and an irreducible module $W$ of it, we identify the bundle ${\mathcal{C}}_{\mathbb{H}}(W)$ of one-point conformal blocks over the upper half-plane $\mathbb{H}$ with the trivial bundle of intertwining operators. In particular, bases of intertwining operators give rise to global frames of ${\mathcal{C}}_{\mathbb{H}}(W)$ via their trace functions. Factorization of conformal-block bundles provides a rank formula used to establish this identification and also yields a geometric proof of basic properties of the fusion ring. Then the modular group $\mathrm{SL}(2,\mathbb{Z})$ acts projectively on such a frame, with matrices independent of $\tau \in \mathbb{H}$. This modular-invariance result recovers that of Zhu for vertex-operator traces and gives a uniform geometric interpretation of the affine and Virasoro intertwining-operator traces studied by Etingof--Kirillov, Krauel--Marks, and Krauel--Shafiq--Wood. The trace frame is horizontal for a natural connection whose projectivization agrees with the pullback of the Damiolini--Gibney--Tarasca connection; after descent, its monodromy gives the modular action. With respect to this natural connection, we characterize global horizontal sections in terms of Miyamoto's four conditions. As an application of our results, we realize the cusp form $\Delta$ as a trace function for the affine VOA $L_{{\widehat{\mathfrak{sl}}}_2}(16,0)$.