A basis theorem for Genus-One Conformal Blocks and modular invariance of intertwining operators

Abstract

We prove that trace functions associated to intertwining operators over a strongly rational vertex operator algebra form a global frame of the conformal block bundle $\mathscr{C}_{\mathbb{H}}(W)$ over $\mathbb{H}$. Consequently, for each $\tau\in\mathbb{H}$, these trace functions, evaluated at $\tau$, form a basis of the fiber $\mathscr{C}(E_\tau,\mathsf{p},z,W)$, and the natural $\mathrm{SL}(2,\mathbb{Z})$-action on the fiber is represented in this basis. This result is both a generalization and a refinement of Zhu's and Dong-Li-Mason's modular invariance theorems for trace functions associated to vertex operators and twisted vertex operators, and a specialization and refinement of Huang's and Miyamoto's modular invariance theorems for (logarithmic) intertwining operators for $C_2$-cofinite vertex operator algebras. The proof combines a new construction of a connection on the bundle $\mathscr{C}_{\mathbb{H}}(W)$, Zhu's recursive formulas for trace functions, Frenkel-Zhu's fusion rules theorem, and recent theorems of Damiolini-Gibney-Krashen-Tarasca on the geometry of sheaves of vertex operator algebra conformal blocks over the moduli spaces $\overline{\mathscr{M}}_{g,n}$.