A construction of vertex algebra bundles on logarithmic smooth curves
Xi-Chuan Tan
TL;DR
The paper extends the Frenkel–Ben‑Zvi framework for vertex algebra bundles to families of logarithmic smooth curves by leveraging log geometry to accommodate singularities. It introduces a coordinate-centric construction using formal discs, the frame space, and the Aut^0 O torsor to build a $V$-bundle $\mathcal{V}$ with a compatible connection, and defines a Lie algebra $Lie_{X^circ}(V)$ that acts on inserted modules to form spaces of coinvariants and conformal blocks $C_V$. The approach generalizes the notion of conformal blocks to log-smooth settings and establishes propagation-of-vacua-type results, while a normal-crossing example demonstrates how log-differentials capture intricate behavior at nodes. Overall, the work provides a systematic framework for vertex algebra bundles and conformal blocks in the presence of logarithmic singularities, with potential applications to nodal degenerations and representation theory of vertex algebras.
Abstract
We give a construction of vertex algebra bundles and spaces of conformal blocks over a family of logarithmic smooth curves, which contains some types of singular algebraic curves in the usual sense. It generalizes some earlier results of Frenkel and Ben-Zvi on complex smooth algebraic curves.