Logarithmic Vertex Algebras
Bojko Bakalov, Juan J. Villarreal
TL;DR
The paper develops a rigorous framework for logarithmic vertex algebras (logVAs), extending ordinary vertex algebras to LCFT by introducing a locally nilpotent braiding map $\mathscr{S}$ (and its derived $\hat{\mathscr{S}}$) that encodes logarithmic singularities in OPEs. It constructs the state-field correspondence $Y$ in this setting, derives a logarithmic Borcherds identity and a Kac-type existence theorem, and establishes equivalence between two complementary logVA formulations. Conformal logVAs with a Virasoro action are treated, including how $L_0^{(n)}$ relates to $\mathscr{S}$ and how twistings tie into the logVA structure. The paper then provides concrete examples—symplectic fermions, free bosons, lattice logVAs, and Gurarie–Ludwig LCFT—to illustrate the theory and yield explicit generator relations, thereby offering a practical algebraic toolkit for LCFT constructions and twisted log modules.
Abstract
We introduce and study the notion of a logarithmic vertex algebra, which is a vertex algebra with logarithmic singularities in the operator product expansion of quantum fields; thus providing a rigorous formulation of the algebraic properties of quantum fields in logarithmic conformal field theory. We develop a framework that allows many results about vertex algebras to be extended to logarithmic vertex algebras, including in particular the Borcherds identity and Kac Existence Theorem. Several examples are investigated in detail, and they exhibit some unexpected new features that are peculiar to the logarithmic case.