Vertex $F$-Algebras and Their Associated Lie Algebra
Markus Upmeier
TL;DR
This work extends Borcherds' Lie-algebra construction from ordinary vertex algebras to vertex $F$-algebras, where a formal group law $F$ replaces the additive law. It develops a full formal-calculus toolkit for $F$-theory—$F$-residues, $F$-binomial coefficients, $F$-delta distributions, and $F$-hyperderivatives—and proves that the bracket $[a,b] = \mathrm{Res}_{z=0}^F Y(a,z)b\,dz$ endows the quotient $V/\sum_{n\ge1}\mathcal{S}^{(n)}(V)$ with a Lie algebra structure. The paper also supplies a concrete Heisenberg-type example, illustrating the construction in a familiar setting, and discusses connections to enumerative geometry and generalized homology theories. By establishing a robust $F$-calculus and a meromorphicity framework, it broadens the applicability of vertex-algebra methods to contexts where the formal group law is non-additive, potentially informing wall-crossing and invariants in generalized homology theories.
Abstract
Vertex $F$-algebras are a deformation of the concept of an ordinary vertex algebra in which the additive formal group law is replaced by an arbitrary formal group law $F$. The main theorem of this paper constructs a Lie algebra from a vertex $F$-algebra - for the additive formal group law, this extends Borcherds' well-known construction for ordinary vertex algebras. Our construction involves the new concept of an $F$-residue and some other new algebraic concepts, which are deformations of familiar concepts for the special case of an additive formal group law.