Meromorphic open-string vertex algebras

TL;DR

This paper introduces meromorphic open-string vertex algebras, a generalization of open-string vertex algebras with relaxed axioms, and constructs explicit examples and modules for these structures.

Contribution

It defines meromorphic open-string vertex algebras, explores their properties, and constructs explicit algebraic examples and modules, expanding the framework of vertex algebra theory.

Findings

Defined meromorphic open-string vertex algebras with relaxed axioms.

Constructed a specific example on the tensor algebra of a Lie algebra's negative part.

Developed the theory of modules for these new algebraic structures.

Abstract

A notion of meromorphic open-string vertex algebra is introduced. A meromorphic open-string vertex algebra is an open-string vertex algebra in the sense of Kong and the author satisfying additional rationality (or meromorphicity) conditions for vertex operators. The vertex operator map for a meromorphic open-string vertex algebra satisfies rationality and associativity but in general does not satisfy the Jacobi identity, commutativity, the commutator formula, the skew-symmetry or even the associator formula. Given a vector space \mathfrak{h}, we construct a meromorphic open-string vertex algebra structure on the tensor algebra of the negative part of the affinization of \mathfrak{h} such that the vertex algebra struture on the symmetric algebra of the negative part of the Heisenberg algebra associated to \mathfrak{h} is a quotient of this meromorphic open-string vertex algebra. We also introduce the notion of left module for a meromorphic open-string vertex algebra and construct left modules for the meromorphic open-string vertex algebra above.