Braided Logarithmic Vertex Algebras

TL;DR

This paper introduces a unified framework for braided logarithmic vertex algebras, connecting them to non-local Poisson vertex algebras, and constructs a new example inspired by the non-linear Schrödinger equation.

Contribution

It develops a method linking braided logarithmic vertex algebras to non-local Poisson vertex algebras and provides a novel example motivated by integrable systems.

Findings

Established a correspondence between braided vertex algebras and non-local Poisson structures.

Constructed a new generalized vertex algebra inspired by the non-linear Schrödinger equation.

Demonstrated the applicability of the framework to integrable models.

Abstract

We study a family of algebras defined using a locally-finite endomorphism called a braiding map. When the braiding map is semi-simple, the algebra is a generalized vertex algebra, while when the braiding map is locally-nilpotent we have a logarithmic vertex algebra. We describe a method that associates to these algebras non-local Poisson vertex algebras, and we use this relation to build a new example of a generalized vertex algebra motivated by the non-linear Schr\"odinger non-local Poisson vertex algebra