arXiv subjects

Juan J. Villarreal

Publications and source records attributed to Juan J. Villarreal.

4 recordsLinked to original sources

arXiv2024

Braided Logarithmic Vertex Algebras

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 Schrödinger non-local Poisson vertex algebra

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arXiv2024

The centre of the modular affine vertex algebra

The Feigin--Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W.~Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the $p$-centre. They conjectured that at the critical level the centre should be generated by the Feigin--Frenkel centre and the $p$-centre. In this paper we prove the conjecture for classical simple Lie algebras for $p$ larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the $p$-centre.

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arXiv2023

Lambda bracket and Intertwiners

We describe the intertwiners between modules of a vertex algebra using the language of lambda bracket. We apply this formalism to obtain some classical results on conformal field theory.

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arXiv2021

Logarithmic Vertex Algebras

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.

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