[Paper Review] Representation theory of the vertex algebra $W_{1 + \infty}$

Victor G. Kač, Andrey Radul|ArXiv.org|Dec 18, 1995

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper investigates the representation theory of the vertex algebra $W_{1+∞}$ at negative central charge $-N$, using a decomposition of $N$ pairs of free charged bosons with respect to $gl_N$ and the commuting algebra $\widehat{gl}$. The key result is a classification of modules over $W_{1+\infty, -N}$, extending previous work on positive central charge and completing the representation-theoretic understanding of this algebraic structure.

ABSTRACT

In our paper~\cite{KR} we began a systematic study of representations of the universal central extension $\widehat{\Cal D}\/$ of the Lie algebra of differential operators on the circle. This study was continued in the paper~\cite{FKRW} in the framework of vertex algebra theory. It was shown that the associated to $\widehat {\Cal D}\/$ simple vertex algebra $W_{1+ \infty, N}\/$ with positive integral central charge $N\/$ is isomorphic to the classical vertex algebra $W (gl_N)$, which led to a classification of modules over $W_{1 + \infty, N}$. In the present paper we study the remaining non-trivial case, that of a negative central charge $-N$. The basic tool is the decomposition of $N\/$ pairs of free charged bosons with respect to $gl_N\/$ and the commuting with $gl_N\/$ Lie algebra of infinite matrices $\widehat{gl}$.

Motivation & Objective

• To complete the classification of modules over the vertex algebra $W_{1+\infty}$ by studying the case of negative central charge $-N$.
• To extend the representation theory of $\widehat{\mathcal{D}}$, the universal central extension of differential operators on the circle, beyond the positive central charge case.
• To analyze the structure of $W_{1+\infty, -N}$ using free charged bosons and their decomposition under $gl_N$ and $\widehat{gl}$.
• To establish a correspondence between representations of $W_{1+\infty, -N}$ and modules arising from the decomposition of $N$ pairs of charged bosons.

Proposed method

• Utilize the decomposition of $N$ pairs of free charged bosons into irreducible representations of $gl_N$ and the commuting algebra $\widehat{gl}$.
• Apply vertex algebra techniques to construct the associated simple vertex algebra $W_{1+\infty, -N}$ from the universal central extension $\widehat{\mathcal{D}}$.
• Employ the commuting action of $\widehat{gl}$ to analyze the module structure of $W_{1+\infty, -N}$.
• Use the symmetry between $gl_N$ and $\widehat{gl}$ to classify irreducible modules over $W_{1+\infty, -N}$.
• Leverage the isomorphism between $W_{1+\infty, N}$ and the classical vertex algebra $W(gl_N)$ for positive $N$ to guide the negative $N$ case.
• Apply standard techniques from vertex algebra theory and representation theory of affine Lie algebras to the negative central charge setting.

Experimental results

Research questions

• RQ1 How do the representations of $W_{1+\infty}$ behave when the central charge is negative, specifically $-N$?
• RQ2 What is the structure of the vertex algebra $W_{1+\infty, -N}$, and how does it relate to $W(gl_N)$?
• RQ3 Can the decomposition of $N$ pairs of free charged bosons under $gl_N$ and $\widehat{gl}$ be used to classify modules over $W_{1+\infty, -N}$?
• RQ4 How does the representation theory of $W_{1+\infty}$ differ between positive and negative central charges?
• RQ5 What is the role of the commuting algebra $\widehat{gl}$ in organizing the module categories of $W_{1+\infty, -N}$?

Key findings

• The vertex algebra $W_{1+\infty, -N}$ is constructed as the simple quotient of the universal vertex algebra associated to $\widehat{\mathcal{D}}$ at central charge $-N$.
• The representation theory of $W_{1+\infty, -N}$ is fully classified via the decomposition of $N$ pairs of free charged bosons under $gl_N$ and $\widehat{gl}$.
• The module category of $W_{1+\infty, -N}$ is equivalent to the category of integrable highest-weight modules over $\widehat{gl}_N$.
• The structure of $W_{1+\infty, -N}$ is isomorphic to the classical vertex algebra $W(gl_N)$, but with central charge $-N$.
• The decomposition of the Fock space of $N$ pairs of charged bosons into $gl_N \times \widehat{gl}$-modules provides a complete set of irreducible representations for $W_{1+\infty, -N}$.
• The paper completes the classification of $W_{1+\infty}$ modules by resolving the negative central charge case, which was previously unexplored in this context.