[Paper Review] W-Algebras Extending Affine gl(1|1)

Thomas Creutzig, David Ridout|arXiv (Cornell University)|Nov 21, 2011

Algebraic structures and combinatorial models25 references14 citations

TL;DR

This paper constructs free field realizations of W-algebras extending the affine superalgebra $ˇ\mathfrak{gl}(1|1)$, identifying infinite families of extensions that contain subalgebras isomorphic to Feigin and Semikhatov's $W^{(2)}_N$ algebras at central charges $C = \pm 1$. The authors compute operator product expansions (OPEs) and verify that the leading OPE structures match those of $W^{(2)}_N$ for specific parameter choices, confirming the conjecture for $N \leq 4$. The work provides explicit realizations and establishes connections between extended $ˇ\mathfrak{gl}(1|1)$ algebras and known $W$-algebras.

ABSTRACT

It was recently shown that gl^(1|1) admits an infinite family of simple current extensions. Here, these findings are reviewed and explicit free field realisations of the extended algebras are constructed. The leading contributions to the operator product algebra are then calculated. Among these extensions, one finds four infinite families that seem to contain, as subalgebras, copies of the W^(2)_N algebras of Feigin and Semikhatov at various levels and central charges +/- 1.

Motivation & Objective

• To construct explicit free field realizations of extended W-algebras arising from simple current extensions of the affine superalgebra $ˇ\mathfrak{gl}(1|1)$.
• To identify infinite families of these extensions that contain subalgebras isomorphic to Feigin and Semikhatov's $W^{(2)}_N$ algebras.
• To compute and analyze the operator product expansions (OPEs) of the extended algebras to verify the presence of $W^{(2)}_N$ structures.
• To verify the conjecture that certain extended algebras contain $W^{(2)}_N$ subalgebras by matching OPE structures up to the fourth term, including the dimension-3 primary field $“—_N$.
• To determine the central charge and level of the $W^{(2)}_N$ subalgebras within the extended $ˇ\mathfrak{gl}(1|1)$ algebras for specific parameter values of $n$ and $\ell$.

Proposed method

• Utilizes a well-known free field realization of $ˇ\mathfrak{gl}(1|1)$ to compute OPEs of fields in the extended algebras.
• Applies the algorithmic method of simple current extensions to generate infinite families of extended W-algebras labeled by $n \in \mathbb{R}$ and $\ell \in \mathbb{Z}$.
• Identifies bosonic subalgebras generated by specific fields $V_{n,\ell}^{\pm}$, which are rescaled to match the $\mathcal{E}^{\pm}_N$ generators of $W^{(2)}_N$.
• Compares the OPEs of the extended algebra fields with the known OPEs of $W^{(2)}_N$ up to the fourth term, matching the energy-momentum tensor, $\mathcal{H}_N$, and the dimension-3 primary field $\mathcal{W}_N$.
• Verifies that the central charge $C$ and level $K$ of the $W^{(2)}_N$ subalgebra match the expressions derived from the OPE analysis for specific $n$ and $\ell$.
• Checks consistency of the $W^{(2)}_N$ structure by confirming that the field $\mathcal{W}_N$ is a Virasoro primary of dimension 3 and $\mathcal{H}_N$-weight 0.

Experimental results

Research questions

• RQ1 Which extensions of the affine superalgebra $\u02c7\mathfrak{gl}(1|1)$ contain subalgebras isomorphic to the $W^{(2)}_N$ algebra of Feigin and Semikhatov?
• RQ2 What are the central charge and level of the $W^{(2)}_N$ subalgebras embedded in the extended $\u02c7\mathfrak{gl}(1|1)$ algebras for specific parameters $n$ and $\ell$?
• RQ3 Do the operator product expansions of the extended algebra fields match the known OPEs of $W^{(2)}_N$ up to the fourth term, including the dimension-3 primary field $\mathcal{W}_N$?
• RQ4 Under what conditions on $n$ and $\ell$ do the fields $\mathcal{E}^{\pm}_N$ form a $W^{(2)}_N$ algebra with the correct dimensions and OPE structure?
• RQ5 Is the conjecture that certain $\u02c7\mathfrak{gl}(1|1)$ extensions contain $W^{(2)}_N$ subalgebras verified for $N \leq 4$?

Key findings

• The extended algebra $\mathfrak{W}_{n,\ell}$ contains a subalgebra isomorphic to $W^{(2)}_N$ when $\ell=1$ and $n=0,1,2,\ldots$, with $N=2n+1$ and level $K = -2(n-1)(2n+1)/(2n-1)$.
• For $\ell=1$ and $n=\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$, the extended algebra contains a $W^{(2)}_N$ subalgebra with $N=2n+1$ and level $K = -(2n^2 - 1)/n$, central charge $C = -1$.
• When $\ell=2$ and $n=-\frac{3}{4}, -\frac{1}{4}, \frac{1}{4}, \ldots$, the extended algebra contains a $W^{(2)}_N$ subalgebra with $N=4(n+1)$ and level $K = -2(n+1)(4n+1)/(2n+1)$, central charge $C = -1$.
• For $n = -\frac{1}{2}(\ell - 1)$ and $\ell = 1,2,3,\ldots$, the extended algebra contains a $W^{(2)}_N$ subalgebra with $N=\ell$ and level $K = -(β^2 - \ell - 1)/\ell$, central charge $C = -1$.
• The dimension-3 primary field $\mathcal{W}_N$ in the extended algebras matches the $W^{(2)}_N$ OPE structure, confirming the conjecture for $N \leq 4$.
• The central charge of the $W^{(2)}_N$ subalgebra is $C = 1$ when $\ell=1$ and $n=\frac{1}{2},\frac{3}{2},\ldots$, and $C = -1$ otherwise, consistent with the OPE analysis.