[Paper Review] Coset Vertex Operator Algebras and $\W$-Algebras

Tomoyuki Arakawa, Cuipo Jiang|arXiv (Cornell University)|Jan 24, 2017

Algebraic structures and combinatorial models46 references3 citations

TL;DR

This paper provides an explicit construction of the weight-three generator for the coset vertex operator algebra $ C_{L_{\widehat{\mathfrak{sl}_n}}(l,0)\otimes L_{\widehat{\mathfrak{sl}_n}}(1,0)}(L_{\widehat{\mathfrak{sl}_n}}(l+1,0)) $, and proves that for $ n=3 $, this coset is isomorphic to the $ \mathcal{W} $-algebra $ \mathcal{W}_{-3 + \frac{l+3}{l+4}}(\mathfrak{sl}_3) $, confirming a conjecture on level-rank duality for $ \mathfrak{sl}_3 $-type $ \mathcal{W} $-algebras.

ABSTRACT

We give an explicit description for the weight three generator of the coset vertex operator algebra $C_{L_{\widehat{\sl_{n}}}(l,0)\otimes L_{\widehat{\sl_{n}}}(1,0)}(L_{\widehat{\sl_{n}}}(l+1,0))$, for $n\geq 2, l\geq 1$. Furthermore, we prove that the commutant $C_{L_{\widehat{\sl_{3}}}(l,0)\otimes L_{\widehat{\sl_{3}}}(1,0)}(L_{\widehat{\sl_{3}}}(l+1,0))$ is isomorphic to the $\W$-algebra $\W_{-3+\frac{l+3}{l+4}}(\sl_3)$, which confirms the conjecture for the $\sl_3$ case that $C_{L_{\widehat{\frak g}}(l,0)\otimes L_{\widehat{\frak g}}(1,0)}(L_{\widehat{\frak g}}(l+1,0))$ is isomorphic to $\W_{-h+\frac{l+h}{l+h+1}}(\frak g)$ for simply-laced Lie algebras ${\frak g}$ with its Coxeter number $h$ for a positive integer $l$.

Motivation & Objective

• To explicitly describe the weight-three generator of the coset vertex operator algebra $ C_{L_{\widehat{\mathfrak{sl}_n}}(l,0)\otimes L_{\widehat{\mathfrak{sl}_n}}(1,0)}(L_{\widehat{\mathfrak{sl}_n}}(l+1,0)) $ for $ n \geq 2, l \geq 1 $.
• To prove that the commutant $ C_{L_{\widehat{\mathfrak{sl}_3}}(l,0)\otimes L_{\widehat{\mathfrak{sl}_3}}(1,0)}(L_{\widehat{\mathfrak{sl}_3}}(l+1,0)) $ is isomorphic to the $ \mathcal{W} $-algebra $ \mathcal{W}_{-3 + \frac{l+3}{l+4}}(\mathfrak{sl}_3) $, verifying a conjecture for the $ \mathfrak{sl}_3 $ case.
• To establish a general isomorphism between coset constructions and $ \mathcal{W} $-algebras for simply-laced Lie algebras, extending level-rank duality in vertex operator algebra theory.

Proposed method

• The authors use the coset construction to define the commutant $ C_V(U) $ as the subalgebra of $ V $ that commutes with a vertex operator subalgebra $ U $, focusing on $ V = L_{\widehat{\mathfrak{sl}_n}}(l,0) \otimes L_{\widehat{\mathfrak{sl}_n}}(1,0) $ and $ U = L_{\widehat{\mathfrak{sl}_n}}(l+1,0) $.
• They explicitly identify the weight-three generator of the coset algebra using the structure of the Virasoro algebra and fusion rules.
• A key technical tool is the use of a map $ \varphi $ that sends elements of the coset algebra to their $ \widetilde{W} $-image in the $ \mathcal{W} $-algebra, preserving the vertex operator algebra structure.
• The proof relies on induction on weight and the use of inner products between monomials in $ W $-fields and their $ \widetilde{W} $-counterparts to show that $ \varphi $ is an isomorphism.
• The authors leverage known results on $ C_2 $-cofiniteness and rationality of parafermion algebras to support the structure of the coset algebra.
• They establish a duality between the coset construction and $ \mathcal{W} $-algebras by comparing characters and using self-duality of $ \mathcal{W}_{k}(\mathfrak{sl}_n) $.

Experimental results

Research questions

• RQ1 What is the explicit form of the weight-three generator in the coset vertex operator algebra $ C_{L_{\widehat{\mathfrak{sl}_n}}(l,0)\otimes L_{\widehat{\mathfrak{sl}_n}}(1,0)}(L_{\widehat{\mathfrak{sl}_n}}(l+1,0)) $?
• RQ2 Is the commutant $ C_{L_{\widehat{\mathfrak{sl}_3}}(l,0)\otimes L_{\widehat{\mathfrak{sl}_3}}(1,0)}(L_{\widehat{\mathfrak{sl}_3}}(l+1,0)) $ isomorphic to a $ \mathcal{W} $-algebra of the form $ \mathcal{W}_{-3 + \frac{l+3}{l+4}}(\mathfrak{sl}_3) $?
• RQ3 Does the isomorphism $ C_{L_{\widehat{\mathfrak{g}}}(l,0)\otimes L_{\widehat{\mathfrak{g}}}(1,0)}(L_{\widehat{\mathfrak{g}}}(l+1,0)) \cong \mathcal{W}_{-h + \frac{l+h}{l+h+1}}(\mathfrak{g}) $ hold for all simply-laced Lie algebras $ \mathfrak{g} $ with Coxeter number $ h $?

Key findings

• The weight-three generator of the coset vertex operator algebra $ C_{L_{\widehat{\mathfrak{sl}_n}}(l,0)\otimes L_{\widehat{\mathfrak{sl}_n}}(1,0)}(L_{\widehat{\mathfrak{sl}_n}}(l+1,0)) $ is explicitly constructed for all $ n \geq 2 $ and $ l \geq 1 $, providing a concrete realization of the algebra's structure.
• For $ n = 3 $, the commutant $ C_{L_{\widehat{\mathfrak{sl}_3}}(l,0)\otimes L_{\widehat{\mathfrak{sl}_3}}(1,0)}(L_{\widehat{\mathfrak{sl}_3}}(l+1,0)) $ is isomorphic to the $ \mathcal{W} $-algebra $ \mathcal{W}_{-3 + \frac{l+3}{l+4}}(\mathfrak{sl}_3) $, confirming the conjecture for the $ \mathfrak{sl}_3 $ case.
• The isomorphism is established via a surjective vertex operator algebra homomorphism $ \varphi $ from the coset algebra to the $ \mathcal{W} $-algebra, which is shown to be an isomorphism by matching characters and using duality.
• The proof relies on a key lemma showing that if a linear combination of monomials in $ W $-fields vanishes, then the corresponding combination in $ \widetilde{W} $-fields also vanishes, preserving the inner product structure.
• The authors demonstrate that the $ \mathcal{W} $-algebra $ \mathcal{W}_{k}(\mathfrak{sl}_n) $ is self-dual and generated by $ \widetilde{W} $, which is essential for the isomorphism proof.
• The result extends level-rank duality to $ \mathcal{W} $-algebras, showing that the coset construction yields $ \mathcal{W} $-algebras with central charge $ c = -3 + \frac{l+3}{l+4} $ for $ \mathfrak{sl}_3 $, and supports a general conjecture for simply-laced Lie algebras.