[Paper Review] Duality via convolution of W-algebras

Thomas Creutzig, Andrew R. Linshaw+2 more|arXiv (Cornell University)|Mar 3, 2022

Advanced Topics in Algebra4 citations

TL;DR

This paper proves a conjecture that the relative semi-infinite cohomology of a hook-type W-algebra tensored with a kernel vertex algebra via convolution yields its Feigin-Frenkel dual W-algebra. The authors establish this duality by constructing a convolution operation using relative semi-infinite cohomology and verifying isomorphism through character calculations and representation theory, extending Feigin-Frenkel duality beyond affine cosets to full W-algebras.

ABSTRACT

Feigin-Frenkel duality is the isomorphism between the principal $\mathcal{W}$-algebras of a simple Lie algebra $\mathfrak{g}$ and its Langlands dual Lie algebra ${}^L\mathfrak{g}$. A generalization of this duality to a larger family of $\mathcal{W}$-algebras called hook-type was recently conjectured by Gaiotto and Rapčák and proved by the first two authors. It says that the affine cosets of two different hook-type $\mathcal{W}$-algebras are isomorphic. A natural question is whether the duality between affine cosets can be enhanced to a duality between the full $\mathcal{W}$-algebras. There is a convolution operation that maps a hook-type $\mathcal{W}$-algebra $\mathcal{W}$ to a certain relative semi-infinite cohomology of $\mathcal{W}$ tensored with a suitable kernel VOA. The first two authors conjectured previously that this cohomology is isomorphic to the Feigin-Frenkel dual hook-type $\mathcal{W}$-algebra. Our main result is a proof of this conjecture.

Motivation & Objective

• To extend Feigin-Frenkel duality from affine cosets to full W-algebras by constructing a convolution operation.
• To prove that the relative semi-infinite cohomology of a hook-type W-algebra with a kernel vertex algebra yields its Langlands dual W-algebra.
• To generalize the duality beyond principal W-algebras to a broader class of W-superalgebras, including subregular and minimal types.
• To establish a uniform framework for duality in W-algebras using the structure of hook-type algebras and their decomposition properties.
• To verify the conjecture via character computations and representation-theoretic arguments involving Verma modules and integrable representations.

Proposed method

• Define a convolution operation using relative semi-infinite cohomology of a W-algebra tensored with a kernel vertex algebra.
• Utilize the Chevalley anti-involution and the action of the loop algebra to construct the cohomological complex.
• Apply the Euler–Poincaré principle to compute the character of the cohomology, linking it to the characters of integrable representations.
• Use the non-degenerate bilinear form on Verma modules to ensure non-degeneracy and compute the cohomology in degree zero.
• Leverage the relation $k + h^ atural = -( ilde{k} + ilde{h}^ atural)$ to relate conformal weights and simplify character expressions.
• Show that the cohomology is one-dimensional and isomorphic to the dual W-algebra by identifying the lowest weight space with the invariant subspace $(L_ u imes L_ u^lat)^rak{g}$.

Experimental results

Research questions

• RQ1 Can Feigin-Frenkel duality be extended from affine cosets to the full W-algebras via a convolution operation?
• RQ2 Is the relative semi-infinite cohomology of a hook-type W-algebra with a suitable kernel isomorphic to its Langlands dual W-algebra?
• RQ3 How does the convolution operation preserve duality in the context of quantum Hamiltonian reduction and vertex algebra structures?
• RQ4 What role do character computations and Verma module representations play in verifying the duality at the level of cohomology?
• RQ5 Can the duality be uniformly formulated across all types of hook-type W-algebras, including Lie superalgebras?

Key findings

• The relative semi-infinite cohomology $H^{rac{ty}{2}+0}_{ ext{rel}}(rak{g}, b{V}^k_ u oxtimes b{V}^ ilde{k}_ u)$ is one-dimensional and of lowest conformal weight, confirming the existence of a unique non-trivial cohomology class.
• The character of the cohomology is $q^{rac{| u+ ho|^2 - | u^lat+ ho|^2}{2(k+h^ atural)}}$, which matches the character of the dual W-algebra under the duality condition.
• The cohomology is isomorphic to the dual W-algebra $\W^\ell({}^L\frak{g}, f_{\frak{a}})$, proving the conjecture that convolution yields the dual algebra.
• The isomorphism holds uniformly across all hook-type W-algebras, including types $A^\pm, B^\pm, C^\pm, D^\pm, O^\pm$, confirming the duality beyond principal cases.
• The proof relies on the complete reducibility of integrable $rak{g}$-modules and the identification of the cohomology with the invariant subspace $(L_ u imes L_ u^lat)^rak{g}$.
• The result generalizes Feigin-Frenkel duality to full W-algebras, not just their cosets, by realizing the dual algebra as a cohomological construction.