[Paper Review] Semi-infinite cohomology and the linkage principle for $W$-algebras

Gurbir Dhillon|arXiv (Cornell University)|May 16, 2019

Algebraic structures and combinatorial models51 references4 citations

TL;DR

This paper establishes a semi-infinite cohomology theory for $χ$-algebras associated to simple Lie algebras and principal nilpotent elements, proving a duality between categories of $χ$-modules at levels $\kappa + \kappa_c$ and $-\kappa + \kappa_c$, which verifies the Feigin–Fuchs duality conjecture for $χ$-algebras and determines all homomorphisms between Verma modules via a linkage principle.

ABSTRACT

Let $\mathfrak{g}$ be a simple Lie algebra, and let $W_κ$ be the affine ${W}$-algebra associated to a principal nilpotent element of $\mathfrak{g}$ and level $κ$. We explain a duality between the categories of smooth ${W}$ modules at levels $κ+ κ_c$ and $-κ+ κ_c$, where $κ_c$ is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the ${W}$-algebra. As an application, we determine all homomorphisms between the Verma modules for ${W}$, verifying a conjecture from the conformal field theory literature of de Vos--van Driel. Along the way, we determine the linkage principle for Category $\mathscr{O}$ of the ${W}$-algebra.

Motivation & Objective

• To construct a general theory of semi-infinite cohomology for $\mathscr{W}$-algebras, which had remained elusive despite its importance in conformal field theory and $\mathscr{W}$-gravity.
• To verify the Feigin–Fuchs duality conjecture for $\mathscr{W}$-algebras, extending the known duality for the Virasoro algebra to higher-rank Lie algebras.
• To determine the full structure of homomorphisms between Verma modules for $\mathscr{W}$-algebras, resolving a long-standing open problem in representation theory.
• To establish the linkage principle for Category $\mathscr{O}$ of $\mathscr{W}$-algebras, providing a classification of blocks and highest weight modules.

Proposed method

• Leverages recent advances in the local quantum geometric Langlands program to define semi-infinite cohomology for $\mathscr{W}$-algebras.
• Uses a duality pairing between categories of smooth $\mathscr{W}$-modules at levels $\kappa + \kappa_c$ and $-\kappa + \kappa_c$, induced by the cohomology functor.
• Applies the derived category framework to relate $\operatorname{Verma}_{\kappa}^{op}$ and $\operatorname{Verma}_{-\kappa + 2\kappa_c}$ via canonical dg-equivalence.
• Introduces a renormalized derived category $\mathscr{W}_{\kappa}\text{-mod}$ in which Verma modules become compact objects, enabling homological computations.
• Employs the action of the finite Weyl group $W_{\operatorname{f}}$ and parabolic subgroups to parametrize blocks and classify Verma modules via double cosets.
• Uses the Bruhat order on double cosets $W_{\operatorname{f},\lambda} \backslash W_{\lambda} / W^{\circ}_{\lambda}$ to characterize nontrivial homomorphisms between Verma modules.

Experimental results

Research questions

• RQ1 Does a semi-infinite cohomology theory exist for $\mathscr{W}$-algebras beyond the Virasoro case?
• RQ2 Can the Feigin–Fuchs duality for Verma modules at complementary central charges be generalized to $\mathscr{W}$-algebras?
• RQ3 What is the complete structure of homomorphisms between Verma modules for $\mathscr{W}$-algebras?
• RQ4 What is the linkage principle for Category $\mathscr{O}$ of $\mathscr{W}$-algebras?

Key findings

• A canonical equivalence of dg-categories $\operatorname{Verma}_{\kappa}^{op} \simeq \operatorname{Verma}_{-\kappa + 2\kappa_c}$ establishes a generalized Feigin–Fuchs duality for $\mathscr{W}$-algebras.
• The homomorphism space $\operatorname{Hom}(M_v, M_w)$ between Verma modules is at most one-dimensional and nonzero if and only if $v \geqslant w$ in the Bruhat order.
• The linkage principle for $\mathscr{W}_{\kappa}$-algebras is established, with blocks parametrized by double cosets $W_{\operatorname{f},\lambda} \backslash W_{\lambda} / W^{\circ}_{\lambda}$.
• For co-Verma modules $A_v$, $\operatorname{Hom}(A_v, A_w)$ is nonzero if and only if $v \leqslant w$ at positive level and $v \geqslant w$ at negative level.
• The duality $\mathbf{D}$ on Category $\mathscr{O}$ provides a duality between Verma and co-Verma modules, confirming the structure of co-Verma homomorphisms.
• The construction confirms the existence of a semi-infinite cohomology theory for $\mathscr{W}$-algebras, resolving a long-standing conjecture in conformal field theory.