On the structure of W-algebras in type A

Abstract

We formulate and prove examples of a conjecture which describes the W-algebras in type A as successive quantum Hamiltonian reductions of affine vertex algebras associated with several hook-type nilpotent orbits. This implies that the affine coset subalgebras of hook-type W-algebras are building blocks of the W-algebras in type A. In the rational case, it turns out that the building blocks for the simple quotients are provided by the minimal series of the regular W-algebras. In contrast, they are provided by singlet-type extensions of W-algebras at collapsing levels which are irrational. In the latter case, several new sporadic isomorphisms between different W-algebras are established.

Figures (7)

• Figure 1: One associates to the $(p,q)$-web diagram the vertex algebra obtained from $V^k(\mathfrak{gl}_N)$ by reduction by stages. The gauge group on each face is given by the unitary group $U(a)$ associated with the number, and the half-BPS interface $\mathcal{B}_{p,q}$ on each line is given by the title angle $p/q$.
• Figure 2: $(p,q)$-web for $\mathcal{W}^k(\mathfrak{sl}_{n+r|n}, f_{n+r|n})$ and its flip.
• Figure 3: The Young diagrams represent the $\mathcal{W}$-algebras of $\mathfrak{sl}_6$. They are sorted according to the closure relations of nilpotent orbits with the smallest (corresponding to $V^k(\mathfrak{sl}_6)$) on the left and the biggest ($\mathcal{W}^k(\mathfrak{sl}_6,f_6)$) on the right. The arrows in blue describe inverse Hamiltonian reductions for hook-type $\mathcal{W}$-algebras given by Feh23. The arrows in red describe the expected embeddings given by Conjectures \ref{['conj:embedding']}.
• Figure 4: Symmetric pyramid and grading associated with the partition $(2,3,4)$
• Figure 5: Grading and positives roots for $x$ and $x_2$.

Theorems & Definitions (52)

• Conjecture A
• Theorem A: Theorem \ref{['thm: the case 2,2']}/\ref{['thm: the case 3,2']}/\ref{['thm: the case 2,2,1']}
• Theorem B: Theorem \ref{['Conj at the level of q']}
• Conjecture B
• Conjecture C
• Conjecture D
• Theorem C: Theorem \ref{['thm: the case 2,2']}/\ref{['thm: the case 3,2']}/\ref{['thm: the case 2,2,1']}
• Conjecture E
• Theorem D
• Conjecture F