Branching rules for winding subalgebras of the affine Kac--Moody algebras $A^{(1)}_1$ and $A^{(2)}_2$

TL;DR

The paper develops a comprehensive framework for branching problems from an affine Kac–Moody algebra $\mathfrak g$ to its winding subalgebras $\mathfrak g[u]$, establishing that the branching support $\Gamma(\mathfrak g,\mathfrak g[u])$ forms a subsemigroup and describing it via the sets $P_u(\Lambda)$, the maximal weights, and the associated shifts $b_{\Lambda,\lambda,u}$. It introduces a Virasoro-coset approach to relate characters of $\mathfrak g$-modules to those of $\mathfrak g[u]$-modules, along with a precise identity of characters and unitarizability criteria that ensure finite multiplicities in the decomposition. The authors provide a detailed general theory and then compute explicit results for the cases $A^{(1)}_1$ and $A^{(2)}_2$, including explicit formulas for the maximal weights and explicit conditions under which the unshifted and shifted $b$-numbers coincide, as well as a saturation description. These results link branching multiplicities to Virasoro representations and offer concrete criteria and explicit data for important affine types, with implications for tensor-product-like phenomena and Frenkel-type conjectures in the affine setting.

Abstract

We study branching problems for affine Kac--Moody algebras. Unlike the finite-dimensional case, an affine Kac--Moody algebra may contain proper subalgebras isomorphic to itself, such as winding subalgebras obtained by rescaling the loop parameter. We investigate the restriction of integrable highest-weight representations to such subalgebras. The restriction remains integrable and decomposes into irreducible components with finite multiplicities, encoded by pairs of highest weights. We show that this set is closed under addition, extending a result of Brion and Knop to the affine setting. We also give a partial description of this set and provide explicit results for types $A^{(1)}_1$ and $A^{(2)}_2$.

Theorems & Definitions (62)

• Theorem 1.1: Theorem \ref{['M1']}
• Theorem 1.2: Theorem \ref{['BAa']}
• Theorem 1.3: Theorem \ref{['b=b.a11']}
• Theorem 1.4: Theorem \ref{['b=b.a22']}
• Corollary 1.5: Corollary \ref{['satured']}
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5