Higher-order Sugawara operators for affine Lie algebras

Abstract:

Let $\hat {\mathfrak {g}}$ be the affine Lie algebra associated to a simple Lie algebra $\mathfrak {g}$. Representations of $\hat {\mathfrak {g}}$ are described by current fields $X(\zeta )$ on the circle ${\mathbf {T}}\;(X \in \mathfrak {g}$ and $\zeta \in {\mathbf {T}})$. In this paper a linear map $\sigma$ from the symmetric algebra $S(\mathfrak {g})$ to (formal) operator fields on a suitable category of $\hat {\mathfrak {g}}$ modules is constructed. The operator fields corresponding to $\mathfrak {g}$-invariant elements of $S(\mathfrak {g})$ are called Sugawara fields. It is proved that they satisfy commutation relations of the form $(\ast )$ \[ [\sigma (u)(\zeta ),X(\eta )] = {c_\infty }D\delta (\zeta /\eta )\sigma ({\nabla _X}u)(\zeta ) + {\text {higher-order}}\;{\text {terms}}\] with the current fields, where ${c_\infty }$ is a renormalization of the central element in $\hat {\mathfrak {g}}$ and $D\delta$ is the derivative of the Dirac delta function. The higher-order terms in $(\ast )$ are studied using results from invariant theory and finite-dimensional representation theory of $\mathfrak {g}$. For suitably normalized invariants $u$ of degree $4$ or less, these terms are shown to be zero. This vanishing is also proved for $\mathfrak {g} = {\text {sl}}(n,{\mathbf {C}})$ and $u$ running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever ${c_\infty }$ is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for $\hat {\mathfrak {g}}$.