Representations of affine Lie algebras, parabolic differential equations, and Lamé functions
Let \(\mathfrak g\) be a finite-dimensional complex simple Lie algebra and \({\mathfrak h} \subset {\mathfrak g}\) a Cartan subalgebra. Let \(\widehat {\mathfrak g}\) be the associated affine Lie algebra, and \(M_{\lambda, k}\) the Verma module for \(\widehat{\mathfrak g}\) with level \(k\) and highest weight \(\lambda \in {\mathfrak h}^*\) for \(\mathfrak g\). The authors consider the \(n\)- point correlation functions `on the torus' \[ \text{trace}_{M_{\lambda_ 0,k}} (\Phi'(\tau_ 1) \dots \Phi^ n(\tau_ n) q^{-\partial} e^ h), \] where \(\Phi^ i(\tau_ i) : M_{\lambda_{i,k}} \to M_{\lambda_{i - 1,k}} \otimes V_ i\) are vertex operators, \(\tau_ 1, \dots,\tau_ n \in \mathbb{C}^ \times\), \(\lambda_ 0 = \lambda_ n\), \(V_ i\) is a finite-dimensional irreducible \(\mathfrak g\)-module, \(\partial\) is the grading operator on Verma modules, and \(h \in {\mathfrak h}^*_ \mathbb{R}\). In the `flat' case, it is well known that the \(n\)-point functions satisfy the Knichnik- Zamolodchikov equation. The authors derive a differential equation satisfied by their functions and study the asymptotics of its solutions. When \({\mathfrak g} = \text{sl}_ 2\), \(V_ 0 = \cdots = V_ n\) and \(k \to -1\) (the critical level), they find that the asymptotics are closely related to Lamé functions.
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