Representations of affine Lie algebras, parabolic differential equations, and Lamé functions
Representations of affine Lie algebras, parabolic differential equations, and Lamé functions
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仿射李代数、抛物型微分方程和 Lamé 函数的表示
DOI：
10.1215/s0012-7094-94-07421-8
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发表时间：
1993
影响因子：
2.5
通讯作者：
A. Kirillov
中科院分区：
文献类型：
--
作者：
P. Etingof;A. Kirillov
我们考虑插入嘉当元素的环面上的 Wess-Zumino-Witten 模型的相关函数；从数学上讲，这意味着我们考虑 $F=\Tr (\Phi_1 (z_1)\ldots \Phi_n (z_n)q^{-\d}e^{h})$ 形式的函数，其中 $\Phi_i$ 是仿射李代数 $\ghat$ 上 Verma 模块和评估模块之间的交织体，$\d$ 是 Verma 模块中的分级运算符，$h$ 位于 $\g$ 的嘉当子代数中。我们推导了这样一个函数满足的微分方程组。特别是，$q\frac{\d} {\d q} F$ 的计算产生与 $\g$ 对应的紧李群上的热方程密切相关的抛物二阶偏微分方程。我们详细考虑$n=1$、$\g = \sltwo$的情况。在这种情况下，我们得到以下微分方程（$q=e^{\pi \i \tau}$）： $ \left( -2\pi\i (K+2)\frac{\d}{\d\tau} +\frac{\d^2}{\d x^2}\right) F = (m(m+1)\wp(x+\frac{\tau}{2}) +c)F$，对于 $K=-2$（临界水平）变为 Lam\'e 方程。对于$m\in\Z$ 的情况，我们推导出$F$ 的积分公式，并发现它们的渐近线为$K\to -2$，从而恢复经典的Lam\'e 函数。
We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form $F=\Tr (\Phi_1 (z_1)\ldots \Phi_n (z_n)q^{-\d}e^{h})$ where $\Phi_i$ are intertwiners between Verma modules and evaluation modules over an affine Lie algebra $\ghat$, $\d$ is the grading operator in a Verma module and $h$ is in the Cartan subalgebra of $\g$. We derive a system of differential equations satisfied by such a function. In particular, the calculation of $q\frac{\d} {\d q} F$ yields a parabolic second order PDE closely related to the heat equation on the compact Lie group corresponding to $\g$. We consider in detail the case $n=1$, $\g = \sltwo$. In this case we get the following differential equation ($q=e^{\pi \i \tau}$): $ \left( -2\pi\i (K+2)\frac{\d}{\d\tau} +\frac{\d^2}{\d x^2}\right) F = (m(m+1)\wp(x+\frac{\tau}{2}) +c)F$, which for $K=-2$ (critical level) becomes Lam\'e equation. For the case $m\in\Z$ we derive integral formulas for $F$ and find their asymptotics as $K\to -2$, thus recovering classical Lam\'e functions.