DEFORMATION QUANTIZATION OF COMPLEX INVOLUTIVE SUBMANIFOLDS
DEFORMATION QUANTIZATION OF COMPLEX INVOLUTIVE SUBMANIFOLDS
复制标题
复卷合子流形的变形量化
DOI：
10.1142/9789812775061_0008
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
P. Polesello
中科院分区：
文献类型：
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作者：
A. D'agnolo;P. Polesello
WKB算子环的层提供了复流形余切丛的变形量子化。在复辛流形X上可能不存在与WKB算子环局部同构的环层。我们的想法是考虑整个家庭的局部定义层的WKB运营商的变形量子化的$X$。为了准确地说明这一点，我们需要一个由Kontsevich引入的代数体栈的概念。特别地，在Polesello-Schapira中定义的$X$上的WKB模的栈（也参见Kashiwara的接触情况）更好地理解为$X$的变形量子化的代数体栈上的模的栈。
设$V$是$X$的一个对合子流形，为简单起见，假设$V$与其双特征叶的商同构于一个复辛流形$Z$。简单WKB模沿着$V$的自同态代数局部（反）同构于$Z$上WKB算子的拉回。因此，我们可以说一个简单的模提供了一个$V$的变形量子化。同样，由于一般不存在全局定义的简单WKB模，我们的想法是考虑局部定义的简单WKB模的代数体堆栈作为$V$的变形量子化。
在本文中，我们首先定义什么是代数体栈，以及如何局部描述它。然后讨论了复辛流形X上WKB算子的代数体叠，并利用沿沿着V的简单WKB模定义了对合子流形V的形变量子化.最后，我们将这种变形量子化的WKB运营商的商$V$的双特征叶。
The sheaf of rings of WKB operators provides a deformation-quantization of the cotangent bundle to a complex manifold. On a complex symplectic manifold $X$ there may not exist a sheaf of rings locally isomorphic to a ring of WKB operators. The idea is then to consider the whole family of locally defined sheaves of WKB operators as the deformation-quantization of $X$. To state it precisely, one needs the notion of algebroid stack, introduced by Kontsevich. In particular, the stack of WKB modules over $X$ defined in Polesello-Schapira (see also Kashiwara for the contact case) is better understood as the stack of modules over the algebroid stack of deformation-quantization of $X$.
Let $V$ be an involutive submanifold of $X$, and assume for simplicity that the quotient of $V$ by its bicharacteristic leaves is isomorphic to a complex symplectic manifold $Z$. The algebra of endomorphisms of a simple WKB module along $V$ is locally (anti-)isomorphic to the pull-back of WKB operators on $Z$. Hence we may say that a simple module provides a deformation-quantization of $V$. Again, since in general there do not exist globally defined simple WKB modules, the idea is to consider the algebroid stack of locally defined simple WKB modules as the deformation-quantization of $V$.
In this paper we start by defining what an algebroid stack is, and how it is locally described. We then discuss the algebroid stack of WKB operators on a complex symplectic manifold $X$, and define the deformation-quantization of an involutive submanifold $V$ by means of simple WKB modules along $V$. Finally, we relate this deformation-quantization to that given by WKB operators on the quotient of $V$ by its bicharacteristic leaves.