Thick morphisms of supermanifolds and oscillatory integral operators
Thick morphisms of supermanifolds and oscillatory integral operators
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超流形和振荡积分算子的厚态射
DOI：
10.1070/rm9725
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发表时间：
2015
影响因子：
0.9
通讯作者：
T. Voronov
中科院分区：
文献类型：
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作者：
T. Voronov
我们证明了光滑（超）流形之间的厚态射（或微形式态射），我们以前介绍过，是“量子厚态射”的经典极限，这里定义为函数上的特殊振荡积分算子。在（3，4）中，我们引入了关于（超）流形的“微形式”或“厚”态射的函数的非线性拉回，这推广了普通的光滑映射。根据定义，这样的态射是一种特殊类型的余切丛之间的形式标准关系，即，由依赖于源上的位置坐标和目标上的动量坐标的生成函数指定。该函数被视为零部分附近的幂展开。厚态射形成一个形式范畴;这意味着生成函数的复合律是一个形式幂级数。同样，函数w.r.t.厚态射由形式幂级数给出，其项是非线性微分算子。有一个平行的构造基于反余切丛产生奇函数的非线性拉回（前一个构造适用于偶函数）。我们的主要应用是同伦Schouten或泊松代数的函数之间的L1-态射。另一个应用是建设一个'伴随运营商'的非线性映射的向量丛。（偶数版本的厚态射接近于Cattaneo-Dherin-Weinstein的辛微态射，参见（1）和后续的工作，定义为拉格朗日子流形上辛流形的芽之间的典范关系的芽;我们的拉回的类似物不会在这样的设置中出现。见（4）中对此的进一步讨论。）我们在这里表明，厚态射的微形几何可以被看作是经典极限的某些“量子厚态射”，这是给定的振荡积分算子的一种特殊的。让我们指出，振荡积分算子（和傅立叶积分算子）是众所周知的，以及众所周知的是它们与余切丛之间的正则关系。粗略地说，每个这样的关系定义一类傅里叶积分算子（参见，例如，（2）译注。然而，我们考虑一个非常特殊的积分算子在这个类中，推广的操作员拉回w.r.t.平滑的地图它是由一个“量子”版本的生成函数定义的，指定了一个厚态射。这里的“量子”是指依赖于~。在经典极限中，这种算子对振荡波函数的作用精确地再现了（3，4）的非线性拉回。这同样适用于我们的操作符的合成：在经典极限中，它减少到厚态射的合成。
We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of 'quantum thick morphisms' defined here as particular oscillatory integral operators on func- tions. In (3, 4) we introduced nonlinear pullbacks of functions with respect to 'micro- formal' or 'thick' morphisms of (super)manifolds, which generalize ordinary smooth maps. By definition, such a morphism is a formal canonical relation between the cotangent bundles of a special kind, namely, specified by a generating function de- pending on position coordinates on the source and momentum coordinates on the target. This function is seen as a power expansion near the zero section. Thick morphisms form a formal category; that means that the composition law for the gen- erating functions is a formal power series. Likewise, the pullback of a function w.r.t. a thick morphism is given by a formal power series whose terms are nonlinear dif- ferential operators. There is a parallel construction based on anticotangent bundles yielding nonlinear pullbacks of odd functions (the former construction applies to even functions). Our main application was to L1-morphisms between homotopy Schouten or Poisson algebras of functions. Another application was the construction of an 'adjoint operator' for nonlinear maps of vector bundles. (Thick morphisms in the even version are close to symplectic micromorphisms of Cattaneo-Dherin-Weinstein, see (1) and subsequent works, defined as germs of canonical relations between germs of symplectic manifolds at Lagrangian submanifolds; analogs of our pullbacks do not arise in such a setting. See further discussion of this in (4).) We show here that thick morphisms of microformal geometry can be seen as the classical limit of certain 'quantum thick morphisms', which are given by oscillatory integral operators of a particular kind. Let us point out that oscillatory integral operators (and Fourier integral operators) are well known, as well as well known is their connection with canonical relations between cotangent bundles. Roughly, each such relation defines a class of Fourier integral operators (see, e.g., (2)). We, however, consider a very special integral operator in this class, generalizing the operator of pullback w.r.t. a smooth map. It is defined by a 'quantum' version of a generating function specifying a thick morphism. 'Quantum' here mean depending on ~. The action of such operators on oscillatory wave functions in the classical limit exactly reproduces the nonlinear pullback of (3, 4). The same holds true for the composition of our operators: in the classical limit it reduces to the composition of thick morphisms.