限制Grassmannian流形上的Dirac算子
Dirac operator on the restricted Grassmannian manifold
摘要：
米克尔森在他的书中指出，西格尔和威尔逊的无限维格拉斯曼流形允许一个自旋^在此之后，他自然考虑了在其上定义Dirac算子的问题。米克尔森给出了这样一个算子的可能候选者，但不幸的是，它被证明是严重发散的，他把它作为一个开放的问题，对他的原始构造进行适当的修改，以获得一个具有预期属性的良好定义（无界）运算符。使用费米Fock空间表示的限制酉群相关的极化希尔伯特空间，在著名的书中介绍了由Pressley和Segal写的循环群，我们构造了一个定义良好的候选人的Dirac型算子的限制格拉斯曼流形上的相关空间的旋量。作为我们的主要结果，我们证明了我们的算子是一个具有有限维核的无界对称算子。
英文摘要：
In his book Mickelsson notices that the infinite-dimensional Grassmannian manifold of Segal and Wilson admits a Spin^c structure and after this he naturally considers the problem of defining a Dirac operator on it. Mickelsson gives a possible candidate for such an operator but unfortunately it proves out to be badly diverging and he leaves it as an open problem to introduce proper modifications to his original construction in order to obtain a well-defined (unbounded) operator with expected properties. Using fermionic Fock space representations of the restricted unitary group associated to a polarized Hilbert space, introduced in the well-known book written by Pressley and Segal on loop groups, we construct a well-defined candidate for a Dirac type operator on the restricted Grassmannian manifold acting on a relevant space of spinors. As our main result we show that our operator is an unbounded symmetric operator with finite-dimensional kernel.
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