On Determinant Line Bundles
On Determinant Line Bundles
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关于行列式线束
DOI：
10.1142/9789812798411_0011
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发表时间：
1987
影响因子：
2.4
通讯作者：
D. Freed
中科院分区：
文献类型：
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作者：
D. Freed
行列式线丛在Quillen [Q]的一篇著名论文中进入了微分几何。他重视一个全纯线丛L的一个特殊家庭的柯西-黎曼算子的黎曼曲面，建造了一个埃尔米特度量L，并计算其曲率。大约在同一时间，Atiyah和Singer [AS 2]将行列式线丛与物理学中的异常联系起来。稍后，维滕[W1]用η-不变量给出了一个“整体反常”的公式。他建议，它可以被解释为一个完整的联系上的行列式线丛。这些思想是由数学和物理工作者发展起来的。我们的目标是调查其中的一些工作。我们考虑光滑紧致流形X上的任意Dirac算子族D。伴随的拉普拉斯算子具有离散谱，这导致行列式线丛L的修补构造。行列式detD是L的一个截面。Quillen利用Ray和Singer [RS 1]的解析挠率定义了L上的一个度量。这些想法的延伸产生一个酉连接，其曲率和holonomy可以明确计算。完整性公式再现了维滕的全局异常。第1节是与Jean-Michel Bismut的共同工作，他对族[B]的指数定理的证明是曲率公式（1 - 30）中的一个重要组成部分。这些基本主题允许有许多变化，我们在第2和第3节中介绍了其中的两个。假设X是一个复流形，Dirac算子族（或Cauchy-Riemann算子）全纯变化。则L具有自然复结构，并且在适当的几何限制下，标准联络与全纯结构相容。适当的几何假设，即由X扫过的整个空间是Kahler的，至少在参数空间中是局部的，也保证了算子全纯地变化。这个结果的许多特殊情况可以在文献中找到;我们证明的版本是由于Bismut，Gillet和Soule [BGS]。这里的一个新奇之处是观察到，如果X的维数等于2模8，则detD具有自然平方根。在拓扑的基础上，人们可以使用与真实的K-理论相联系的洛林定理来论证L1/2的存在性。然而，我们需要微分几何来看到detD也允许平方根。有一个完整性定理的推广到L1/2。在§4中，我们研究黎曼曲面。这是Quillen最初考虑的情况。Faltings [法]认为决定因素的黎曼曲面在算术方面。这些决定因素也构成了
Determinant line bundles entered differential geometry in a remarkable paper of Quillen [Q]. He attached a holomorphic line bundle L to a particular family of Cauchy-Riemann operators over a Riemann surface, constructed a Hermitian metric on L, and calculated its curvature. At about the same time Atiyah and Singer [AS2] made the connection between determinant line bundles and anomalies in physics. Somewhat later, Witten [W1] gave a formula for “global anomalies” in terms of η-invariants. He suggested that it could be interpreted as the holonomy of a connection on the determinant line bundle. These ideas have been developed by workers in both mathematics and physics. Our goal here is to survey some of this work. We consider arbitrary families of Dirac operators D on a smooth compact manifold X. The associated Laplacian has discrete spectrum, which leads to a patching construction for the determinant line bundle L. The determinant detD is a section of L. Quillen uses the analytic torsion of Ray and Singer [RS1] to define a metric on L. An extension of these ideas produces a unitary connection whose curvature and holonomy can be computed explicitly. The holonomy formula reproduces Witten’s global anomaly. Section 1 represents joint work with Jean-Michel Bismut, whose proof of the index theorem for families [B] is a crucial ingredient in the curvature formula (1.30). These basic themes allow many variations, two of which we play out in §2 and §3. Suppose X is a complex manifold and the family of Dirac operators (or Cauchy-Riemann operators) varies holomorphically. Then L carries a natural complex structure, and under appropriate restrictions on the geometry the canonical connection is compatible with the holomorphic structure. The proper geometric hypothesis, that the total space swept out by X be Kahler, at least locally in the parameter space, also ensures that the operators vary holomorphically. Many special cases of this result can be found in the literature; the version we prove is due to Bismut, Gillet, and Soule [BGS]. One novelty here is the observation that detD has a natural square root if the dimension ofX is congruent to 2 modulo 8. On topological grounds one can argue the existence of L1/2 using Rohlin’s theorem, which is linked to real K-theory. However, one needs the differential geometry to see that detD also admits a square root. There is an extension of the holonomy theorem to L1/2. In §4 we study Riemann surfaces. This is the case originally considered by Quillen. Faltings [Fa] considered determinants on Riemann surfaces in an arithmetic context. These determinants also form the