阿贝尔共形域论与行列式丛
Abelian Conformal Field theories and Determinant Bundles
摘要：
本文是从[TUY]发展的共形域论出发，构造模函子和拓扑量子场论的系列论文的第一篇。其基本思想是，与定向尖曲面的Teichmüller空间上的简单李代数相关联的真空层的协变常数部分给出了模函子与定向尖曲面相关联的向量空间。然而，真空层上的联络只是射影平坦的，所以我们需要找到一个合适的具有联络的线丛，使得两者的张量积具有平坦的联络。我们将构造一个线丛，它在任何具有形式坐标的点曲线族上有一个联络。通过计算这个线丛的曲率，我们得出结论，我们实际上需要这个线丛的一个分数幂，以便在张紧之后获得平坦连接。为了函数地提取这个分数幂，我们需要构造线丛的优选部分。我们将通过使用所谓的$bc$ -鬼系统（Faddeev-Popov鬼）首先在协变量化[FP]中引入来构造线丛。我们遵循[KNTY]的思想，但从[TUY]的角度描述它。
英文摘要：
The present paper is the first in a series of papers, in which we shall construct modular functors and Topological Quantum Field Theories from the conformal field theory developed in [TUY]. The basic idea is that the covariant constant sections of the sheaf of vacua associated to a simple Lie algebra over Teichmüller space of an oriented pointed surface gives the vectorspace the modular functor associates to the oriented pointed surface. However the connection on the sheaf of vacua is only projectively flat, so we need to find a suitable line bundle with a connection, such that the tensor product of the two has a flat connection. We shall construct a line bundle with a connection on any family of pointed curves with formal coordinates. By computing the curvature of this line bundle, we conclude that we actually need a fractional power of this line bundle so as to obtain a flat connection after tensoring. In order to functorially extract this fractional power, we need to construct a preferred section of the line bundle. We shall construct the line bundle by the use of the so-called $bc$-ghost systems (Faddeev-Popov ghosts) first introduced in covariant quantization [FP]. We follow the ideas of [KNTY], but decribe it from the point of view of [TUY].
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