Conformal field theory and the cohomology of the moduli space of stable bundles
Conformal field theory and the cohomology of the moduli space of stable bundles
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共形场论与稳定丛模空间的上同调
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发表时间：
1992
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通讯作者：
M. Thaddeus
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作者：
M. Thaddeus
设λ g是亏格g ≥ 2的紧致黎曼曲面，λ是λ g上的1次线丛.然后Seshadri [19]证明了满足ΛV_n = Λ的N_g上秩为2的稳定丛V的模空间是一个非奇异的投射簇Ng。它的有理上同调环H（Ng）是非常丰富的，尽管多年的研究从来没有完全计算。本文的目的是给这个环的一个基本上完全的特征，或者至少把这个问题归结为一个线性代数问题。特别是，我们找到了一个证明纽斯特德的猜想，即pg 1（Ng）= 0。我们还得到了Ng的体积公式，它可以被看作是维滕最近宣布的0度模空间的公式的扭曲版本。我们将采用的方法不是来自代数几何，而是来自数学物理：它依赖于SU（2）Wess-Zumino-维滕模型，这是一个将有限维向量空间与每个带有标记点的黎曼曲面相关联的函子Zk。与模空间的关系是，当函子的“水平”k是偶数时，与没有标记点的Ng相关联的向量空间可以用H（Ng ;L）标识，其中L是Ng上的固定线丛。现在的工作Verlinde为我们提供了一种手段来计算的维数的任何向量空间所产生的我们的函子，特别是dimH（NG ;L），我们将表示D（g，k）。另一方面，Newstead [14]发现了H（Ng）的显式生成元，我们也可以使用Riemann-Roch定理来表示D（g，k）。使这两个公式相等使我们能够在Ng的基本类上计算生成元中的任何单项式，并且通过庞加莱对偶，这至少在原则上足以确定H（Ng）的环结构。在上面的讨论中，有一个重要的问题被忽略了。SU（2）WZW模型当然是与0度而不是1度的丛相关联的，所以为了利用Verlinde的工作，有必要制定一个“扭曲”版本的场论。这是在§2中进行的，但扭曲理论的关键性质，类似于那些使普通理论成为模函子的性质，在本文中没有得到证明。相反，我们将仅限于探索这些主张的后果，并希望在以后的论文中重新证明它们。其余各节的概要如下。在§3中，我们回顾了Verlinde的工作中我们需要的那些部分，展示了在扭曲的情况下它们必须如何修改，并为D（g，k）制定了一些明确的公式。在§4中，我们研究了Ng的上同调，我们通过Narasimhan和Seshadri定理将其视为一个表示空间。我们定义了H（Ng）的Newstead生成元α，β和β i
Let Σg be a compact Riemann surface of genus g ≥ 2, and let Λ be a line bundle over Σg of degree 1. Then the moduli space of rank 2 stable bundles V over Σg such that ΛV ∼= Λ was shown by Seshadri [19] to be a nonsingular projective variety Ng . Its rational cohomology ring H(Ng ) is exceedingly rich and despite years of study has never quite been computed in full. The object of this paper is to give an essentially complete characterization of this ring, or at least to reduce the problem to a matter of linear algebra. In particular, we find a proof of Newstead’s conjecture that pg1(Ng ) = 0. We also obtain a formula for the volume of Ng , which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by Witten. The approach we shall take is not from algebraic geometry but from mathematical physics: it relies on the SU(2) Wess-Zumino-Witten model, which is a functor Zk associating a finite-dimensional vector space to each Riemann surface with marked points. The relationship with the moduli space is that when the “level” k of the functor is even, the vector space associated to Σg with no marked points can be identified with H(Ng ;L), where L is a fixed line bundle over Ng . Now the work of Verlinde provides us with a means of calculating the dimension of any vector space arising from our functor, and in particular dimH(Ng ;L), which we shall denote D(g, k). On the other hand Newstead [14] found explicit generators for H(Ng ), and we can also express D(g, k) in terms of them using a Riemann-Roch theorem. Equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of Ng , and by Poincare duality this is sufficient, at least in principle, to determine the ring structure of H(Ng ). In the discussion above one important point has been skated over. The SU(2) WZW model is of course associated to bundles of degree 0, not degree 1, so in order to make use of Verlinde’s work it is necessary to formulate a “twisted” version of the field theory. This is carried out in §2, but the crucial properties of the twisted theory, analogous to those which make the ordinary theory a modular functor, are not proved in this paper. Rather, we will confine ourselves to exploring the consequences of these claims, and hope to return to justify them in a later paper. An outline of the remaining sections goes as follows. In §3 we review those parts of Verlinde’s work we shall need, show how they must be modified in the twisted case, and work out some explicit formulas for D(g, k). In §4 we study the cohomology of Ng , which we regard throughout as a space of representations via the theorem of Narasimhan and Seshadri. We define Newstead’s generators α , β , and ψi of H (Ng )