On elliptic genera and theta-functions
On elliptic genera and theta-functions
复制标题
DOI：
10.1016/0040-9383(95)00042-9
复制
发表时间：
1996-07
期刊：
影响因子：
--
通讯作者：
Kefeng Liu
中科院分区：
文献类型：
--
作者：
Kefeng Liu
本文的主要目的是给出Witten猜想并由Taubes[25]、Bott-Taubes[S]、Hirzebruch Cl23和Krichever[15]证明的Witten刚性定理的一个简单而统一的新证明。我们的证明表明，椭圆代数的固有对称性--模不变性实际上隐含着它们的刚性。还讨论了椭圆亏格的一些新性质及其与theta-函数的关系。我们指出，我们的证明本质上利用了循环群和循环空间的新特征--模不变性。我们注意到，在模群的帮助下，我们可以通过简单地处理有限维流形来捕捉循环空间上的拓扑信息。通过进一步发展这一思想，在文[21]中，我们证明了Dirac算子在环空间上被更高级别的循环群表示扭曲的刚性，而Witten里迪奇定理是水平1的特例。文[21]还通过提炼本文的论证，得到了许多拓扑消零定理，特别是环空间的一个‘&-消零定理。在文[19]中，我们再次利用模不变性建立了一个一般的奇妙消去公式，将Hirzebruch型的L-型与某些扭曲的‘&-型联系起来，得到了许多有趣的拓扑学结果。这些结果发表在文献[20]中，设X是允许圆作用的光滑紧自旋流形，P是X上与圆作用可交换的椭圆算子。则P的核和余核都是S模。Lefschetz数，或P在g E S‘处的特征标号定义为
THE MAIN PURPOSE of this paper is to give a simple and unified new proof of the Witten rigidity theorems, which were conjectured by Witten and first proved by Taubes [25], Bott-Taubes [S], Hirzebruch Cl23 and Krichever [15]. Our proof shows that the modular invariance, which is the intrinsic symmetry of elliptic genera, actually implies their rigidity. Some new properties of elliptic genera and their relationships with theta-functions are also discussed. We remark that our proof makes essential uses of the new feature of loop groups and loop spaces, the modular invariance. We note that, with the help of the modular group, we can catch the topological information on loop space by simply working on finite-dimensional manifold. By developing this idea further, in [21] we have proved the rigidity of the Dirac operator on loop space twisted by higher-level loop group representations, while the Witten ridigity theorems are the special cases of level 1. Many topological vanishing theorems are also derived in [21] by refining the argument in this paper, especially an ‘&-vanishing theorem for loop space. In [19] modular invariance is used again to establish a general miraculous cancellation formula, relating the Hirzebruch L-form to certain twisted ‘&-forms, which has many interesting topological results as consequences. These results were announced in [20].Let X be a smooth compact spin manifold admitting a circle action and P be an elliptic operator on X commuting with the action. Then both the kernel and the cokernel of P are S’-modules. The Lefschetz number, or the character-valued index of P at g E S’, is defined to be