Invariants of higher-dimensional knots and topological quantum field theories
Invariants of higher-dimensional knots and topological quantum field theories
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高维结的不变量和拓扑量子场论
DOI：
10.5167/uzh-36005
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发表时间：
2001
影响因子：
1.1
通讯作者：
C. Rossi
中科院分区：
文献类型：
--
作者：
C. Rossi
本文的工作可分为两个主要部分：第一部分研究任意维BF理论的BV形式的泛函积分量子化，第二部分研究欧几里德空间中高维k点的（同位素-）不变量。我们简要地评论一下第一部分和第二部分的动机和结果。三维BF理论（加上所谓的“宇宙学术语”）只是chen - simons作用泛函的另一种写法。人们立即看到，与chen - simons理论相反，三维BF理论可以直接推广到任意维度。Cattaneo， Cotta-Ramusino和Longoni明确地给出了圆嵌入R的s步的上同类，对于m bbbb3；这些类是来自于chen - simons理论在三维中的微扰展开的微扰结不变量在任何维度上的自然推广。因此，高维的BF理论似乎是自然的拓扑量子场论（简称TQFT），将这些上同调类解释为微扰展开。然而，由于可约对称性的存在，任意维BF理论的泛函积分量化需要比三维更小心；因此，我们不得不求助于所谓的BV形式主义。在这一点上，我们可以为BF理论在与高维结相关的所有维度上产生一个B - v可观测值（即，通常的规范不变泛函的推广），即余维2的sp点嵌入到r中。我们讨论了这种n个可观测值的函数积分量化，它有望产生嵌入的不变量。最后，我们显式地计算了该观测值的v. v.的微扰展开式的二阶和三阶项。第ii)部分与第i)部分直接相关，尽管我们指出数学结果独立于tqft框架。在这里，我们研究了由上述可观测值的真空期望值（简称v.e.v.）的摄动不变量所产生的嵌入空间上的函数的性质。对于余维数为2的奇球，二阶项可以用博特不变量Θ2来标识。3阶项在余维为2的偶球嵌套空间上得到functionΘ3；对于m = 4，该函数的一个修正是同位素不变量。还给出了一般情况下的表征。
The present work can be divided into two main parts: i) the first deals with the functionalintegral quantization via BV formalism of BF theories in any dimension, and ii) the second with (isotopy-) invariants of higher-dimensional k nots in Euclidean spaces. We briefly comment on the motivations and the results of part i) andii). BF theory in3 dimensions (with the addition of a so-called “cosmological term”) is just another way of writing the Chern–Simons action funct ional. One sees immediately that, on the contrary to Chern–Simons theory, BF theory in3 dimensions admits a straightforward generalizations to arbitrary dimension . Cattaneo, Cotta-Ramusino and Longoni explicitly produced cohomology classes of the s pace of imbeddings of the circleS into R, for m > 3; these classes are the natural generalizations in any dimension of the perturbative knot invariants coming from t he perturbative expansions in Chern–Simons theory in3 dimensions. Therefore, BF theories in higher dimensions seem to be the natural Topological Quantum Field Theor ies (shortly, TQFT) to interpret such cohomology classes as perturbative expansi ons. However, functionalintegral quantization of BF theories in arbitrary dimensions requires more care than in 3 dimensions, due to the presence of reducible symmetries; he nce, we have to resort to the so-called BV formalism. At this point we may produce a B V-observable (i.e., a generalization of usual gauge-invariant functionals) for BF theories in all dimensions related to higher-dimensional knots, i.e. imbeddings of sp heres of codimension 2 into R. We discuss the functional-integral quantization of such a n observable which is expected to yield an invariant of the imbedding. Finally, we compute explicitly the terms of order2 and3 of the perturbative expansion of the v.e.v. of this observab le. Part ii) is directly linked to parti), although we point out that the mathematical results are independent of the TQFT-framework. Here we stud y he properties of the functions on the space of imbeddings coming from perturbati ve invariants of the Vacuum Expectation Value (shortly v.e.v.) of the observable de scribed above. The term of order2 can be identified with the Bott invariant Θ2 for odd spheres of codimension 2. The term of order3 yields a functionΘ3 on the space of imbeddings of even spheres of codimension2; a modification of this function is shown to be an isotopy-inv ariant for m = 4. A characterization of the general case is also given.