Topological Auantum Field Theories derived from the Kauffman bracket-猫眼课题宝

DOI：

10.1016/0040-9383(94)00051-4

发表时间：

1995-10

期刊：

Topology

影响因子：

通讯作者：

中科院分区：

文献类型：

作者：

C. Blanchet;N. Habegger;G. Masbaum;P. Vogel

在[41]中，维滕发现了琼斯多项式[15，163]和规范理论之间的复杂关系。(See也预言文章由Atiyah [2]。虽然他的方法使用费曼路径积分的量子场论，维滕给出了令人信服的论点，一个可行的组合方法可以作出严格的使用方法的手术。他的发现包括新的3-流形不变量（有时称为Jones-W&en不变量），其存在性首先由Reshetikhin和Turaev [30]使用量子群和Kirby手术演算[19]证明（参见[20]）。Kohno [21]、Turaev和Viro 1361、Lickorish [22，23]、作者[lo]、Morton和Strickland [29]、Wenzl [40]、Turaev和Wenzl [35]开发了相关不变量的其他组合方法。根据维滕，他的不变量应该属于拓扑量子场理论（TQFT）。这个概念是公理化的Atiyah等人。[38][39][39]。特别地，流形C的状态形成埃尔米特向量空间V（E）（更一般地，V（x）是交换环k上具有单位和对合的模），并且从x1到&的配边M诱导从V（&）到V（&）的转换（k-线性映射），记为Zy。我们知道V（0）是基环k，所以如果aA= C（即A4是从0到x的配边），我们得到V（x）中的向量Z（M），由Z（M）= Z，（1）给出。因此，M诱导8 M的状态。特别地，如果M是封闭的，则Z（M）（也用（M）表示以与物理学家的期望值符号保持一致）位于V（0）= k中，使得TQFT根据其本质产生

IN [41], Witten has made the remarkable discovery of an intricate relationship between the Jones polynomial [15, 163 and gauge theory.(See also the prophetical article by Atiyah [2].) Although his approach uses the Feynman path integral of Quantum Field Theory, Witten gave convincing arguments that a viable combinatorial approach could be made rigorous using the method of surgery. His discovery includes new 3-manifold invariants (sometimes called Jones-W&en invariants), whose existence was first proven by Reshetikhin and Turaev [30] using quantum groups and Kirby’s surgery calculus [19](see also [20]). Other combinatorial approaches for related invariants were developed by Kohno [21], Turaev and Viro 1361, Lickorish [22, 23], the authors [lo], Morton and Strickland [29], Wenzl [40], Turaev and Wenzl [35].According to Witten, his invariants should belong to a topological quantumfield theory (TQFT). This notion was axiomatized by Atiyah et al.[6, 3](see also [38]). In particular, the states of a manifold, C, form a hermitian vector space, V (E)(more generally V (x) is a module over a commutative ring k with unit and involution), and a cobordism M from x1 to & induces a transition (k-linear map), denoted Zy, from V (&) to V (&). One has that V (0) is the ground ring k, so that if aA= C (ie, A4 is a cobordism from 0 to x), one obtains a vector Z (M) in V (x), given by Z (M)= Z,(l). Thus, M induces a state of 8M. In particular, if M is closed, Z (M)(also denoted by (M) in keeping with the physicists’ expectation value notation) lies in V (0)= k, so that TQFTs, by their very nature, yield

Topological Auantum Field Theories derived from the Kauffman bracket

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