纽结不变量与拓扑量子场论
Knot Invariants and Topological Quantum Field Theory
摘要：
本文介绍了纽结理论及其与量子场论的联系，旨在提供一些在教科书中不容易找到的基本计算的细节。本文概括性地介绍了用规范群$\mathcal{G}$研究Chern-Simons理论的方法，以及Wilson线的某些表象，并对Chern-Simons传播子进行了重要的计算。对$U(1)$ Chern-Simons理论进行了显式计算，得到了拓扑不变量，并最终得到了纽结不变量.进一步利用这一结果和高斯连接数公式计算了威尔逊圈的期望值。有色结不变量也与使用同调理论获得的更高级的结不变量一起讨论，即，琼斯和HOMFLY多项式的分类。介绍了$SU(N)$规范群的各种纽结不变量，并简要介绍了A-多项式和超A-多项式。本文讨论了这一领域的最新进展，给出了色超多项式的一个简化公式，HOMFLY多项式的一个封闭表达式，以及无重数情况下$U_q(\mathfrak{sl}_N)$的$6j$符号的一个简化表达式.此外，还开发了一个基于该公式的MATHEMATICA程序，它可以计算使用HOMFLY多项式的闭式表达式所需的$6j$ -符号和所需的对偶矩阵。
英文摘要：
An elementary introduction to knot theory and its link to quantum field theory is presented with an intention to provide details of some basic calculations in the subject, which are not easily found in texts. Study of Chern-Simons theory with gauge group $\mathcal{G}$, along with the Wilson lines carrying some representation is explained in generality, and a vital calculation of the Chern-Simons propagator is done. Explicit calculation for $U(1)$ Chern-Simons theory is presented, which leads to the topological invariants, and finally to knot invariants. Further, using this result along with the Gauss linking number formula, the expectation value of Wilson loops are calculated. Colored knot invariants are also discussed along with more advanced knot invariants which are obtained using Homology theory, i.e., categorification of Jones and HOMFLY polynomials. Various knot invariants for $SU(N)$ gauge group are also introduced, along with a brief introduction to A-polynomials and super A-polynomials. Recent developments in the field are explored, and we discuss a conjectured formula for colored superpolynomials, closed-form expression for HOMFLY polynomials, and conjectured expression for $6j$ symbol for $U_q(\mathfrak{sl}_N)$ for multiplicity free case. Also, a MATHEMATICA program based on the conjectured formula had been developed, which can compute the $6j$-symbols and the desired duality matrices which are needed to use the closed-form expression for HOMFLY polynomials.
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