Refined Chern-Simons Theory and Knot Homology
Refined Chern-Simons Theory and Knot Homology
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完善的陈-西蒙斯理论和结同调
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Shamil Shakirov
中科院分区：
文献类型：
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作者：
Mina Aganagic;Shamil Shakirov
精化的Chern-Simons理论是Seifert流形上普通Chern-Simons理论的单参数变形。它是通过N个M5膜上的理论的一个指数来定义的，其中相应的单参数变形是几何背景的自然变形。类似于未加细的情形，加细的Chern-Simons理论的解是用S和T矩阵给出的，它们是通常矩阵的适当Macdonald变形。这提供了一种直接的方法来计算一个广泛的三流形和结的精化陈-西蒙斯不变量。本文证明了精化Chern-Simons理论的纽结不变量与三阶化纽结同调理论的纽结超多项式-- Poincare多项式是一致的。这个猜想在S^3中的大量环面纽结中得到验证，这些纽结被基本表示着色。这是arXiv：1105.5117的简短版本，包含了一些新的结果。
The refined Chern-Simons theory is a one-parameter deformation of the ordinary Chern-Simons theory on Seifert manifolds. It is defined via an index of the theory on N M5 branes, where the corresponding one-parameter deformation is a natural deformation of the geometric background. Analogously with the unrefined case, the solution of refined Chern-Simons theory is given in terms of S and T matrices, which are the proper Macdonald deformations of the usual ones. This provides a direct way to compute refined Chern-Simons invariants of a wide class of three-manifolds and knots. The knot invariants of refined Chern-Simons theory are conjectured to coincide with the knot superpolynomials -- Poincare polynomials of the triply graded knot homology theory. This conjecture is checked for a large number of torus knots in S^3, colored by the fundamental representation. This is a short, expository version of arXiv:1105.5117, with some new results included.