We formulate large $N$ duality of $U(N)$ refined Chern–Simons theory with a torus knot/link in $S^3$. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the $\mathrm{\Omega }$-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in $S^3$ to refined BPS invariants in the resolved conifold. Assuming that the extra $U(1)$ global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

我们制定大$氮$的二元性$U（氮）$改进的陈-西蒙斯理论，带有环面结/链接$S^3$。通过研究M理论中细化的BPS态，我们提供了具有D4膜的IIA型弦理论低能有效作用的显式形式$\mathrm{\Omega }$-背景。这种形式使我们能够将环面结/链接的精炼陈-西蒙斯不变量联系起来$S^3$到已解析conifold 中的精炼BPS 不变量。假设额外的$U（1）$全局对称性对 BPS 状态的作用微不足道，对偶性预测与解析圆锥折叠中的环面结/链接相关的 M2-M5 束缚态模空间上同调群的分级维数。因此，这个公式也可以解释为精炼的陈-西蒙斯环面结/环不变量的正猜想。我们还讨论了非环面结的扩展。