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Chern-Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix. (English) Zbl 1204.81097
Summary: In temporal gauge $$A_{0}=0$$ the $$3d$$ Chern-Simons theory acquires quadratic action and an ultralocal propagator. This directly implies a 2d R-matrix representation for the correlators of Wilson lines (knot invariants), where only the crossing points of the contours projection on the $$xy$$ plane contribute. Though the theory is quadratic, $$P$$-exponents remain non-trivial operators and $$R$$-factors are easier to guess then derive. We show that the topological invariants arise if additional flag structure $${\mathbb R}^3\supset {\mathbb R}^2\supset {\mathbb R}^1$$ ($$xy$$ plane and a $$y$$ line in it) is introduced, $$R$$ is the universal quantum $$R$$-matrix and turning points contribute the “enhancement” factors $$q^{\rho}$$.

MSC:
 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 17B37 Quantum groups (quantized enveloping algebras) and related deformations 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
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