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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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