Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix

TL;DR

This paper demonstrates how Chern-Simons theory in the temporal gauge leads to a 2D R-matrix representation of knot invariants, with the universal quantum R-matrix playing a central role and topological invariants arising from additional flag structures.

Contribution

It establishes a direct link between Chern-Simons theory in the temporal gauge and knot invariants via the universal quantum R-matrix, introducing a new geometric interpretation.

Findings

R-matrix representation simplifies knot invariant calculations

Topological invariants emerge from flag structures and R-matrix properties

Turning points contribute specific enhancement factors q^{ ho}

Abstract

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 (xy plane and an y line in it) is introduced, R is the universal quantum R-matrix and turning points contribute the "enhancement" factors q^{\rho}.