Kauffman Knot Invariant from SO(N) or Sp(N) Chern-Simons theory and the Potts Model

TL;DR

This paper derives and extends the relationship between SO(N) and Sp(N) Chern-Simons theories and knot invariants, establishing dualities and connecting to the Potts model through perturbative analysis.

Contribution

It introduces a simple variational method to derive the Kauffman polynomial from SO(N) and Sp(N) Chern-Simons theories and explores dualities and connections to the Potts model.

Findings

First-order derivation of Kauffman polynomial from SO(N) and Sp(N) theories

Duality relation between SO(N) and Sp(N) invariants established

Connection between Chern-Simons perturbation theory and Potts model partition function

Abstract

The expectation value of Wilson loop operators in three-dimensional SO(N) Chern-Simons gauge theory gives a known knot invariant: the Kauffman polynomial. Here this result is derived, at the first order, via a simple variational method. With the same procedure the skein relation for Sp(N) are also obtained. Jones polynomial arises as special cases: Sp(2), SO(-2) and SL(2,R). These results are confirmed and extended up to the second order, by means of perturbation theory, which moreover let us establish a duality relation between SO(+/-N) and Sp(-/+N) invariants. A correspondence between the firsts orders in perturbation theory of SO(-2), Sp(2) or SU(2) Chern-Simons quantum holonomies and the partition function of the Q=4 Potts Model is built.