Tetrahedron Diagram and Perturbative Calculation in Chern-Simons-Witten Theory

TL;DR

This paper explores the tetrahedron Wilson loop operator in (2+1)-dimensional Chern-Simons-Witten theory, providing both non-perturbative and perturbative calculations that agree up to third order, advancing understanding of topological quantum field theory.

Contribution

It introduces the tetrahedron operator in Chern-Simons-Witten theory and demonstrates its evaluation through both non-perturbative and perturbative methods, confirming their consistency.

Findings

Non-perturbative evaluation of the tetrahedron operator.

Perturbative calculation matches non-perturbative results up to third order.

Insights into Wilson loop operators in topological quantum field theory.

Abstract

We investigate extended Wilson loop operators, in particular tetrahedron operator in (2 + 1)-dimensional Chern-Simons-Witten theory. This operator emerges naturally from the contribution terms in twoparticle scattering amplitude. We evaluate this diagram non-perturbatively in terms of vacuum expectation values of Wilson loop operators, especially for gauge group SU(N) with specific choices of representations. On the other hand, we also discuss the perturbative calculation of vacuum expectation value in this theory. We show that, up to the third order, this values of unknotted Wilson loop operators are identical to the non-perturbative result.