On the Calculation of Invariant Tensors in Gauge Theories

TL;DR

This paper introduces an efficient method to identify independent invariant tensors in gauge theories, leveraging a theorem connecting invariant tensors with D-flat directions, and demonstrates its application across various gauge groups and representations.

Contribution

The paper presents a novel, efficient approach for calculating invariant tensors in gauge theories, linking them to D-flat directions, with practical examples across different gauge groups.

Findings

Successfully identified invariant tensors for SO(3), SU(2), and SU(3) gauge theories.

Demonstrated the method's efficiency and applicability to various representations.

Provided explicit sets of invariant tensors for the studied theories.

Abstract

We present an efficient method for finding the independent invariant tensors of a gauge theory. Our method uses a theorem relating invariant tensors and D-flat directions in field space. We apply our method to several examples-- SO(3) with symmetric tensors, SU(2) with a dimension 4 representation, and SU(3) with matter in the sextet-- and find the set of independent invariant tensors in these theories.