Classifying Invariants in $SU(3)$ Theories with Adjoints

The author discusses a method of determining the independent invariants of a gauge theory, by gauge-fixing field configurations. The author applies this methodology to the example of SU(3) gauge theory with adjoint matter.

The work seems technically correct, and could be useful for generating the independent basis of invariants of a group. The calculations are presented clearly and in detail.

My main criticism of the work is that it is not clear what is conceptually new relative to the author’s previous works. In particular, Reference [24] seems to propose the main method being used, and this work focuses mainly on an example application of this method to invariants made from N adjoints of SU(3). This example is already studied for N=1,2 (e.g. by reference [27]), and so the new results are for N>2.

I would therefore recommend that the author clarifies and/or improves upon the following points in their article, before its publication is considered.

(1) What are the new conceptual points explained / explored in this work relative to Reference [24]?


(2) Can the author please clarify in more detail whether they have a specific application in mind by studying the SU(3) + adjoints example, or whether they view this as a proof of concept? For example, the author references dualities in theories with adjoints, but to my knowledge these only usefully apply to theories with N <= 2 where results on the gauge-invariant operators were already known, and where furthermore the open questions concern quantum modifications of the classical spectrum which are outside the scope of this work. They also mention a possible application to invariant tensors of exceptional groups — can they explain what is known about the invariant tensors in these cases, and what might be gained by applying their method? Answering these points in more detail would substantiate why this work is an important stepping stone to future developments in this field.

(3) It would be helpful if the author clarifies how their main approach is related to Reference [25]. In particular, that reference relies on the fact that they are considering supersymmetric theories which are invariant under the complexified gauge group. (For a non-supersymmetric theory, the statement is that all constant field configurations lie in a gauge equivalence class of vacua.) It is not clear to me whether the author’s method requires this, or whether they differentiate between supersymmetric / non-supersymmetric theories. 


(4) As the author discusses, the Hilbert series approach can be used to count gauge invariant operators of a given degree in the fields. It seems that an important check of the author’s claims would be to compare their constructed basis with the number predicted by the Hilbert series for this example (at least to low degree). Can the author perform this check, or else explain further why expanding to the requisite degree 7 in the fields to fully corroborate their results is too challenging?