# Wilson loops and gauge invariant operators (Part 2)

These questions are sort of a continuation of this previous question.

• I would like to know of the proof/reference to the fact that in a pure gauge theory Wilson loops are all the possible gauge invariant operators (... that apparently even the local operators can be gotten from the not-so-obviously-well-defined infinitesimal loop limit of them!..)

• If the pure gauge theory moves into a confining phase then shouldn't there be more observables than just the Wilson loops... like "glue balls" etc? Or are they also somehow captured by Wilson loops?

• If matter is coupled to the gauge theory are the so called chiral primary" operators, $Tr[\Phi_{i1}\Phi_{i2}\cdots\Phi_{im}]$ a separate class of observables than either the baryons or mesons (for those field which occur in the fundmanetal and the anti-fundamental of the gauge group) or Wilson loops?..is there a complete classification of all observables in the confined phase?

{..like as I was saying last time..isn't the above classification the same thing as what in Geometric Invariant Theory is well studied as asking for all G-invariant polynomials in a polynomial ring (..often mapped to $\mathbb{C}^n$ for some $n$..) for some group $G$?..)

• But can there be gauge invariant (and hence colour singlet?) operators which are exponential in the matter fields?

• In the context of having new colour singlet( or equivalently gauge invariant?) observables in the confining phase I would like to ask the following - if one is working on a compact space-(time?) then does the Gauss's law (equation of motion for $A_0$) somehow enforce the colour-singlet/confinement condition on the matter observables?..hence possibly unlike on flat space-time, here even at zero gauge coupling one still has to keep track of the confinement constraint?

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Wilson loops cannot be all the possible gauge-invariant operators. What about surface operators or 't Hooft loops? –  Ryan Thorngren Aug 1 '12 at 22:48
As for classifying all gauge invariant observables, you are right about the relationship to GIT. BRST is effectively Lie algebra cohomology and classifies (polynomial?) observables. I wonder about more complicated observables like Wilson loops though. –  Ryan Thorngren Aug 1 '12 at 22:52
@Anirbit: It's customary on stackexchange sites to ask one question per post. You are asking several unrelated questions here. Question 2, for example, seems to me to be basically asking "How is a glueball represented in terms of the standard gauge field observables?" –  user1504 Aug 1 '12 at 23:52
@user404153 Thanks for your replies! Can you give a reference for what you are saying? Is there any classification of all the gauge invariant operators in a pure gauge theory? Or somewhere it might be explained in details as to how(if!) the Wilson loops "generate" all possible gauge invariant observables like the surface operator, t'Hooft loops etc.. –  user6818 Aug 2 '12 at 20:56

The correct statement is that the Wilson loop observables generate via limits and algebraic operations all of the other observables in pure gauge theory. Obviously the finite size Wilson loops are not the only observables. As user404153 points out, there are surface operators and also point-like operators that create glueballs.

The relevant mathematical fact is that a connection (a classical gauge field) is determined up to gauge transformation by its holonomies (the Wilson loops).

You can probably find a proof of this in Kobayashi & Nomizu, but you really ought to just prove this for yourself. (I'm assuming that you are a mathematician, since you seem to be asking questions relevant to supersymmetric and topological field theory without knowing basic facts about gauge theory).

Given this fact about classical gauge fields, the same claim follows for the quantum theory via the path integral construction. Anything you can write down in terms of the basic gauge fields $A$, you can also write down in terms of the Wilson loops.

Disorder observables like the 't Hooft loops are a funny special case. These observables are usually defined in the path integral by altering the boundary conditions in the path integral to include interior singularities. However, the 't Hooft loops can be constructed algebraically by solving $dA = const *dB$ for the dual gauge field $B$ in terms of $A$. Then the 't Hooft observable is simply the holonomy of $B$. Presumably, similar reasoning holds for other disorder observables.

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Your replies are extremely condescending and presumptious! What do you mean by "not knowing basic facts about gauge theory"!? Who decides what is basic!? At least no QFT course I know or any standard QFT book I have seen even touches on these issues that I asked about here. It would be helpful if you can give references. –  user6818 Aug 2 '12 at 20:31
@Anirbit: I'm sorry I offended you. My point is this: Right now, the only way to define pure Yang-Mills theory is to take the continuum limit of lattice gauge theory. If you don't know this language, then you don't know what Yang-Mills theory is. That's going to make it hard for you to understand the answers to your questions. (And that's what I mean by 'basic': knowing the definition of the things you're discussing.) –  user1504 Aug 2 '12 at 22:30
For a list of references on lattice gauge theory, see: physik.uni-graz.at/~cbl/library/lgt-reviews.html –  user1504 Aug 2 '12 at 22:36
I don't know what you mean by "only way to define". I simply don't buy the argument about lattice gauge theory being a way to "define" Yang-Mills! There are just so many awesome verifiable results which can be gotten in Yang-Mill's theory without ever needing to talk of a lattice formulation. I absolutely see no fundamental relevance of the lattice formulation unless you want to say that everytime I use a momentum cut-off I am "actually" using a lattice..that may be true in one sense but far too contrived. –  user6818 Aug 3 '12 at 15:15
I am asking my questions from the point of view of the enormous literature that exists about Wilson loops in Yang-Mill's theory and more so from the point of view of supersymmetric field theory. –  user6818 Aug 3 '12 at 15:16