I want to understand the proof of the claims (of the construction as well as of its uniqueness) of gauge singlet states given around equation 2.13 (page 10) of this paper.

  • Also does the listing of gauge singlet states there depend on the fact that these are superconformal primaries? (Are they claiming that any gauge singlet state is a primary?)

    What exactly is the connection between the construction of the gauge singlets and them being superconformal primaries?

Let me repeat the claims here again,

  • If you have $N_f$ fields in the fundamental representation of $U(N_c)$ then apparently these can't be combined (tensored?) into an $U(N_c)$ invariant (gauge singlet).

  • But $N_f$ in the fundamental of $SU(N_c)$ can be combined into "baryons" - gauge singlets of $SU(N_c)$ as, $\epsilon_{i_1\dots i_{N_f-N_c}j_1\dots j_{N_c}}\epsilon^{a_1\dots a_{N_c}}$ $\prod_{k=1}^{N_c} \phi^{j_k}_{a_k}$

  • If with the same $SU(N_c)$ the $N_f$ fields happen to be in the adjoint of $SU(N_c)$ then there exists forms invariant under $SU(N_c)$ given as $Tr[\prod_{k=1}^n \phi_{i_k}]$ (for any $n$ of these $N_f$ fields)

  • If one has a pair of fields in the fundamental and the anti-fundamental of $SU(N_c)$ then the gauge invariant operators under $SU(N_c)$ are given as the "mesons" - $\phi^i_a \bar{\phi}^a_j$ (where $a$ is the $N_c$ index and $i,j$ is the $N_f$ index)


I wonder if some Hilbert's invariant theory has gone into these claims. If yes, how? I guess somewhere it is being used that the gauge invariant states are finitely generated since these are invariant under the action of these reductive gauge groups.

  • 4
    $\begingroup$ Hi user6818 - this is a lot of different questions. Could you edit your post to reduce it to ask only one thing, or a couple of closely related ones? You can always post the others separately. $\endgroup$
    – David Z
    Feb 1, 2013 at 5:44
  • $\begingroup$ I may be out of my depth here, but I think the requirement for field unitarity is related to the invariance of the scalar observables, i.e., the field energy density. Non-unitary transformations could break this conservation, and would lead to non-physical results. $\endgroup$
    – KDN
    Feb 13, 2013 at 12:54

1 Answer 1


There is nothing fancy going on here. The question can be re-phrased as a question(s) in group theory.

  1. Starting from an object in the fundamental representation of $SU(N_c)$, can one obtain a single by considering arbitrary tensor products of the the fundamental representation with itself. The answer is that one needs to take $N_c$-th power to find a singlet. The easy (physics) way to see this is that there are only two isotropic tensors in $\mathbb{C}^{N_c}$: $\delta^{a\bar{b}}$ and $\epsilon^{a_1a_2\ldots a_{N_c}}$, where I use unbarred indices $a$ ($a_i$)for the fundamental and barred indices for the anti-fundamental representations of $SU(N_c)$. These lead to the "baryons" where we follow a convention that the field has a "lower" index.
  2. If you have one field in the fundamental and the other in the anti-fundamental, then a singlet may be formed using the invariant tensor $\delta^{a\bar{b}}$.
  3. An adjoint field may be considered to be a bi-fundamental i.e., having one fundamental and one anti-fundamental index. You can easily see that the trace can be re-written as a bunch of contractions involving $\delta^{a\bar{b}}$.

None of these make any reference to whether these operators are conformal or superconformal primaries. That is an independent question which I have not answered.


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