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The basic question I guess can be formulated as - given two integers $N_f$ and $N_c$ what are the ways in which the fundamental and the anti-fundamental representations of $U(N_f)$ be combined to get 1-dimensional representations of either $U(N_c)$ or $SU(N_c)$.

Apparently that is what goes into the following classification of particle physics as,

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

  • But the same $N_f$ fields 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 $U(N_f)$ 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 $U(N_f)$ then the gauge invariant operators under $U(N_c)$ 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 guess there is an uniqueness about the gauge invariant objects created for each flavour combination given. I guess this "fundamental theorem of invariant theory" in some sense guarantees this uniqueness.

It would be great if someone can explain this.

(.googling "fundamental theorem of invariant theory" didn't yield any clear answer..)

user6818
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  • Your question doesn't contain enough information for a sensible answer until you also specify the $SU(N_c)$ representation of the fields and also their statistics (bosons or fermions). – Jeff Harvey May 13 '13 at 22:48
  • @Jeff Harvey Thanks for your comments. The question arises from trying to understand the classification given on the top of page 10 of arXiv:0704.3740 There it doesn't seem that they have specified the gauge representation of the matter fields. (..also I wonder if this uniqueness of the gauge singlet combinations has anything to do with the fact that there they also want the states to be BPS/superconformal primary...) I didn't understand why they need the chiral primary to be a homogeneous polynomial in such gauge singlets. It would be great if you can fill in what is being kept implicit there – user6818 May 13 '13 at 23:08
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    This question reminds me of the Jabberwocky poem :-) – Mariano Suárez-Álvarez May 13 '13 at 23:24
  • @Mariano Suarez Alvarez As in? The question isn't well framed? (I have put in some clarifying comments also) – user6818 May 13 '13 at 23:27
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    @user6818 As in the question is not well posed to start with and second it is not written in language that most mathematicians will understand. I partially understand what you are asking because I happen to be a physicist. I'd suggest that you either ask the question on physics stack exchange or make the effort to translate your question into a precise mathematical question framed in language that mathematicians will understand. Otherwise your question will be and should be closed since this is a site for research level math questions. – Jeff Harvey May 14 '13 at 00:05
  • @Jeff Harvey Can you point out if there are physics errors in the framing of the question? Then I can correct those errors and repost it in the physicsstackexchange. Or if you can tell me of a physics reference which derives the concept - since I guess you can see what I am really confused about. (...also if you can add if the context of superconformal primaries in the original paper reference has any bearing on the question and the answer..) I did long time ago ask a similar question on physicsstackexchange - http://physics.stackexchange.com/questions/52727/about-defining-baryons-and-mesons – user6818 May 14 '13 at 04:27
  • @Jeff Harvey The Gaiotto-Xi Yin paper I referred to seems to claim that any homogeneous polynomial made up of gauge singlet monomials is a superconformal primary. Am I reading this right? If this is the claim then can you kindly refer me to proof of it? – user6818 May 14 '13 at 04:32
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    I agree that the question resembles Jabberwocky. What is the relation between representations of $U(N_f)$ and $U(N_c)$? Standard linear algebra operations (direct sums, tensor products, symmetrizations, etc) would transform representations of a group $G$ (here $U(N_f)$) into representations of the same group $G$. – Victor Protsak May 14 '13 at 05:46
  • @Victor Protsak You can notice my conventon - where for the fields $\phi$ I have two indices. It is in a representation of both the N_f and the N_c group. You can see in my comment to Jeff Harvey, I have pointed out to the original paper reference that the N_c representation has been kept implicit. (I wonder if the analysis is independent of what it is exactly - and probably thats why the authors have kept it implicit) – user6818 May 14 '13 at 06:44
  • @Victor Protsak It would be helpful if you can point out as to what is the "fundamental theorem of invariant theory" which apparently helps gets 1-dimensional representations of a group from a given collection of representations of that group. – user6818 May 14 '13 at 06:48
  • @Jeff Harvey I have edited the question on the physicsstackexchange. It would be a great favour if you can kindly lend your expertise there. http://physics.stackexchange.com/questions/52727/about-defining-baryons-and-mesons – user6818 May 14 '13 at 07:26
  • A small collection of "fundamental theorem's of invariant theory" can be found here: http://www.math.lsa.umich.edu/~hderksen/math711.w01/quivers/lecture7.pdf – Dietrich Burde May 14 '13 at 07:56
  • The group $SU(N)$ is a simple Lie group, so has no nontrivial 1-dimensional representations. – Ben McKay May 14 '13 at 10:59
  • @user6818 I think you need to read the paper you are citing more carefully. They do specify the $SU(N_c)$ representations on the bottom of p. 9 and top of p.10. They consider four cases and in each case they specify the $SU(N_c)$ representation content (the $R_i$ in their notation). – Jeff Harvey May 14 '13 at 15:23
  • @Jeff Harvey I have also been confused about this very difficult paper. Thanks for clarifying the notation! So the "a" indices on the fields ϕ are the gauge indices corresponding to the representation Ri that has been specified. And i,j go from 1..Nf corresponding to how many flavours they have? So they are never talking of any flavour representation here? So the USp(2Nf) falvour representation specified in the Appendix on page 31 comes from what? But in the very next line they seem to claim that the N=3 fields are in the fundamental of USp(2Nf) - how do they arrive at that? – Anirbit May 14 '13 at 18:12
  • @Jeff Harvey Looking at their equation A.2 it seems that they have twisted the SU(2) R-symmetry into the flavour symmetry in some curious way. Even if I take on faith that a N=3 hyper splits into 2 N=2 chirals in conjugate representations of the gauge group the equation A.2 doesn't seem to follow automatically. I think their A.2 is very crucial since thats the complete specification of the flavour symmetry - it would be great if you can help understand its derivation – Anirbit May 14 '13 at 18:13

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Normally, the first fundamental theorem of invariant theory (due to Cayley and Clebsch in the mid 19th century) says that all invariants can be obtained as contractions of elementary tensors like the epsilon expression in your question. See my answer to MO 121715 for an example of how that works. Coincidentally, the latter is not far from your question since there are also two Lie groups acting. However, if you want more help, you need to follow Jeff's advice and formulate your question with more mathematical precision.

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Abdelmalek Abdesselam
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