# Reference for the ${\cal N}=3$ Chern-Simons Lagrangian at general $N_c$, $N_f$

I was wondering if someone could give me a reference where someone has explicitly written the Lagrangian for ${\cal N}=3$ $SU(N_c)$ Chern-Simons theory coupled to $N_f$ fundamental hypermultiplets.

Any degree of explicitness would be great given that I haven't been able to trace it anywhere. If its written out in component fields with the matter and multiplet content made clear then thats really great!

Reference to any other Lagrangian which is "close" to this would also be fine.

• Appendix A on page 31 of this paper seems to do what I want but the notation is highly unusual as in that $s_{ab}$ and the symplectic form and the $US_p(2N_f)$, the $Q,\tilde{Q}$ (and the $A1$, $A2$) are very unfamiliar coming from a Wess-and-Bagger and Weinberg-vol-3 backgorund. May be it will help if someone can reference instead a pedagogic background literature which will help parse the statement in this appendix.

• Lastly is this theory somehow automatically superconformal?

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Well, I was kindly thanked on the very page 31 so it should be fair for me to try to offer an answer, however imperfect:

Construction of the Lagrangian

I feel that the Lagrangian in components is clearly described in appendix A, especially equation (A.4), but the fact that the structure of the terms looks laborious isn't an illusion; it is a complicated enough calculation. Their theory is a Chern-Simons theory so it has the gauge field with the usual Chern-Simons Lagrangian term that has to be included in the total Lagrangian. Everything else are terms for the extra matter fields.

There are $N_f$ "generations" of matter fields. The matter fields include scalars $q$ and fermions $\psi$: these are parts of a supermultiplet under the ${\mathcal N}=2$ subalgebra of the supersymmetry which is equivalent to four real supercharges, much like the minimal ${\mathcal N}=1$ in four dimensions.

The supercharges in 3D transform as 2-component real spinors, no chirality here, so if you have ${\mathcal N}=n$ SUSY in 3D, the R-symmetry is inevitably $SO(n)$ at the level of Lie algebra. For ${\mathcal N}=3$, we therefore get $SO(3)$ R-symmetry which is better written as $SU(2)$. So all the added matter fields actually have to carry $a,b=1,2$ indices of this $SU(2)$ R-symmetry; the supercharges constructed mainly as bilinear objects from these fields therefore transform as a triplet of this $SU(2)=SO(3)$, as required by our triply extended SUSY.

The fermionic matter fields also carry a spinor index $\alpha=1,2$ for obvious reasons: fermions are spacetime spinors.

The remaining index carried by scalars $q$ as well as fermions $\psi$ is the capitalized index $A,B=1,\dots 2N_f$ labeling the fundamental representation of $USp(2N_f)=U(N_f,{\mathbb H})$. This is the global symmetry you get from $N_f$ flavors here. A priori, you could think that you get just $SU(N_f)$ as the global flavor symmetry. However, that would be an underestimate; the total flavor symmetry is extended to the symplectic one. Why?

This is pretty much answered by the equation (A.1) and maybe at other places. The components of the matter fields are complex but one may still impose a reality condition. However, the reality condition involves the conjugation by the $\epsilon_{ab}$ symbol of the $SU(2)$ R-symmetry; that's needed for complex conjugation of doublets to preserve $SU(2)$. Due to this epsilon, one may add another antisymmetric object, the $\omega_{AB}$ invariant of the symplectic group, and impose a reality condition (A.1) on the matter fields (with an extra spinorial epsilon for the fermions, needed to preserve the Lorentz symmetry in 3D).

One couldn't replace $\omega$ by $\delta$ because the 2-dimensional representation of $SU(2)$ isn't real; but it's pseudoreal and the tensor product of a pseudoreal representation of $SU(2)$ and the pseudoreal representation of $USp(2N_f)$ does give us a real representation (that's a basic fact on representation theory: the $j=-1$ structure map that exists for each pseudoreal representation get multiplied to yield a $j=+1$ structure map on the tensor product, proving it's real: I don't really know whether you want similar things to be explained as well or you know them) – one that may be constrained by a reality condition. The extra epsilon for the spacetime Lorentz indices doesn't change anything about the reality conditions because that 2-dimensional spinor representation of $Spin(2,1)$ is real.

The rest of the Lagrangian (A.4) is just obtained as the rewriting of the superspace Lagrangian parts such as (2.3) and (2.5) as well as the superpotential term (2.9), a special case of (2.10), to the language of components. The objects such as $s$ (a bilinear one) are bookkeeping devices to simplify the structure of the interacting Lagrangian which includes things like the sixth-order interactions (if written in terms of components) so these interaction terms may be rewritten as cubic ones in $s$, at least some of them. Some of those constructions – and hopefully all of them that have a high chance not to be self-explanatory – are explained in the paper and if something isn't detailed enough, you should specify what the point of the confusion is because otherwise you're really asking the Stack Exchange users to write a "more detailed (i.e. probably longer) version of a 47-page paper" which may be too much to ask for.

It shouldn't be surprising that Weinberg or Wess and Bagger don't discuss this particular case of 3D supersymmetry with these particular matter fields. The papers about the superconformal 3D Chern-Simons theories with matter are discoveries of the recent 5 years or so, parts of the membrane minirevolution, which were unknown decades ago when Wess and Bagger and Weinberg wrote their textbooks on supersymmetry. However, it's clear that a user of Xi's and Davide's paper – and others – has to know things such as the symplectic symmetry. Your misspelling of $USp$ as $US_p$ does offer some hint that you don't really know the group and one can't study advanced things such as 3D Chern-Simons theory coupled to matter without a good knowledge of as basic pieces of maths as the symplectic symmetry.

Conformal invariance

In such contexts, the conformal invariance of the quantum theory may be proved by proving the classical conformal invariance; and the vanishing of the quantum corrections that could break the scale invariance. Quite generally, it is enough to prove the scale invariance – the theory's being a fixed point – and that's enough for the theory to have the full conformal symmetry, too. If a theory has a scale invariance and supersymmetry, it's enough to prove the superconformal invariance because the extra fermionic superconformal generators may be obtained as commutators of conformal generators and SUSY generators.

There exist exceptions – scale-invariant theories that are not conformal – but these exceptions can't occur in most physics contexts such as this one. I've really forgotten what the exceptions require.

The ordinary pure 3D Chern-Simons theory is topological – observables only depend on the spacetime (or world volume) topology – so it's of course exactly conformal, too. When one adds matter, the theory ceases to be topological but with some good choices, it may stay conformal. In the paper by Xi and Davide, the conformal symmetry of the complicated theory is demonstrated around page 8. They demonstrate it in two ways. In the ${\mathcal N}=2$ language, they are adding a superpotential with coefficient $\alpha$. To verify the scale invariance or its failure, it's enough to compute the RG running of this new parameter $\alpha$, i.e. the beta-function, and it's given by equation (2.11).

Quite generally, it's enough to verify that the theory is renormalizable and the RG running of all the renormalizable dimensionless couplings vanishes. For the right choices of the couplings, this was done in their paper. The only correction of the quantum loops to the parameters in these theories is a fixed shift to the Chern-Simons level $k$.

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Thanks for the answer. I will may be later ask for some more explanations for some of the points you have made. Also thought of pointing out that I had initially typed it as $USp$ and that QMechanic eventually changed it to $US_p$. And its not clear to me as to how "symplectic symmetry" is a basic mathematics structure. I haven't seen any education system where that is taught as a part of any undergrad or grad course! –  user6818 May 24 '12 at 22:31
I was wondering if you could may be give a reference to the ${\cal N}=3$ superconformal algebra in $2+1$ dimensions. I haven't seen any place where that algebra is written down and if you could indicate as to how one can check that this Lagrangian on page 31 actually has that symmetry. If you could indicate what that algebra is and how to go about doing such a check. –  user6818 May 24 '12 at 22:35
And if you know of computer algebra packages which can help handle such large Lagrangians. Typically decomposing a superfield language expression into component fields is a very laborious calculation - especially the $W^2$ or the $\Phi e^V \Phi$ kind of terms! I was wondering if there is a way out of this. –  user6818 May 24 '12 at 22:39
Dear user, I wasn't taught USp anywhere at school either - not even from "textbooks" of advanced topics such as string theory - and it's too bad because it's needed. Sometimes... I am not sure whether there is any preexisting literature on N=3. It's a very special topic. The people who work on it usually check the N=2 symmetry i.e. 4 supercharges equivalent to N=1 d=4 (guaranteed by superspace etc.) and then they impose one more SUSY. The algebra is completely regular, as expected from extended SUSY, with the SO(3) R-symmetry added. –  Luboš Motl May 29 '12 at 3:50
There are surely macros in Mathematica that do similar algebras but I am not really familiar with them. From similar packages, I have at least heard about FeynRules, see feynrules.irmp.ucl.ac.be - However, how to decompose $W^2$ or $\Phi e^V \Phi$ is about a one-time learning what's going on. And what remains laborious really is laborious but you need it to get the coefficients. The general form of the terms generated are all the terms from non-SUSY theories at the right order with the right symmetries... –  Luboš Motl May 29 '12 at 3:53