# Construction of the $\cal{N}=3$ supersymmetric Yang-Mills Chern-Simons theory in $2+1$ dimensions

I am stuck with understanding the following construction. I am breaking it up into segments which I think can be separately answered.

This is related to an earlier question of mine. Note that this previous question was edited a lot after the initial comments came in.

• Firstly I would like to know how the following field multiplet is determined,

The gauge multiplet of $\cal{N}=3$ in $2+1$ dimensions is apparently ,

$A_\mu$, a massive vector field of spin $1$

$\lambda_a$, $3$ Majorana fermions of spin $\frac{1}{2}$

$C_a$, $3$ neutral scalar bosons

$\chi$, a Majorana fermion of spin $\frac{1}{2}$

(I guess there should be another scalar field $D_a$ not mentioned in the above list)

• Its not clear to me as to what are the superfields one has in mind with the above multiplet structure that helps construct the lagrangians below.

• Like in the linked earlier question above one had to "know" what exactly is the meaning of doing the dimensional reduction on a given set of supersymmetric transformations, here its not clear to me as to what are the corresponding starting points. (I wonder if I can start from the $\cal{N}=2$ SYM in $3+1$ lagrangian that is given in say equation $27.9.3$ in Weinberg's QFT book)

Then dimensional reduction of $\cal{N}=2$ SYM in $3+1$ apparently gives the following $\cal{N}=3$ SYM in $2+1$,

$L_{\cal{N}=3,2+1,SYM} = \frac{1}{g^2}Tr\left [ -\frac{1}{2}F_{\mu \nu} ^2 + (D_\mu C_a)^2 + (D_a)^2 + i \bar{\lambda_a}\gamma^\mu D_\mu \lambda_a + i \bar{\chi}\gamma^\mu D_\mu \chi • i\epsilon_{abc}\bar{\lambda_a}[\lambda_b,C_c] -2i\bar{\lambda_a}[\chi,C_a] - \frac{1}{2}[C_a,C_b][C_b,C_a] \right ]$

where the notation is, $D_\mu = \partial_\mu -iA_\mu$, $ab,c=1,2,3$. The gauge group generators in the fundamental representation satisfy $[T^m,T^n]=if^{lmn}T^l$ with normalization as $Tr{T^mT^n}=\frac{1}{2}\delta ^{mn}$ and $f^{kmn}f^{lmn}=c_v\delta^{kl}$. ($c_v$ being the quadratic Casimir) The fields belong to the adjoint representation and $A_\mu = A_\mu^mT^m$. The metric is given by $\eta_{\mu \nu} = diag(1,-1,-1)$. The purely imaginary gamma matrices satisfy, $\gamma^\mu \gamma ^\nu = \eta^{\mu \nu}-i\epsilon^{\mu \nu \rho}\gamma_\rho$.

• It is not clear to me as to how does one argue that the above has $\cal{N}=3$ supersymmetry.

• The dimensional reduction of the $\cal{N}=2$ supersymmetry somehow implies that the theory has the not-so-obvious $\cal{N}=4$ supersymmetry.

One also notes that supersymmetrizing the Chern-Simon's terms one is supposed to get the Chern-Simon's lagrangian as,

$L_{\cal{N}=3,2+1,SCS} = kTr\left [ \epsilon^{\mu \nu \rho}(A_\mu \partial_\nu A_\rho - \frac{2}{3}iA_\mu A_\nu A_\rho) - \bar{\lambda_a}\lambda_a + \bar{\chi}\chi • 2C_aD_a + \frac{i}{3}\epsilon_{abc}C_a[C_b,C_c] \right ]$

• In all these lagrangians apparently the corresponding $\cal{N}=2$ theories can be obtained by setting $C_1=C_2=D_1=D_2=\lambda_3 = \chi = 0$ and the $\cal{N}=1$ theories can be obtained from the $\cal{N}=2$ theories by further setting $C_3=\lambda_2=0$.

(I am unable to get the equations to wrap around properly!)

## 1 Answer

Dear Anirbit, the field content of a vector multiplet of $N=3$ in $d=3$, if it exists, is obvious. There's one transverse polarization of the gauge boson in $d=3$. The 3 supersymmetries may be used to get 3 different minimal spinorial fermions - Majorana fermions in $d=3$ - out of the vector boson's polarization. That's the $\lambda$ fields. However, in SUSY theories, the multiplets have dimensions that are powers of two. So far we have 3 fermionic fields, we clearly need 4, so one more Majorana spinor, $\chi$, has to be included as well. Now, SUSY has to match bosons and fermions so we must also add 3 extra bosons to the gauge field - they must be spinless e.g. scalars.

The relationship of this field content to the Lagrangian you wrote below it is that the Lagrangian you wrote is the Lagrangian that uses the fields you used above, and produces the physical states enumerated in the previous paragraph; a simple check of all the bilinear terms in the Lagrangian is enough to see its free spectrum. The kinetic terms have been supplemented with the interactions with the gauge field to keep the Lagrangian gauge-invariant.

The starting point in constructing the theory is to decide about its particle spectrum - see the first paragraph. Then one constructs the Lagrangian that produces the required physical states - it's essentially a simple exercise in free field theory. Then one constructs the interacting Lagrangian with all the possible or renormalizable interactions that preserve the pre-determined symmetries.

You also ask how one "argues" that the Lagrangian is $N=3$ supersymmetric. When, one "argues" in such a way that he must calculate and prove that it is. It means that one needs to find the appropriate SUSY variations of the elementary fields and prove that the action is invariant under those transformations.

A theory with $N=2$ SUSY, when dimensionally reduced to a lower dimension where the minimal spinor is smaller, automatically has $N=4$ SUSY in the lower-dimensional notation. This simply follows from the spinor's decomposition into two spinors under the smaller group. In your first Lagrangian, $N=3$ SUSY was pretty manifest because the 3-valued index was added to the superpartners of the gauge field. However, such theories often have an automatic higher degree of supersymmetry, too.

The simplest proof to show the $N=4$ SUSY is to prove the $N=2$ SUSY of the higher-dimensional "mother" theory. You may also check the symmetry under the fourth supersymmetry in $d=3$ explicitly.

At any rate, I think that your "primary" questions have nothing to do with $N=3$ SUSY in $d=3$ or any of these obscure topics. You shouldn't study these obscure topics before you learn basics of physics and field theory - such as what it means for a theory to have a certain symmetry; certain spectrum; what is the relationship between the Lagrangian and the free spectrum; how one chooses a notation to describe a particular set of fields in particular representations; what it means for objects to transform as representations of symmetries; what happens with the representations if we only look at or preserve a smaller symmetry (subgroup), and so on. As far as I can say, you must be misunderstanding pretty much all these elementary questions which simply makes it inappropriate if not impossible to study vector multiplets in $d=3$, $N=3$ SUSY.

• Thanks for your reply. Can you give me a reference or a method to determine the $\cal{N}=2$ supersymmetry transformations in $3+1$ dimensions? Like I knew the $\cal{N}=1$ supersymmetry transformations in $3+1$ dimensions and hence could systematically dimensionally reduce it to get $\cal{N}=2$ in $2+1$. Here the analogous starting point is not clear. (which if I knew I could do the analogous dimensional reduction to get $\cal{N}=3$ in $2+1$) – user6818 Mar 8 '11 at 18:25
• Hi Anirbit, see a paper for some $N=3$ $d=3$ theories such as arxiv.org/abs/0808.0864 - the SUSY transformations may not be explicit but they're always of the same form, $\delta \psi = \epsilon \times$ combination of derivatives of fundamental bosons and potentials, with the right covariantization, contractions of indices, gamma matrices, etc., and $\delta$ bosons = $\epsilon \psi$ for fermions $\psi$, and you just adjust the coefficients. It's a lot of hard work. I don't know any reference with your exact task and it's hard to see why you think it's important. – Luboš Motl Mar 10 '11 at 21:22