# The ${\cal N} = 3$ Chern-Simons matter lagrangian

This question is sort of a continuation of this previous question of mine.

I would like to know of some further details about the Lagrangian discussed in this paper in equation 2.8 (page 7) and in Appendix A on page $31$.

• On page 7 the authors introduce the idea of having a pair $(Q, \tilde{Q})$ for the matter and that these are ${\cal N}= 2$ chiral multiplets" transforming in adjoint representations of the gauge group. But on page $8$ they seem to refer to the same matter content as being $N_f$ hypermultiplets.

What is the relation between these two ways of thinking about it?

I haven't seen a definition of a "chiral multiplet" and a "hypermultiplet" for $2+1$ dimensions.

• If the gauge group is $U(n)$ and we are working in a representation $R$ of it then should I be thinking of $Q$ and $\tilde{Q}$ as matrices with two indices $i$ and $a$, $Q_{ia}, \tilde{Q}_{ia}$, such that $1 \leq i \leq N_f$ and $1 \leq a \leq dim(R)$?

And their transformations are like (for a matrix $U$ in the $R$ representation of $U(n)$), $Q'_{ia} = U_{ab} Q_{ib}$ and $\tilde{Q}'_{ia} = U^*_{ab}\tilde{Q}_{ib} = \tilde{Q}_{ib}U^{\dagger}_{ba}$

Is the above right?

• In appendix $A$ (page 31) what is the explicit form of the fundamental representation of $USp(2N_f)$" that is being referred to? Is that the matrix $T$ as used in $A.3$ on page 31?

• In equation $A.3$ I guess the notation $()$ means symmetrization as in,

$s^{m}_{ab} = \frac{4\pi}{k} q_{A(a} T^m q^A_{b)} = \frac{4\pi}{k} (q_{Aa}T^mq^A_b + q_{Ab}T^m q^A_a)$

I guess it is similarly so for $\chi ^m_{ab}$ ?

• In equation 4.4 (page 31) is the first factor of $\frac{k}{4\pi}CS(A)$ equal to equation 2.4 on page 5?

• In the same expression for $A.4$, what is the meaning of the quantities

• $D^{ab}$ as in $Tr[D^{ab}s_{ab}]$?
• $\chi _{ab}$ (..what is the explicit expression which is represented by $Tr[\chi ^{ab} \chi _{ab}]$ ?

• $\chi$ as in $Tr[\chi \chi]$ and the last term $iq_{Aa}\chi \psi ^{Ab}$ ?

(..is this $\chi$ the fermionic component of the gauge superfield and different from $\chi ^{ab}$?..)

This Lagrangian is notationally quite unobvious and it would be great if someone can help decipher this.