This is related to this earlier question I had asked.
I am using the so called ``Majorana" representation of gamma matrices in $2+1$ dimensions in which everything is real. After doing the dimensional reduction of the $\cal{N}=1$ supersymmetry transformations of the components of the vector superfield in $3+1$ dimensions the supersymmetry transformations of the resulting $\cal{N}=2$ components of the vector superfield in $2+1$ dimensions are,
$$\delta F_A = i \bar{\alpha}^a\lambda_{Aa}$$ $$\delta D_A = i \bar{\alpha}^a\gamma_3^\mu D_\mu \lambda_{Aa}$$ $$\delta V_{A\mu} = i \bar{\alpha}^a\gamma_{3\mu}\lambda_{Aa}$$ $$\delta \lambda_{Aa} = -\frac{1}{2}f^3_{A\mu \nu}\gamma_3^{\mu \nu}\alpha_a + D_A\alpha^a + \gamma_3^\mu D_\mu F_A\alpha ^a$$
where $A,B,..$ are the gauge group indices, $f^3_{A\mu \nu}$ is the non-Abelian field strength and $\alpha$ is a spinor parameter whose components are raised and lowered as, $\alpha^1 = \alpha_2$ and $\alpha^2 = -\alpha_1$.
Using the above one can derive the following transformations for the possible terms in the intended super-Chern-Simons' theory,
$$\delta(Tr[FD]) = Tr[t_At_B] \{ i\bar{\alpha}^a \lambda _{Aa}D_B - i \bar{\alpha}^a \gamma_3^\mu\lambda_{Ba}\partial_\mu F_A + i\bar{\alpha}^a \gamma_3^\mu \lambda_{Ca} C_{BB'C}V_{B'\mu}F_A\}$$
$$\delta(Tr[\bar{\lambda}_a\lambda_a]) = 2Tr[t_A t_B]\{ \frac{1}{2}\bar{\alpha}_a\gamma_{3\rho}\lambda_{Ba}f^3_{A\mu\nu}\epsilon ^{\mu \nu \rho} + \bar{\alpha}^a\lambda_{Ba}D_A - \bar{\alpha}^a\gamma_3^\mu \lambda_{Ba}\partial_\mu F_A$$
$$- \bar{\alpha}^a\gamma_3^\mu \lambda_{Ba}C_{AB'C}V_{B'\mu}F_C \}$$
$$\delta(Tr[\epsilon^{\mu \nu \rho}(V_\mu \partial_\nu V_\rho)]) = -i\epsilon^{\mu \nu \rho}Tr[t_At_B]\bar{\alpha}^a\gamma_{3\rho}\lambda_{Aa}(\partial_\mu V_{B\nu} - \partial_\nu V_{B\mu})$$
$$\delta(Tr[\epsilon ^{\mu \nu \rho}V_\mu V_\nu V_\rho]) = -\frac{3}{2}\epsilon^{\mu \nu \rho}Tr[t_At_D]C_{DBC}\bar{\alpha}^a\gamma_{3\mu}\lambda_{Aa}V_{B\nu}V_{C\rho}$$
(where $t_A$ are a chosen basis in the lie algebra of the gauge group such that the structure constants are defined as, $[t_A,t_B]=iC_{DAB}t_D$)
It is clear that by choosing a coefficient of $-2$ for the $Tr[FD]$ and $i$ for the $Tr[\bar{\lambda}_a\lambda_a]$ some of the terms the variation of the auxiliary fields can be cancelled and some of the remaining terms of the variation of the fermionic term totally cancel the supersymmetric variation of the kinetic term of the gauge fields.
What remains are,
$$\delta(Tr[\epsilon^{\mu \nu \rho}V_\mu\partial _ \nu V_\rho + i \bar{\lambda}_a\lambda_a - 2FD]) = Tr[t_At_B]\{i\bar{\alpha}_a\gamma_{3\rho}\lambda_{Aa}C_{BCD}V_{C\mu}V_{D\nu}\epsilon^{\mu \nu \rho}$$ $$ - 2i\bar{\alpha}^a\gamma_3^\mu \lambda_{Ba}C_{AB'C}V_{B'\mu}F_C - 2i\bar{\alpha}^a\gamma_3^\mu \lambda_{Ca}C_{BB'C}V_{B'\mu}F_A\}$$
and
$$\delta(Tr[\epsilon ^{\mu \nu \rho}V_\mu V_\nu V_\rho]) = -\frac{3}{2}Tr[t_At_B]\bar{\alpha}^a\gamma_{3\mu}\lambda_{Aa}C_{BCD}V_{C\nu}V_{D\rho}\epsilon^{\mu \nu \rho}$$
- Its not clear that a coefficient can be chosen for the last term so that the supersymmetric variation of the sum of the LHSs go to zero.
The above terms seem to be structurally very different and hence its not clear how they will cancel. Like the variation of the fermionic self-coupling term produces a coupling of the fermionic component, gauge field and auxiliary field. Such a term is not produced by the variation of the gauge field cubed term!
One expects the lagrangian should look something like,
$$Tr[\epsilon^{\mu \nu \rho}(V_\mu\partial _\nu V_\rho -i\frac{2}{3}V_\mu V_\nu V_\rho ) + i \bar{\lambda}_a\lambda_a - 2FD ]$$
I would like to get some help in establishing the above!
- One progress would be if the two terms with the structure constant actually cancel i.e,
if
$$Tr[t_At_B]\{C_{AB'C}\lambda_{Ba} V_{B'\mu}F_C + C_{BB'C}\lambda_{Ca}V_{B'\mu}F_A\} = 0$$
But the above is not clear!
NB. My structure constants are defined as $[t_A,t_B]=iC_{DAB}t_D$
