State counting in the d = 1+2, $\cal{N} = 2$ vector multiplet

Question

The question is from Box 8.2, page 282 of the book "Gauge Gravity Duality" by Ammon and Erdmenger. The link to the specific page from Google Books is here.

According to the authors, a $\mathcal{N} = 2$ vector superfield includes a vector potential $A_\mu$, a real scalar field $\sigma$, two real (Majorana) gauginos, and an auxiliary real scalar field $D$, all in the adjoint representation of the gauge group.

I am not sure how the counting works:

• Vector potential $A_\mu$ has $(3-2) = 1$ (bosonic) degree of freedom, as a gauge field in $d = 1 + 2$ dimensions.

• A real scalar field has $1$ (bosonic) degree of freedom.

• Two real Majorana gauginos have $2 \times 2^{(3-1)/2}$ (real) fermionic degrees of freedom, i.e. $4$ fermionic degrees of freedom.

• An auxiliary real field has $1$ bosonic degree of freedom.

The number of fermionic and bosonic components do not match.

Answer 1

You are performing an on-shell counting for an off-shell multiplet. The off-shell vector multiplet has

$\sigma [1], A_{\mu} [2], D[1], \lambda [4]$

thus $4+4$ degrees of freedom. The on-shell vector multiplet consists of the scalar $\sigma$, the vector $A_\mu$ and the Dirac fermion $\lambda$. In that case, the counting is

$\sigma [1], A_{\mu} [1], \lambda [2]$

so you have $2+2$ degrees of freedom.