# About the definition/motivation/properties of the twisted chiral superfield in ${\cal N}=2$ theories in $1+1$ dimensions

The following is in the context of the ${\cal N}=2$ supersymmetry in $1+1$ dimensions - which is probably generically constructed as a reduction from the ${\cal N}=1$ case in $3+1$ dimensions.

• In the $\pm$ notation what is the definition of ${\cal D}_+$ and ${\cal D}_{-}$, which I understand from context to be the gauge covariant superderivatives. (..It would be great if someone can relate them to the usual definition in the notation of say Wess and Bagger..)

• So what is the meaning/motivation of defining a twisted chiral superfield as, $\Sigma = \{\bar{{\cal D}}_{+}, {\cal D}_{-}\}$ (..naively this looks like an operator and not a field - I guess there is some way of arguing that the derivative terms which are not evaluated on something actually go to zero..)

I am guessing that in the above context it will be helpful if someone can explain as to what is meant by the following decomposition/reduction of the gauge field from $3+1$ dimensions,

$\sum _ {\mu = 0}^3 A_\mu dx^\mu = \sum _{\mu =0} ^1 A_\mu dy^\mu + \sigma (dy^2-idy^3) + \bar{\sigma}(dy^2+idy^3)$ ?

• From the above (does it/how does it) follow that one can write $\Sigma$ as,

$\Sigma = \sigma + \theta\lambda + \theta \bar{\theta}(F+iD)$

(..where I am not sure if $F,D,\sigma$ are real or complex scalar fields...and $\lambda$ is a Weyl fermion..)

• What is the R-charge of this twisted chiral super field? (..from some consistency conditions I would think that its 2..but I am not sure..)

I guess that the R-symmetry transformations act as,

• The "right" R symmetry keeps $\theta^-$s invariant and maps, $\theta^+ \mapsto e^{i\alpha}\theta^+$, $\bar{\theta}^+ \mapsto e^{-i\alpha}\bar{\theta}^+$

• The "left" R-symmetry keeps $\theta^+$ invariant and maps, $\theta^- \mapsto e^{-i\alpha}\theta^-$, $\bar{\theta}^- \mapsto e^{i\alpha}\bar{\theta}^+$.

Though I am not sure and like to understand as to why one wants to think of these two different R-symmetry groups as having two different origins - one coming from the rotation symmetry of the two spatial dimensions of the original $\cal{N}=1$, $1+3$ theory and another coming from R-symmetry of the $\cal{N}=1$, $U(1)$ gauge theory.

-

After dimensional reduction from 4 to 2 dimensions, it is convenient to simply label the last two remaining dimensions as $+$ and $-$ instead of 1 and 2. So, basically you have ${\cal D}_- = {\cal D}_1$ and ${\cal D}_+ = {\cal D}_2$.