$\pm$ (light-cone?) notation in supersymmetry

Question

• I would like to know what is exactly meant when one writes $\theta^{\pm}, \bar{\theta}^\pm, Q_{\pm},\bar{Q}_{\pm},D_{\pm},\bar{D}_{\pm}$.

{..I typically encounter this notation in literature on $2+1$ dimensional SUSY like super-Chern-Simon's theory..}

• I guess that when one has only half the super-space (i.e only the $+$ of the above or the $-$) it is called the $(0,2)$ superspace compared to the usual $(2,2)$ superspace. In this case of $(0,2)$ SUSY I have seen the following definitions,

$Q_+ = \frac{\partial}{\partial \theta^+} + i \bar{\theta}^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

$\bar{Q}_+ = -\frac{\partial}{\partial \bar{\theta}^+} - i \theta^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

which commute with,

$D_+ = \frac{\partial}{\partial \theta^+} - i \bar{\theta}^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

$\bar{D}_+ = -\frac{\partial}{\partial \bar{\theta}^+} + i \theta^+\left( \frac{\partial}{\partial y^0} + \frac{\partial}{\partial y^1} \right)$

• I am guessing that there is an exactly corresponding partner to the above equations with $+$ replaced by $-$. Right?

How does the above formalism compare to the more familiar version as,

$Q_\alpha = \frac{\partial}{\partial \theta^\alpha} - i\sigma^\mu_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial x^\mu}$

$\bar{Q}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} + i\sigma^\mu_{\alpha \dot{\alpha}}\theta^{\alpha}\frac{\partial}{\partial x^\mu}$

which commute with,

$D_\alpha = \frac{\partial}{\partial \theta^\alpha} + i\sigma^\mu_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial x^\mu}$

$\bar{D}_\dot{\alpha} = -\frac{\partial}{\partial \bar{\theta}^\dot{\alpha}} - i\sigma^\mu_{\alpha \dot{\alpha}}\theta^{\alpha}\frac{\partial}{\partial x^\mu}$

{..compared to the above conventional setting, in the $\pm$ notation among many things the most perplexing is the absence of the Pauli matrices!..why?..}

I would be very grateful if someone can explain this notation.

{..often it turns out that not just the Qs and the Ds but also various superfields also acquaire a $\pm$ subscript and various usual factors of Pauli matrices look missing..it would be great if someone can help clarify this..}

Answer 1

It's hard to answer because you don't give many details, but since $\tfrac \partial {\partial y^2}$ does not appear in the SUSY generators, I think it's safe to assume that this is some light-cone superspace. In this kind of superspace one projects the spinors to plus and minus components with $\gamma_+ \gamma_-$ (I omitted a numerical factor which makes this a projector and which depends on your conventions).

Then the superspace is realized with only half of the odd variables, $\theta_+$ or $\theta_-$. So the only supercharges (or covariant derivatives) which are considered in this formalism are $\theta_+$ in your case. You don't need to include their analogs with $\theta_-$. If you insist on including them, you will generally run into problems with the closure of the algebra.

You had another question about the absence of the $\gamma$ matrices. They are there, just written out explicitly.