$\pm$ (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..}

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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.

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Comments

Can you give some more explicit expressions about the projections you mentioned or a reference(pedagogic?)? I don't get it when you say that the $\theta_1,\theta_2$ superspace is equally captured by just $\theta_+$ or $\theta_-$. I guess you mean to say that $\theta_+$ and $\bar{\theta}_+$ is the same space as $\theta_1$ and $\theta_2$. So what is the $(2,2)$ superspace. As for extra details - well - much of my immediate notation is from this paper - http://arxiv.org/abs/hep-th/9301042 - and one can see this $\pm$ notation also in this paper - http://arxiv.org/abs/1104.0680.

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@user6818 The first paper you cite deals with two dimensions. So your comment about Chern-Simons definitely doesn't apply. In the second reference I can't see any equation of the kind you asked about. If your question is really about 2D, try this [reference](http://arxiv.org/abs/hep-th/0504147), starting at page 52.

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Thanks for the reference. Let me see if that helps completely understand this $\pm$ notation. About my second reference looks at page 29, 37 and 40 for example - you can see this curious $\pm$ notation - I don't understand what they call as the supersymmetry transformation in that language - like say on page 37.

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