# What is kappa symmetry?

On page 180 David McMohan explains that to obtain a (spacetime) supersymmetric action for a GS superstring one has to add to the bosonic part

$$S_B = -\frac{1}{2\pi}\int d^2 \sigma \sqrt{h}h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu}$$

the fermionic part

$$S_1 = -\frac{1}{2\pi}\int d^2 \sigma \sqrt{h}h^{\alpha\beta}\Pi_{\alpha}^{\mu}\Pi_{\beta}{\mu}$$

plus a long and unwieldy term $S_2$ due to the so called local kappa symmetry which has to be preserved. This $S_2$ term is not further explained or derived.

So can somebody at least roughly explain to me what this kappa symmetry is about and what its purpose is from a physics point of view?

• Be warned, it's a technically very complex thing with limited physical implications. See e.g. intro to arxiv.org/abs/hep-th/9908045 for some background. Surprising that David McMahon chose this topic/formalism in a "demystified book". The kappa-symmetry is a local fermionic symmetry on the world sheet whose task is to remove the excessive number of spinor components of the Green-Schwarz "covariant" string down to 8 physical transverse fermions (8+8 on left/right). It may be done in some backgrounds - in others, the right known constructions don't start with a manifestly covariant start. – Luboš Motl Apr 16 '13 at 5:51
• Thanks @LubošMotl for this comment and link. David McMohan just said that there is this additional $S_2$ contribution to the action due to kappa symmetry but considered it to be inappropriate to explain this further in a demystivied book ... ;-). This picked me and is why I asked here to see and give it a try if not somebody, like you for example :-P, could explain it in a way such that I can get it. – Dilaton Apr 16 '13 at 10:13
• An interesting discussion of this can be found at Becker, Becker, Schwarz . (pg 156 and beyond . ) – Abhimanyu Pallavi Sudhir Aug 2 '13 at 9:27
• Thanks for the hint @Dimension10, I have even downloaded that :-) – Dilaton Aug 2 '13 at 16:42
• @Dimension10 yep, the term is not unwieldy to look at , but unwieldy to type without typo ... ;-) – Dilaton Aug 9 '13 at 7:25

On general super-target spaces the $\kappa$-symmetry of the Green-Schwarz action functional is indeed a bit, say, in-elegant. But a miracle happens as soon as the target space has the structure of a super-group (notably if it is just super-Minkowski spacetime with its canonical structure of the super-translation group over itself): in that case the Green-Schwarz action functional is just a supergeometric analog of the Wess-Zumino-Witten functional with a certain exceptional super-Lie algebra cocycle on spacetime playing the role of the B-field in the familiar WZW functional. It turns out that this statement implies and subsumes $\kappa$-symmetry in these cases.

Moreover, this nicely explains the brane scan of superstring theory: a Green-Schwarz action functional for super-$p$-branes on super-spacetime exists precisely for each exceptional super-Lie algebra cocycle on spacetime. Classifying these yields all the super-$p$-branes...

... or almost all of them. It turns out that some are missing in the "old brane scan". For instance the M2-brane is there (is given by a $\kappa$-symmetric Green-Schwarz action functional) but the M5-brane is missing in the "old brane scan". Physically the reason is of course that the M5-brane is not just a $\sigma$-model, but also carries a higher gauge field on its worldvolume: it has a "tensor multiplet" of fields instead of just its embedding fields.

But it turns out that mathematically this also has a neat explanation that corrects the "old branee scan" of $\kappa$-symmetric Green-Schwarz action functional in its super-Lie-theoretic/WZW interpretation: namely the M5-brane and all the D-branes etc. do appear as generalized WZW models as soon as one passes from just super Lie algebras to super Lie n-algebras. Using this one can build higher order WZW models from exceptional cocycles on super-$L_\infty$-algebra extensions of super-spacetime. The classification of these is richer than the "old brane scan", and looks like a "bouquet", it is a "brane bouquet"... and it contains precisely all the super-$p$-branes of string M-theory.

This is described in a bit more detail in these notes:

The brane bouquet diagram itself appears for instance on p. 5 here. Notice that this picture looks pretty much like the standard "star cartoon" that everyone draws of M-theory. But this brane bouquet is a mathematical theorem in super $L_\infty$-algebra extension theory. Each point of it corresponds to precisely one $\kappa$-symmetric Green-Schwarz action functional generalized to tensor multiplet fields.

Since no other answer has turned up so far, I decided the Lubos Motl's comment is good enough to make a start and I hope he does not mind when I make what he said a CW answer:

Be warned, it's a technically very complex thing with limited physical implications. See e.g. [this](Be warned, it's a technically very complex thing with limited physical implications. See e.g. this intro for some background. Surprising that David McMahon chose this topic/formalism in a "demystified book". The kappa-symmetry is a local fermionic symmetry on the world sheet whose task is to remove the excessive number of spinor components of the Green-Schwarz "covariant" string down to 8 physical transverse fermions (8+8 on left/right). It may be done in some backgrounds - in others, the right known constructions don't start with a manifestly covariant start.