# D0-branes and counting

Hi,

I'm reading Becker2Schwarz, chapter 5.1, about D0 branes in the GS formalism. They introduce kappa-symmetry, and end the section with "without this symmetry there would be the wrong number of propagating degrees of freedom".

I'm trying to understand that. The fermions $$\Theta^a$$ have, for D=10, $$2^5=32$$ complex components. But they are Majorana-Weyl, so this brings this number back to $$\frac{32}{4}=8$$ complex components. Kappa-symmetry implies that half of these fermions are gauge degrees of freedom, giving us 8 real components.

However, for the D0-brane, which is a particle, we start with 10 real components $$X^{\mu}$$. Choosing e.g. the static gauge brings this back to 9 components. Obviously, to have as many bosonic degrees of freedom as fermionic (8 real), I need to get rid of another bosonic degree of freedom. How do I do that?

Perhaps I'm also confused by worldsheet SUSY versus target space SUSY; to realize target space SUSY on $$\{\Theta^a,X^{\mu}\}$$ one doesn't need this kappa symmetry, right?

Any help is appreciated :)

Let me add something: for the massless case m=0, one can write down the superparticle in both RNS and GS formalism, and also realize both worldline- and targetspace SUSY. (this is also treated in chapter 4 of BBS). In this case the counting works: a massless particle has 8 bosonic degrees of freedom, and so has the spinor.

For the massive case one realizes target space SUSY via

$$\delta \Theta^A = \epsilon^A, \ \ \ \ \delta X^{\mu} = \bar{\epsilon}^A\Gamma^{\mu}\Theta^A$$
where A=1,...,N labels the amount of SUSY. This algebra implies that a translation P on Theta is zero, so Theta is a zero eigenvector of P, and hence that P is not invertible. Then one doesn't need to have the same amount of X-components and Theta-components, right? I wouldn't know how to realize now wordline supersymmetry.

Maybe this section of BBS is a bit unclear; it now seems to me that this kappa symmetry is not per se about obtaining the same amount of X- and Theta components, but just about obtaining the right amount of SUSY. Does that make sense?

"without this symmetry there would be the wrong number of propagating degrees of freedom"

The number of propagating degrees of freedom are determined by the equation of motion (5.18).