About ghost D-branes and wrong statistics

Question

I'm currently reading this article, which is about Ghost D-branes in string theory. The authors define these objects as something that cancels the effects of an ordinary D-brane. However, there is something I do not understand.

On page 6, the authors ask us to consider N $Dp$-branes and $M$ ghost $Dp$-branes on top of each other. They claim the field content can be summarized by a hermitian supermatrix $$ \Phi = \begin{pmatrix} \phi_1 & \psi \\ \psi^{\dagger} & \phi_2 \end{pmatrix}. $$ Here $\phi_1$ and $\phi_2$ are boson hermitian matrices, while $\psi$ is a complex fermionic matrix. They say the following:

The diagonal part $\phi_1$ (or $\phi_2$) corresponds to the open strings between $D$-branes (or ghost $D$-branes). The off-diagonal part $\psi$ corresponds to the open strings between $D$-branes and ghost $D$-branes and thus they have the opposite statistics.

Now does anyone know why (1) the off-diagonal part would represent open strings stretched between $D$-branes and ghost $D$-branes? Why should I believe this? And also (2), how does this imply that the states of this open string have opposite statistics? I think this is far from trivial.

In the boundary state formalism, the amplitude for closed string propagation is then $$ \langle gD \mid \Delta \mid D \rangle = - \langle D \mid \Delta \mid D \rangle. $$ Here $\Delta$ is the closed string propagator. The minus sign is because, according to the authors, the boundary state of a ghost $D$-brane is minus the whole boundary state of an ordinary $D$-brane. In the open string channel, these amplitudes are interpreted as one-loop partition functions of open strings stretched between branes. Is it because I can attribute the minus sign to a fermionic path integral that this wrong statistics results? This is not clear to me.

Answer 1

An elementary way to motivate the "wrong statistics" of the off diagonal terms goes as follows:

The diagonal terms of the super-connection $\Phi$, denoted by $|i,j>_{NS}$, $|g(i),g(j)>_{NS}$ are just the usual Yang-Mills gluons arising from the NS sector of open superstrings with endpoints at brane-brane or ghost brane-ghost brane respectively.

Now, by definition, the wavefunction of an open superstring with an endpoint on a ghost brane changes the sign of the whole gluon wavefunction $|\psi,i,g(j)>$ as $$|\psi,i,g(j)>=|\psi,i,g(j)>_{NS} +|\psi,i,g(j)>_{R}=-|\psi,i,j>_{NS} -|\psi,i,j>_{R},$$

where $\psi$ denotes the gluon wave function (see Zwiebach, A first course in string theory, second edition, quick calculation 14.3 where the photon wavefunction is shown as the open superstring NS-state at level $N=\frac{1}{2}$).

Now notice that if $(-1)^{F}$ is the fermion number operator, then $(-1)^{F}|\psi,i,j>_{NS}=|\psi,i,j>_{NS}$ because the Yang-Mills gluon $|\psi,i,j>_{NS}$ is a bosonic state and arises from the NS sector of the open superstring.

The point is that the off-diagonal elements of the super-connection obey $$(-1)^{F}|\psi,i,g(j)>_{NS} = (-1)^{F}(-|\psi,i,j>_{NS}) = -|\psi,i,j>_{NS},$$

so they are fermions.