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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.

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