Negative open string norms after BRST cohomology?
The question http://physics.stackexchange.com/q/43545 made me think.
Suppose we have a Dp-brane completely wrapped over a $T^d$ compactification with $p\leqslant d$. Look at an open string worldsheet with both ends on this wrapped brane. A Fourier decomposition can be made for the $X$ fields over this worldsheet. Look at the mode with no worldsheet Fock excitations for worldsheet momenta 2 or greater, a single excitation for n=1, and consider the values for $P_\mu$ for the worldsheet n=0 mode. Assume in addition the winding numbers for the d-p compactified dimensions normal to the wrapped brane are zero. For those $9-p$ dimensions normal to the Dp-brane, $P_\mu=0$. That leaves us with p spatial dimensions (let's give them the indices i,j,k,...) and 1 time dimension, index 0. The string modes we're interested in are massless. This means $P_0^2=P_i P_i$. Let $\epsilon^\mu$ be the orientation for the excited n=1 mode. The Lorentz invariant norm is given by $\langle \epsilon^\mu | \epsilon^\nu \rangle = \eta^{\mu\nu}$ ($-+...+$ signature). 669
The condition for BRST-closure is $\epsilon^0 P_0 + \epsilon^i P_i =0$. The null BRST-exact state has $\epsilon^\mu$ parallel to $P^\mu$. Because of the compactification, $P_i = n_i/R_i$ where $R_i$ is the radius along the ith dimension and the $n_i$'s are integers. In other words, the spectrum for the $P_\mu$'s is discrete along all directions. If there is at least one nonzero $n_i$, the BRST-closed subspace has positive semidefinite norm, and after quotienting over the BRST-exact null state, we are left with a positive definite space.
What if all the $n_i$'s are zero? Then, $P_0=0, P_i=0$, all polarizations are BRST-closed, and there is no nontrivial BRST-exact subspace. So, the BRST cohomological space has an indefinite norm! Can someone please help me here?
Suppose we have two D0-branes. They are separated by a distance $L$. Consider an open string connecting both branes. Assume it has the same properties as given in the previous section. As long as $L$ is nonzero, the BRST cohomology has positive definite norm. What happens when $L=0$? An indefinite norm again! 346