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Hello this is my first question. For an open string you can pick different boundary conditions for the endpoints along different directions of space time. For example, you can choose Dirichlet conditions in some directions and Neumann along others. This gives us D-branes. My question is, why can't you choose Neumann b.c. along some directions, and closed string b.c. along some others? Is there some easy way to see this is nonsense? |
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A good idea. ;-) Let's look at it. You have, for example, boundary conditions $$X^1(\sigma+\pi)=X^1(\sigma), \quad X^{\prime 2}(\sigma=0)=X^{\prime 2}(\sigma=\pi)=0$$ The boundary terms in the variation of the action cancel separately for the different coordinates. You see that it's not quite periodic, so the string still has preferred points, $$\sigma=0, \quad \sigma=\pi$$ We may still call them the "end points" of the hybrid string. So the coordinate $X^2$ behaves like for an open string. But $X^1$ has to be periodic. When you quantize the string, you get left-movers and right-movers for $X^1$, but only one set of standing waves for $X^2$ etc. So this will also produce some bizarre zero-point energy. But I think that at the level of the free strings, the theory could be well-defined. However, I think you won't be able to define any consistent interactions although I don't have a strict proof. Your hybrid string is neither open nor periodic. So the world sheet has boundaries that are "partly identified" with other points of the boundary. It's like a closed world sheet with a preferred line drawn on it. When you try to define the interacting theory, you have to decide whether the number of these "bizarre lines" on the world sheet - or bizarre, partly identified pairs of end-points, is conserved or not. If it is conserved, you couldn't pair-create such hybrid strings, so they wouldn't be too interesting. I think that a sensible theory would have to allow these strange pairs of end-points to be created and destroyed. But if it were so, then the world sheet would include singular points - the points in which the cut disappears and your hybrid string transmutes into a normal closed string (or vice versa). My guess is that this process would create a singular points on the world sheet that could return all the ultraviolet divergences. Also, I am not sure whether you could preserve the conformal symmetry. In particular, I don't understand how the state-operator correspondence would work. Closed string states are dual to operators in the bulk of the world sheet; open string states are equivalent to operators on the boundary of the world sheet. But what about your hybrid strings? Maybe, they are dual to the operators at the points where the cut ends. ;-) This could return some unacceptable UV problems to the world sheet, or not. I am not sure. You should try to think about it. If you could find such a new bizarre object that may actually exist in string theory, you could become as important as Joe Polchinski who found D-branes. It's unlikely but you should try. ;-) |
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It looks like open string with kinda "Dirichlet" boundary condition in some direction - in the one where string is closed - but when D-brane which enables this "Dirichlet" b.c. is allowed to move. This movement would be center-of-mass movement plus rippling - when we consider free string. |
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I think conformal symmetry is OK, since this open/closed worldsheet is simply a tensor product of CFTs. However, I'm not sure how the reparameterization ghosts would work. It seems you can either put the bc ghosts on a disk or on a sphere, and there is no hybrid possibility like there is for the embedding coordinate. One compromise would be to include two bc CFTs, one for the open string boundary conditions and another for the closed string boundary conditions. Nilpotency of the BRST charge would probably then require that there are 26 closed string coordinates and 26 open string coordinates. The worldsheet lagrangian would be a sum of an open string Lagrangian and a closed string Lagrangian, so the resulting theory would be better interpreted as describing two independent strings, rather than one composite string. It would be nice if there was a more interesting possibility though :) |
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