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In my String Theory course, we introduced D$p$-branes, and I'm having trouble picturing them as dynamical objects.
We analyzed the open and closed bosonic strings, found the EoM, quantized them, and found out that their oscillations can be seen as the various fields that we already know, so their interpretation is, to me, relatively simple. For D$p$-branes, we introduced them as the hypersurfaces where one can fix the extremal points of open strings using the Dirichlet boundary condition: so far so good, they're just mathematical objects, special hypersurfaces in spacetime. But when attempting quantization I think I lost track of what was going on, and couldn't find an illuminating picture of D$p$-branes as dynamical objects on my books.
There are many points that I don't understand: are D$p$-branes ($p+1$) dimensional objects where open strings live? Are they fundamental? Can they exist without open strings, can open strings exist without them? Are their oscillations the open strings themselves, or maybe the oscillations of the open strings that live on them? Are the branes a high-energy interpretation of something that we know today?

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Remember how you found that, for a closed string propagating freely, that there was a graviton mode, and then later realized that coherent states of string graviton modes can be used to "make" different (asymptotically flat) spacetimes? It's the same exact thing going on here. Assuming we are using the bosonic string, the first thing you realize is that on the Dp-brane you have 26-p-1 spin 0 string modes. These strings, which have both ends that start and end on the Dp-brane and oscillate in the directions transverse to the Dp-brane, are rather amazingly re-interpreted as the oscillations of the Dp-brane itself! There's no way to understand this classically. Classically, there's no way that a string wiggling on a brane can be considered a wiggle of the higher dimensional brane itself. However, like with almost everything, it is really the quantum mechanics that brings the little string to life. If you are just thinking about these spin 0 modes of the string, then from standard QFT class we know that we can reinterpret spin 0 particles as a scalar field. This scalar field lives on the Dp-brane, i.e. its base space is p+1 dimensional. If we now make coherent states of many of these spin 0 particles, we can say that the expectation value for the field operator gives us the target space embedding coordinate of the Dp-brane in one of its transverse dimensions! So now, we have this picture where the Dp-brane is a quantum object, a plane embedded in space where the exact value of its target space coordinates has some quantum fuzziness, much like the string itself. Anyway, the coherent state picture of the transverse brane oscillations is really the main thing going on here. It's why we can use string scattering to study Dp-brane interactions and what not.

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    $\begingroup$ Could you provide the exact expression for the vertex operator involving the scalar fields? $\endgroup$ – Nihar Karve Apr 25 at 1:21
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D-branes, at first sight, look pretty useless - who cares about the hypersurface on which an open string can end? Classically this would probably be the case, but on quantisation of the oriented bosonic string, we see that the massless level-1 states created by oscillators longitudinal to the brane can be interpreted as the quanta of a dim-p spin-1 gauge field lying on the Dp-brane, and those by oscillators transverse to the brane can be interpreted as $(D-p-1)$ scalar fields on the brane. These are the Goldstone modes from breaking the translational part of Poincaré symmetry, forming a singlet under the Lorentz group of the brane. A hint that these scalar fields should correspond to fluctuations of the D-brane in space, and hence that D-branes should be dynamical, is that string theory is a theory of gravity, and so no object, non-perturbative or otherwise, should be perfectly rigid.

This is the commonly presented method for realising the existence of Dp-branes, but they can also be presented by T-dualising a theory of open superstrings on $(9-p)$ toroidally compactified dimensions, which is arguably more enlightening. T-duality is an exact perturbative duality of the theory, and in the open bosonic case, it is easily seen that it swaps Dirichlet and Neumann boundary conditions on the dualised directions. The gauge field in the compact direction turns into the transverse position of the D-brane in the dual theory. This confirms our previous hypothesis and just as certain closed string states correspond to fluctuations of the background geometry, certain open string states correspond to fluctuations in the shape of the D-brane. (see Polchinski, chapter 8)

The most solid confirmation of this fact came from Polchinski's insight in 1995 that D-branes carry unit charge and act as sources for electric and magnetic RR gauge fields, in addition to the fact that they have the correct properties to be the necessary partners of strings and solitons under certain string dualities.

The original definition of D-branes simply being the endpoints of open strings under the Dirichlet boundary condition fails to encompass the existence of a stack of D-branes - this is remedied by considering the Dirichlet boundary conditions as arising from T-dualised Neumann boundary conditions after introducing a Wilson line to break the Chan-Paton factor group, as in this answer, after which we must associate an additional piece of data to distinguish each D-brane in the stack. As to whether D-branes can exist without open strings, the answer is no - the two heterotic string theories have neither, and in the other three string theories - type I, type IIA and type IIB - D-branes are always associated to the endpoints of open strings. In fact, for type II the situation is somewhat reversed: you first find that D-branes exist and this must mean that open strings must also exist, as mentioned here. The "oscillations" of the D-branes (a term that should be interpreted very lightly for an object with quantised modes) are not the open strings; rather, the strings couple to the background gauge fields on the D-brane, which enters as a term in the worldsheet action.

Whether or not D-branes are "fundamental" depends on whether we are looking at the theory perturbatively or not. In perturbative string theory, only the strings are fundamental, and D-branes arise as solitons. In general, all p-branes with $p>1$ become infinitely heavy as $g_s\to0$ and so are not seen by perturbation theory. On the other hand, in the non-perturbative limit $g_s\to\infty$, all p-branes (including Dp-branes) are all equally fundamental (brane democracy).

D-branes can be interpreted as analogous to the quarks on which QCD flux tubes with dynamical boundary conditions end - indeed, historically this was probably the first inkling that D-branes in string theory might themselves be dynamical. There is a more concrete interpretation of domain walls in SQCD as D-branes, with the incantation "domain walls are semi-classical D-branes for the vortex strings" which is a nice throwback to the fact that (perturbatively) D-branes are solitonic - you can read more about this in Tong's TASI Lectures on Solitons.

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Even if the two already presented answers are fine, looking further into the subject I found on Polchinski "String theory vol.1 - Introduction to the bosonic string" a great concise explanation that could help others studying this subject:

This is the same phenomenon as with spacetime itself. We start with strings in a flat background and discover that a massless closed string state corresponds to fluctuations of the geometry. Here we found first a flat hyperplane, and then discovered that a certain open string state corresponds to fluctuations of its shape. We should not be surprised that the hyperplane has become dynamical. String theory contains gravity. A gravitational wave passing through the hyperplane would warp spacetime itself, so the hyperplane could hardly remain rigid.

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