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A condensate of open strings with both ends attached to the same D-brane can be equivalent to a displacement of the D-brane with no open string condensate.

A solution to the D-brane Born-Infeld action gives rise to a semi-infinite string extension which is entirely equivalent to an open string with one end attached to the D-brane.

Why aren't D-branes and strings independent degrees of freedom?

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D-branes and fundamental strings are not independent degrees of freedom because D-branes are solitons or topological defects constructed out of closed string fields. At weak coupling, the closed string degrees of freedom are the only genuinely elementary fields; everything else, including D-branes, may be thought of as being constructed out of them.

Equivalently, it's because the string world sheets that determine all the dynamics of the D-branes are the same world sheets that also determine other things, and these world sheets - histories of strings - can't be double-counted.

The reason why the degrees of freedom are not independent of each other is the same as the reason (known as the "modular invariance") why the torus with the complex structure $\tau$ is equivalent to a torus with the complex structure $-1/\tau$: they only differ in the way how we "read them" or "slice them". One is not allowed to double-count $\tau$ and $-1/\tau$ because they represent two configurations related by a large world sheet diffeomorphism, and all world sheet diffeomorphisms - including the large ones - have to be counted as gauge symmetries (for consistency).

In the same way, a cylinder with circumference of the circular boundaries $2\pi t$ and with the distance $\pi$ between the two boundaries may be thought of as a loop of an open string - if we slice the cylinder into line intervals, i.e. $${\rm Tr}_{\rm open} \exp(-\beta H_{\rm open}),$$ or as a closed string propagation between two boundary states (closed string states encoding the full information about the D-brane), $$\langle B_1|\exp(-\beta H_{\rm closed})|B_2\rangle.$$ This "failure to be independent" between the degrees of freedom is the basis of the UV-IR connection and gauge-gravity or open-closed dualities.

Any configuration of a D-brane or D-branes that solves the stringy equations of motion may be represented by the "boundary state", the closed string state determining the allowed boundary conditions of a world sheet. It must obey various conditions. On the other hand, every perturbation of a D-brane state is also generated by the quanta of open strings attached to this D-brane. These descriptions are mathematically different but physically equivalent and the corresponding degrees of freedom coincide - they shouldn't be double counted.

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