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

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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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.