The vector $V^\mu$ was related to the Lorentz contravariant components $V^a(x)$ \begin{equation} V^a(x)=e^a_\mu(x) V^\mu \end{equation} where $V^\mu$ was contravariant under the general coordinate transformation. (D-branes, Clifford Johnson Page 67)

Notice that $a$ and $\mu$ does not have to have the same dimension, and from the context it seemed that $V^a$ had less degree of freedom?($|\{a\}|\leq|\{\mu\}|$)

This looked like a dimension reduction procedure, but was different from the compatification, i.e. with the compatification procedure, the extra index($\mu=25$) was still there, just not observed under the low energy. But the Lorentz contravariant components eliminated the index completely. The similar thing was the basis from the world sheet to target space, but $V^a(x)$ and $V^\mu$ were obviously both the concept of the target space.

On the other hand, the $V^a(x)$ now dependent on the coordinate $x$, but not for $V^\mu$?(or was it an abbreviation?)

Could you give a hint of how the vector $V^\mu$ and the Lorentz contravariant components $V^a(x)$ was used?

Related:What is the Difference between Lorentz Invariant and Lorentz Covariant?