I'm just going to consider the bosonic sector in the following. I think your issue comes from this:
the fundamental degrees of freedoms are from string worldsheets (so at most they are fundamentally 2d).
Recall that the sigma model $X^i :\Sigma \hookrightarrow M$ describes the embeding of strings into a spacetime $M$. The degrees of freedom are the fields $X^i$ themselves and not the coordinates on the worldsheet that are mapped to the target space.
The massless field content of type IIB string theory is described by type IIB supergravity in ten dimensions, whose bosonic field content in the Neveu-Schwarz– Neveu-Schwarz (NS-NS) and Ramond–Ramond (R-R) sectors.
These fields are a backreation on the target space $M$ of the first excitation states and are described by an action on the target and not the worldsheet.
For example, the R-R p-forms $C_p \in \Omega^p(M)$. For a complete dictionary of the field content, see the democratic formulation.