Why does T-duality not create consistent string theories below the critical dimension? As I know it, T-duality essentially tells us that if we compactify a superstring theory on a circle of radius $R$, it is equivalent to a string theory compactified on a circle of radius $\tfrac{\alpha'}{R}$. So, take $R\to\infty$. Then $R\to\infty$ and $\tfrac{\alpha'}{R}\to 0$. Hence, this should tell us that a string theory in 10 (noncompact) dimensions is equivalent to one in 9D.
However, I know that superstring theory is only non-anomalous and free of ghosts in 10D, so where does this reasoning break down?
 A: Well, I think I finally understand why my argument is flawed. Consider the case $R\to 0$ for closed strings; all of the momentum states, i.e., states with KK mode $n\ne 0$, become infinitely massive. If we were studying field theory we would stop here, as this would be all that would happen—the surviving fields would simply be independent of the compact coordinate, and so we would have performed a dimension reduction down to 9D. However, in string theory, things are different: the pure winding states (i.e., $n = 0$, winding mode $w \ne 0$ states) form a continuum as $R \to 0$, since it is very "cheap" to wind around the small circle. Therefore, in the $R \to 0$ limit, an effective uncompactified dimension reappears.
Let us now consider the $R \to 0$ limit of the open string spectrum. Open
strings do not have a conserved winding around the periodic dimension and so
they have no quantum number comparable to $w$, so something different must
happen, as compared to the closed string case. In fact, it is more like field
theory: when $R \to 0$ the states with nonzero momentum go to infinite mass,
but there is no new continuum of states coming from winding. So we are left with a theory in one dimension fewer. A puzzle arises when one remembers that theories with open strings have closed strings as well, so that in the $R \to 0$ limit the closed strings live in 10 spacetime dimensions but the open strings only in 9D.
This is perfectly fine, though, since the interior of the open string is indistinguishable from the closed string and so should still be vibrating in 10
dimensions. The distinguished part of the open string is the endpoints, and
these are restricted to a 9-dimensional hyperplane. In particular, this is a D9-brane.
