# T duality under a small fluctuation of the compact dimension

How do small perturbations around the compact dimension affect T duality. What happens if I chose a compactification of the nature $r+\delta r$. And what keeps the compact dimension stable, i.e from expanding into a large dimension or imploding into itself.

T-duality says that the radius $r$ is equivalent to $\alpha' / r$. So $r+\delta r$ is also equivalent to $\alpha'/(r+\delta r)$, too. If the radius fluctuates, so does its T-dual radius.
The radius itself, usefully written as $\sqrt{\alpha'}\exp(\phi_R)$, is a "modulus", a scalar field that has no potential (i.e. all conceivable values are equally allowed: this implies that the scalar field is massless around any point and produces new long-range forces that would invalidate the equivalence principle - the experimentally verified principle that all objects accelerate by the same acceleration in a gravitational field) in non-realistic vacua but a potential is generated in realistic ("stabilized") vacua. In the latter vacua, $r=r_0$ chooses a particular value for which $V'(r_0)=0$, the potential has to be minimized.
One may still describe the same point using $\tilde r_0 =\alpha' / r_0$ but we usually pick the larger among these two T-dual values because large compactification radii (longer than the string length) are those in which string theory agrees with the low-energy effective field theory more directly (the excited strings may be neglected for many purposes).