T-duality

Let us now consider the behaviour of the spectrum in regimes of large and small radii.

• In the limit $R\to \infty$, states with non-zero winding number become infinitely massive, while the states with $\omega =0$ become numerous and the KK momentum becomes a continuous spectrum. This reproduces the de-compactified $D$-dimensional theory.
• In the limit $R\to 0$, states with non-zero KK momentum become infinitely massive, while the spectrum of winding states with $n=0$ approach a continuum. Therefore, it seems like the theory grows an additional dimension as $R\to 0$.

While the first limit is familiar from QFT, the second limit is radically different from QFT expectations. This above result is due to the symmetry between KK momentum and the winding number. Indeed, the spectrum is invariant under simultaneously changing KK momentum and winding number and inverting the radius of the circle,

$n\leftrightarrow \omega R\leftrightarrow \frac{{\alpha }^{\prime }}{R}$.

This extends to an equivalence of the full interacting string theory. First, notice that it transforms the left- and right- moving momenta as $p_L^d\to p_L^d$ and $p_R^d\to -p_R^d$, which naturally extends to a transformation $X_R^d({\sigma }^{-})\to -X_R^d({\sigma }^{-})$ of the entire right-moving mode expansion. This could be viewed as a space-time parity operation acting only on the right-moving modes. The eom, energy-momentum tensor and correlation functions are invariant under this transformation and gives rise to a quantum equivalence between interacting string theories. This is known as T-duality.

Since string theories related by T-duality are physically equivalent, the parameter space of physically distinct string theories may be taken to be $R\ge \sqrt{{\alpha }^{\prime }}=\frac{\ell }{\sqrt{2}}$.

If we try to probe $R<\sqrt{{\alpha }^{\prime }}$, string theory is basically equivalent to another string theory at radius $R^{\prime }>\sqrt{{\alpha }^{\prime }}$. In other words, there is a minimum radius in string theory. This is profound and suggests that string theory probes geometry in an entirely different way to particles.