T Duality


The basic idea of T duality(for a recent discussion see [5]) can be illustrated by considering a compact dimension consisting of a circle of radius R.
In this case there are two kinds of excitations to consider. The first, which is not special to string theory, are Kaluza--Klein momentum excitations on the circle, which contribute (n/R)2 to the energy squared, where n is an integer. Winding-mode excitations, due to a closed string winding m times around the circular dimension, are special to string theory. If

denotes the string tension (energy per unit length), the contribution to the energy squared is Em=2pmRT.
T duality exchanges these two kinds of excitations by exchanging m with n and

This is part of an exact map between a T-dual pair A and B.
One implication is that usual geometric concepts break down at short distances, and classical geometry is replaced by "quantum geometry," which is described mathematically by 2D conformal field theory. It also suggests a generalization of the Heisenberg uncertainty principle according to which the best possible spatial resolution Dx is bounded below not only by the reciprocal of the momentum spread, Dp, but also by the string scale Lst. (Including non-perturbative effects, it may be possible to do a little better and reach the Planck scale.)
Two important examples of superstring theories that are T-dual when compactified on a circle are the IIA and IIB theories and the HE and HO theories. These two dualities reduce the number of distinct theories from five to three.


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