The basic idea of T duality can be illustrated by considering a compact dimension consisting of a circle of radius . 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 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
T duality exchanges these two kinds of excitations by mapping
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
is bounded below not
only by the reciprocal of the momentum spread,
, but also by the
string
scale
. (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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