Re: T-duality

From: Michael Mandelberg (mmandelberg_at_comcast.net)
Date: 07/21/04 

Date: 21 Jul 2004 13:02:40 -0400

"Nikolaas Van der Heyden" <nikolaas.vanderheyden@ugent.be> wrote in message news:<20040718121920.DB7FA4C40E9@astra.telenet-ops.be>...

> I have some difficulties understanding the concept of T-duality in string
> theory.

> Can You explain me what this means. It must have something to do with the
> fact that small scale theories are equivalent to large scale theories, due
> to the T-duality.

Your conceptual understanding is correct. In the simple case of a
closed string with one compact dimesion (a circle), there is a
critical value for the circle's radius, R*, such that the the spectrum
of the theory for R is the same as that for:

 (R*)^2/R.

Note that if R > R* then (R*)^2/R < R* and vice versa.

The mapping between the states of the two theories is quite
straightforward in this case. The theory with R>R* has states that
have energies above the ground state due to the compact dimension for
two reasons. The first is that they can have momentum in the compact
dimension. These momenta are "quantized" to insure that the wave
function for the string is single valued on the circle (usual Fourier
series analysis). So there is a sequence of states with ever
increasing momenta in the compact dimension.

However, the string also stores energy in the tension of the string.
There are a sequence of states for which the string wraps around the
circle. These states are also quantized, since a string must wrap an
integral number of times around the circle. Each winding adds energy
to the state as the string must stretch more, thus increasing the
tension.

In general, a string can have an arbitrary combination of momentum and
wrapping around the circle, leading to states labeled by two numbers:

     | k, i >
             R

where k is the wave number and i the winding number and the subscript
R indicate that this is a state in a universe where the circle has
radius R.

Now, what happens when you switch to (R*)^2/R? Well, it is fairly
intuitive (if you are used to this sort of thing), that the smaller
the radius, the more energetic the momentum modes become (because they
are shorter wavelength/higher frequency), and the less energetic the
winding modes become (because the string has to stretch less to go
around the circle). So it is not implausible that the following two
states might have the same energy:

   | k, i > |i, k >
           R (R*)^2/R

A calculation shows that this is the case: a state in a theory with
radius R, wave number k, and winding number i has the same energy as a
state in a theory with radius (R*)^2/R, wave number i, and winding
number k.

Of course while this is suggestive of a true duality, this is not
enough. It is also necessary to show that the dynamics of the two
theories are the same. In other words, you need to show that all
scattering processes between any set of states is the same in the two
theories, after exchanging wave number for winding number.

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