T-duality ("T" is for target space) is a symmetry of quantum field theories with differing classical descriptions, of which the relationship between small and large distances in various string theories is a special case. Discussion of the subject originated in a paper by T. H. Buscher and was further developed by Martin Rocek and Erik Verlinde. T-duality is not present in ordinary particle theory, indicating that strings would experience spacetime in a way that is fundamentally distinct than the way particles do. It relates different string theories that were thought to be unrelated before T-duality was understood. T-duality preceded the Second Superstring Revolution.
Qualitative description 
String theory predicts the existence of extra dimensions in addition to the usual three space and single time dimensions. Different shapes and sizes of these extra dimensions result in different forces and particles appearing in the four-dimensional low energy physics and so universes with different shapes for the extra dimensions will have different physics. However, many of these geometries result in the same physics and this is the basis of T-duality.
For example, consider the case where one of the dimensions is a circle of radius R. A particle or string can carry momentum in this direction, and such a state is called a Kaluza-Klein mode. The momentum in this direction is quantized in that the momentum satisfies
As the radius R becomes smaller, it costs more energy to excite one of these modes. On the other hand, as R gets larger, the spacing between Kaluza-Klein states becomes smaller and in the limit of infinite radius, momentum is no longer quantized.
Unlike particles, a closed string can also wrap around the extra dimension. Such a state is called a winding mode. The energy to excite a winding mode is quantized proportional to the radius R, so that as the radius gets smaller, the spacing between winding modes gets smaller, and in the limit of zero radius is no longer quantized while as the radius gets larger, it costs more energy to excite the winding modes. This is the opposite behavior of the Kaluza-Klein modes and suggests that the small and large radius behavior of the closed string is the same if we interchange winding modes and Kaluza-Klein modes. One can show that the physics at radius R is the same as the physics at radius α'/R. This relationship is an example of T-duality.
Bosonic string 
To illustrate the ideas of T-duality, consider the bosonic string compactified on a circle of radius R. The string may carry a net momentum p in the compactified dimension. As is the case for particles, the momentum must be quantized in units of 1/R,
where n is an integer. However, unlike the case for particles, a closed string may also wrap around the compactified dimension. The number of times that the closed string wraps around that dimension is called the winding number w. The mass of a closed string is then
where N and Ñ are the excitations for the left- and right-movers of the closed string, and α' is the slope parameter. This spectrum is invariant under the interchange
That is, the spectrum of the closed string is the same spectrum of a closed string in a background with radius α'/R. One can similarly show that the interactions of closed strings are also invariant under this interchange. This implies that the closed bosonic string compactified on a radius R is equivalent to the theory with radius α'/R.
The idea of T-duality can be extended to more general backgrounds and even to superstring theories. T-duality interchanges the type II superstrings with each other and also heterotic strings. For example, one might begin with a IIA string wrapped once around the direction in question. Under T-duality, it will be mapped to a IIB string which has momentum in that direction. A IIA string with a winding number of two (wrapped twice) will be mapped to a IIB string with two units of momentum, and so on.
Open strings and D-branes 
T-duality acting on D-branes changes their dimension by +1 or -1.
Mirror symmetry 
- "nlab page on T-duality".
- "Generalised complex geometry and T-duality".
- superstringtheory article Looking for extra dimensions by Patricia Schwarz
See also 
- Becker, K., Becker, M., and Schwarz, J. H. (2007). String Theory and M-Theory: A Modern Introduction. Cambridge, UK: Cambridge University Press.
- Polchinski, J. (1998). String Theory. Cambridge, UK: Cambridge University Press.
- Buscher, T.H. (1987), "A symmetry of the string background field equations", Phys. Lett. B, 194(1):59-62
- Rocek, M.; Verlinde, E. (1992), "Duality, quotients and currents", Nuclear Phys. B, 373(3):630-646