T-duality
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T-duality is a symmetry of string theory, relating type IIA and type IIB string theory, and the two heterotic string theories. T-duality transformations act on spaces in which at least one direction has the topology of a circle. Under the transformation, the radius R of that direction will be changed to 1/R, and "wrapped" string states will be exchanged with high-momentum string states in the dual theory.
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.
The total squared mass of a closed string
is invariant under the exchange 
, and the interactions and all other physical phenomena can be proved invariant under this operation, too. T-duality acting on D-branes changes their dimension by +1 or -1.
Andrew Strominger, Shing-Tung Yau, and Eric Zaslow have showed that mirror symmetry can be understood as T-duality applied to three-dimensional toroidal fibres of the Calabi-Yau space.
