In theoretical physics, T-duality (short for target-space duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories describes strings propagating in spacetime shaped like a circle of some radius R {\displaystyle R} , while the other theory describes strings propagating on a spacetime shaped like a circle of radius proportional to 1 / R {\displaystyle 1/R} . The idea of T-duality was first noted by Bala Sathiapalan in an obscure paper in 1987. The two T-dual theories are equivalent in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, momentum in one description takes discrete values and is equal to the number of times the string winds around the circle in the dual description. The idea of T-duality can be extended to more complicated theories, including superstring theories. The existence of these dualities implies that seemingly different superstring theories are actually physically equivalent. This led to the realization, in the mid-1990s, that all of the five consistent superstring theories are just different limiting cases of a single eleven-dimensional theory called M-theory. In general, T-duality relates two theories with different spacetime geometries. In this way, T-duality suggests a possible scenario in which the classical notions of geometry break down in a theory of Planck scale physics. The geometric relationships suggested by T-duality are also important in pure mathematics. Indeed, according to the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, T-duality is closely related to another duality called mirror symmetry, which has important applications in a branch of mathematics called enumerative algebraic geometry.
Overview
Strings and duality T-duality is a particular example of a general notion of duality in physics. The term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena. Like many of the dualities studied in theoretical physics, T-duality was discovered in the context of string theory. In string theory, particles are modeled not as zero-dimensional points but as one-dimensional extended objects called strings. The physics of strings can be studied in various numbers of dimensions. In addition to three familiar dimensions from everyday experience (up/down, left/right, forward/backward), string theories may include one or more compact dimensions which are curled up into circles. A standard analogy for this is to consider multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions. Such extra dimensions are important in T-duality, which relates a theory in which strings propagate on a circle of some radius R {\displaystyle R} to a theory in which strings propagate on a circle of radius 1 / R {\displaystyle 1/R} .
Winding numbers
In mathematics, the winding number of a curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The notion of winding number is important in the mathematical description of T-duality where it is used to measure the winding of strings around compact extra dimensions. For example, the image below shows several examples of curves in the plane, illustrated in red. Each curve is assumed to be closed, meaning it has no endpoints, and is allowed to intersect itself. Each curve has an orientation given by the arrows in the picture. In each situation, there is a distinguished point in the plane, illustrated in black. The winding number of the curve around this distinguished point is equal to the total number of counterclockwise turns that the curve makes around this point.
When counting the total number of turns, counterclockwise turns count as positive, while clockwise turns counts as negative. For example, if the curve first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three. According to this scheme, a curve that does not travel around the distinguished point at all has winding number zero, while a curve that travels clockwise around the point has negative winding number. Therefore, the winding number of a curve may be any integer. The pictures above show curves with winding numbers between −2 and 3:
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