# T-duality

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.^{[2]} 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.

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.^{[3]} 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.

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:

In the situation described above, the total energy, or Hamiltonian, of the string is given by the expression

Up until the mid 1990s, physicists working on string theory believed there were five distinct versions of the theory: type I, type IIA, type IIB, and the two flavors of heterotic string theory (SO(32) and E_{8}×E_{8}). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries.

In the mid 1990s, physicists noticed that these five string theories are actually related by highly nontrivial dualities. One of these dualities is T-duality. For example, it was shown that type IIA string theory is equivalent to type IIB string theory via T-duality and also that the two versions of heterotic string theory are related by T-duality.

The existence of these dualities showed that the five string theories were in fact not all distinct theories. In 1995, at the string theory conference at University of Southern California, Edward Witten made the surprising suggestion that all five of these theories were just different limits of a single theory now known as M-theory.^{[5]} Witten's proposal was based on the observation that different superstring theories are linked by dualities and the fact that type IIA and E_{8}×E_{8} heterotic string theories are closely related to a gravitational theory called eleven-dimensional supergravity. His announcement led to a flurry of work now known as the second superstring revolution.

In string theory and algebraic geometry, the term "mirror symmetry" refers to a phenomenon involving complicated shapes called Calabi–Yau manifolds. These manifolds provide an interesting geometry on which strings can propagate, and the resulting theories may have applications in particle physics.^{[6]} In the late 1980s, it was noticed that such a Calabi–Yau manifold does not uniquely determine the physics of the theory. Instead, one finds that there are *two* Calabi–Yau manifolds that give rise to the same physics.^{[7]} These manifolds are said to be "mirror" to one another. This mirror duality is an important computational tool in string theory, and it has allowed mathematicians to solve difficult problems in enumerative geometry.^{[8]}

One approach to understanding mirror symmetry is the SYZ conjecture, which was suggested by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow in 1996.^{[9]} According to the SYZ conjecture, mirror symmetry can be understood by dividing a complicated Calabi–Yau manifold into simpler pieces and considering the effects of T-duality on these pieces.^{[10]}

The SYZ conjecture generalizes this idea to the more complicated case of six-dimensional Calabi–Yau manifolds like the one illustrated above. As in the case of a torus, one can divide a six-dimensional Calabi–Yau manifold into simpler pieces, which in this case are 3-tori (three-dimensional objects which generalize the notion of a torus) parametrized by a 3-sphere (a three-dimensional generalization of a sphere).^{[11]} T-duality can be extended from circles to the three-dimensional tori appearing in this decomposition, and the SYZ conjecture states that mirror symmetry is equivalent to the simultaneous application of T-duality to these three-dimensional tori.^{[12]} In this way, the SYZ conjecture provides a geometric picture of how mirror symmetry acts on a Calabi–Yau manifold.