T-duality

In the simplest example of this relationship, one of the theories describes strings propagating in a spacetime shaped like a circle of some radius

, while the other theory describes strings propagating on a spacetime shaped like a circle of radius proportional to

The existence of these dualities implies that seemingly different superstring theories are actually physically equivalent.

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.

The term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way.

Like many of the dualities studied in theoretical physics, T-duality was discovered in the context of string theory.

[4] If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length.

Such extra dimensions are important in T-duality, which relates a theory in which strings propagate on a circle of some radius

The pictures above show curves with winding numbers between −2 and 3: The simplest theories in which T-duality arises are two-dimensional sigma models with circular target spaces, i.e. compactified free bosons.

These are simple quantum field theories that describe propagation of strings in an imaginary spacetime shaped like a circle.

The strings can thus be modeled as curves in the plane that are confined to lie in a circle, say of radius

denotes the winding number of the string around the circle, and the constant mode

Since this expression represents the configuration of a string at a fixed time, all coefficients (

One can show, using the fact that the strings considered here are closed, that this momentum can only take on discrete values of the form

In the situation described above, the total energy, or Hamiltonian, of the string is given by the expression Since the momenta of the theory are quantized, the first two terms in this formula are

In the mid 1990s, physicists noticed that these five string theories are actually related by highly nontrivial dualities.

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.

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.

[10] The simplest example of a Calabi–Yau manifold is a torus (a surface shaped like a donut).

In this case, mirror symmetry is equivalent to T-duality acting on the longitudinal circles, changing their radii from

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.