page_title: T-duality
section_title: Mirror symmetry
hierarchical_section_title: T-duality / Mirror symmetry
context_page_description: In theoretical physics, T-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 an imaginary 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 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.
context_section_description: 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. 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. 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.
  One approach to understanding mirror symmetry is the SYZ conjecture, which was suggested by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow in 1996. 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.
  The simplest example of a Calabi-Yau manifold is a torus (a surface shaped like a donut). Such a surface can be viewed as the product of two circles. This means that the torus can be viewed as the union of a collection of longitudinal circles (such as the red circle in the image). There is an auxiliary space which says how these circles are organized, and this space is itself a circle (the pink circle). This space is said to parametrize the longitudinal circles on the torus. In this case, mirror symmetry is equivalent to T-duality acting on the longitudinal circles, changing their radii from  to , with  the inverse of the string tension.
  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). 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. In this way, the SYZ conjecture provides a geometric picture of how mirror symmetry acts on a Calabi-Yau manifold.
caption: A torus is the cartesian product of two circles.
exif: {}