Editing Mirror symmetry (string theory) (section)

==Approaches==

===Homological mirror symmetry===
{{main article|Homological mirror symmetry}}

[[File:D3-brane et D2-brane.PNG|thumb|right|alt=A pair of surfaces joined by wavy line segments.|Open strings attached to a pair of [[D-brane]]s]]

In string theory and related theories in physics, a ''[[brane]]'' is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.<ref>{{harvnb|Moore|2005|page=214}}.</ref>

In string theory, a string may be open (forming a segment with two endpoints) or closed (forming a closed loop). [[D-brane]]s are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to a condition that it satisfies, the [[Dirichlet boundary condition]].<ref>{{harvnb|Moore|2005|page=215}}.</ref>

Mathematically, branes can be described using the notion of a [[category (mathematics)|category]].<ref>{{harvnb|Aspinwall et al.|2009|p={{pn|date=August 2023}}}}.</ref> This is a mathematical structure consisting of ''objects'', and for any pair of objects, a set of ''[[morphisms]]'' between them. In most examples, the objects are mathematical structures (such as [[set (mathematics)|sets]], [[vector spaces]], or [[topological spaces]]) and the morphisms are [[function (mathematics)|functions]] between these structures.<ref>A basic reference on category theory is {{harvnb|Mac Lane|1998}}.</ref> One can also consider categories where the objects are D-branes and the morphisms between two branes <math>\alpha</math> and <math>\beta</math> are [[wavefunction|states]] of open strings stretched between <math>\alpha</math> and <math>\beta</math>.<ref name=autogenerated11>{{harvnb|Zaslow|2008|page=536}}.</ref>

In the B-model of topological string theory, the D-branes are [[complex manifold|complex submanifold]]s of a Calabi–Yau together with additional data that arise physically from having charges at the endpoints of strings.<ref name=autogenerated11 /> Intuitively, one can think of a submanifold as a surface embedded inside the Calabi–Yau, although submanifolds can also exist in dimensions different from two.<ref name=autogenerated7 /> In mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi–Yau.<ref name=autogenerated2>{{harvnb|Aspinwall et al.|2009|p=575}}.</ref> In the A-model, the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold. Roughly speaking, they are what mathematicians call [[special Lagrangian submanifold]]s.<ref name=autogenerated2 /> This means among other things that they have half the dimension of the space in which they sit, and they are length-, area-, or volume-minimizing.<ref name=autogenerated12>{{harvnb|Yau|Nadis|2010|p=175}}.</ref> The category having these branes as its objects is called the Fukaya category.<ref name=autogenerated2 />

The derived category of coherent sheaves is constructed using tools from [[complex geometry]], a branch of mathematics that describes geometric curves in algebraic terms and solves geometric problems using [[algebraic equation]]s.<ref>{{harvnb|Yau|Nadis|2010|pages=180–181}}.</ref> On the other hand, the Fukaya category is constructed using [[symplectic geometry]], a branch of mathematics that arose from studies of [[classical physics]]. Symplectic geometry studies spaces equipped with a [[symplectic form]], a mathematical tool that can be used to compute [[area]] in two-dimensional examples.<ref name=autogenerated5 />

The homological mirror symmetry conjecture of Maxim Kontsevich states that the derived category of coherent sheaves on one Calabi–Yau manifold is equivalent in a certain sense to the Fukaya category of its mirror.<ref>{{harvnb|Aspinwall et al.|2009|p=616}}.</ref> This equivalence provides a precise mathematical formulation of mirror symmetry in topological string theory. In addition, it provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.<ref>{{harvnb|Yau|Nadis|2010|page=181}}.</ref>

===Strominger–Yau–Zaslow conjecture===
{{main article|SYZ conjecture}}
[[File:Torus cycles2.svg|thumb|right|alt=A donut shape with two circles drawn on its surface, one going around the hole and the other going through it.|A [[torus]] can be viewed as a [[union (set theory)|union]] of infinitely many circles such as the red one in the picture. There is one such circle for each point on the pink circle.]]
Another approach to understanding mirror symmetry was suggested by Andrew Strominger, Shing-Tung Yau, and [[Eric Zaslow]] in 1996.<ref name=autogenerated3 /> According to their conjecture, now known as the SYZ conjecture, mirror symmetry can be understood by dividing a Calabi–Yau manifold into simpler pieces and then transforming them to get the mirror Calabi–Yau.<ref>{{harvnb|Yau|Nadis|2010|page=174}}.</ref>

The simplest example of a Calabi–Yau manifold is a two-dimensional [[torus]] or donut shape.<ref>{{harvnb|Zaslow|2008|page=533}}.</ref> Consider a circle on this surface that goes once through the hole of the donut. An example is the red circle in the figure. There are infinitely many circles like it on a torus; in fact, the entire surface is a [[union (set theory)|union]] of such circles.<ref>{{harvnb|Yau|Nadis|2010|pages=175–176}}.</ref>

One can choose an auxiliary circle <math>B</math> (the pink circle in the figure) such that each of the infinitely many circles decomposing the torus passes through a point of <math>B</math>. This auxiliary circle is said to ''parametrize'' the circles of the decomposition, meaning there is a correspondence between them and points of <math>B</math>. The circle <math>B</math> is more than just a list, however, because it also determines how these circles are arranged on the torus. This auxiliary space plays an important role in the SYZ conjecture.<ref name=autogenerated12 />

The idea of dividing a torus into pieces parametrized by an auxiliary space can be generalized. Increasing the dimension from two to four real dimensions, the Calabi–Yau becomes a [[K3 surface]]. Just as the torus was decomposed into circles, a four-dimensional K3 surface can be decomposed into two-dimensional tori. In this case the space <math>B</math> is an ordinary [[sphere]]. Each point on the sphere corresponds to one of the two-dimensional tori, except for twenty-four "bad" points corresponding to "pinched" or [[mathematical singularity|singular]] tori.<ref name=autogenerated12 />

The Calabi–Yau manifolds of primary interest in string theory have six dimensions. One can divide such a manifold into [[3-torus|3-tori]] (three-dimensional objects that generalize the notion of a torus) parametrized by a [[3-sphere]] <math>B</math> (a three-dimensional generalization of a sphere). Each point of <math>B</math> corresponds to a 3-torus, except for infinitely many "bad" points which form a grid-like pattern of segments on the Calabi–Yau and correspond to singular tori.<ref>{{harvnb|Yau|Nadis|2010|page=175–177}}.</ref>

Once the Calabi–Yau manifold has been decomposed into simpler parts, mirror symmetry can be understood in an intuitive geometric way. As an example, consider the torus described above. Imagine that this torus represents the "spacetime" for a [[physical theory]]. The fundamental objects of this theory will be strings propagating through the spacetime according to the rules of [[quantum mechanics]]. One of the basic dualities of string theory is T-duality, which states that a string propagating around a circle of radius <math>R</math> is equivalent to a string propagating around a circle of radius <math>1/R</math> in the sense that all observable quantities in one description are identified with quantities in the dual description.<ref name=autogenerated4>{{harvnb|Zaslow|2008|page=532}}.</ref> For example, a string has [[momentum]] as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the [[winding number]]. If a string has momentum <math>p</math> and winding number <math>n</math> in one description, it will have momentum <math>n</math> and winding number <math>p</math> in the dual description.<ref name=autogenerated4 /> By applying T-duality simultaneously to all of the circles that decompose the torus, the radii of these circles become inverted, and one is left with a new torus which is "fatter" or "skinnier" than the original. This torus is the mirror of the original Calabi–Yau.<ref>{{harvnb|Yau|Nadis|2010|page=178}}.</ref>

T-duality can be extended from circles to the two-dimensional tori appearing in the decomposition of a K3 surface or to the three-dimensional tori appearing in the decomposition of a six-dimensional Calabi–Yau manifold. In general, the SYZ conjecture states that mirror symmetry is equivalent to the simultaneous application of T-duality to these tori. In each case, the space <math>B</math> provides a kind of blueprint that describes how these tori are assembled into a Calabi–Yau manifold.<ref>{{harvnb|Yau|Nadis|2010|pages=178–179}}.</ref>