Editing Mirror symmetry (string theory) (section)

===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>