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In string theory and related theories (such as supergravity), a brane is a physical object that generalizes the notion of a zero- point particle, a one-dimensional string, or a two-dimensional membrane to higher-dimensional objects. Branes are dynamical system objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge.
Mathematically, branes can be represented within categories, and are studied in pure mathematics for insight into homological mirror symmetry and noncommutative geometry.
The word "brane" originated in 1987 as a contraction of "membrane".
A p-brane sweeps out a ( p+1)-dimensional volume in spacetime called its worldvolume. Physicists often study fields analogous to the electromagnetic field, which live on the worldvolume of a brane.Moore 2005, p. 214
One crucial point about D-branes is that the dynamics on the D-brane worldvolume is described by a gauge theory, a kind of highly symmetric physical theory which is also used to describe the behavior of elementary particles in the Standard Model of particle physics. This connection has led to important insights into gauge theory and quantum field theory. For example, it led to the discovery of the AdS/CFT correspondence, a theoretical tool that physicists use to translate difficult problems in gauge theory into more mathematically tractable problems in string theory.Moore 2005, p. 215
In one version of string theory known as the topological B-model, the D-branes are complex manifold of certain six-dimensional shapes called Calabi–Yau manifolds, together with additional data that arise physically from having charges at the endpoints of strings.Zaslow 2008, p. 536 Intuitively, one can think of a submanifold as a surface embedded inside of a Calabi–Yau manifold, although submanifolds can also exist in dimensions different from two.Yau and Nadis 2010, p. 165 In mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaf on the Calabi–Yau.Aspinwal et al. 2009, p. 575 In another version of string theory called the topological 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 submanifolds.Aspinwal et al. 2009, p. 575 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.Yau and Nadis 2010, p. 175 The category having these branes as its objects is called the Fukaya category.Aspinwal et al. 2009, p. 575
The derived category of coherent sheaves is constructed using tools from complex geometry, a branch of mathematics that describes geometric shapes in Algebra and solves geometric problems using algebraic equations.Yau and Nadis 2010, pp. 180–1 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.Zaslow 2008, p. 531
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 a completely different Calabi–Yau manifold.Aspinwall et al. 2009, p. 616 This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.Yau and Nadis 2010, p. 181
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