{{short description|Extended physical object in string theory}} {{Other uses}}{{Distinguish|Brain}}{{String theory}} {{wikt|brane}}
In string theory and related theories (such as supergravity), a '''brane''' is a physical object that generalizes the notion of a zero-dimensional point particle, a one-dimensional string, or a two-dimensional membrane to higher-dimensional objects. Branes are dynamical 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".<ref>{{oed|brane}}</ref>
== ''p''-branes == A point particle is a 0-brane, of dimension zero; a string, named after vibrating musical strings, is a 1-brane; a membrane, named after vibrating membranes such as drumheads, is a 2-brane.<ref>Moore 2005, p. 214</ref> The corresponding object of arbitrary dimension ''p'' is called a ''p''-brane, a term coined by M. J. Duff ''et al.'' in 1988.<ref>M. J. Duff, T. Inami, C. N. Pope, {{Interlanguage link multi|Ergin Sezgin|lt=E. Sezgin|de}}, and K. S. Stelle, "Semiclassical quantization of the supermembrane", ''Nucl. Phys.'' '''B297''' (1988), 515.</ref>
A ''p''-brane sweeps out a (''p''+1)-dimensional volume in spacetime called its '''worldvolume'''<!--boldface per WP:R#PLA-->. Physicists often study fields analogous to the electromagnetic field, which live on the worldvolume of a brane.<ref>Moore 2005, p. 214</ref>
== D-branes == {{Main|D-brane}}
[[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-branes]]
In string theory, a string may be open (forming a segment with two endpoints) or closed (forming a closed loop). D-branes 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 the Dirichlet boundary condition, which the D-brane satisfies.<ref>Moore 2005, p. 215</ref>
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.<ref>Moore 2005, p. 215</ref>
== Categorical description == Mathematically, branes can be described using the notion of a category.<ref>Aspinwall et al. 2009</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 sets, vector spaces, or topological spaces) and the morphisms are functions between these structures.<ref>A basic reference on category theory is Mac Lane 1998.</ref> One can likewise consider categories where the objects are D-branes and the morphisms between two branes <math>\alpha</math> and <math>\beta</math> are states of open strings stretched between <math>\alpha</math> and <math>\beta</math>.<ref>Zaslow 2008, p. 536</ref>
[[Image:Calabi yau.jpg|right|thumb|alt=Visualization of a complex mathematical surface with many convolutions and self intersections.|A cross section of a Calabi–Yau manifold ]]
In one version of string theory known as the topological B-model, the D-branes are complex submanifolds of certain six-dimensional shapes called Calabi–Yau manifolds, together with additional data that arise physically from having charges at the endpoints of strings.<ref>Zaslow 2008, p. 536</ref> 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.<ref>Yau and Nadis 2010, p. 165</ref> 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>Aspinwal et al. 2009, p. 575</ref> 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.<ref>Aspinwal et al. 2009, p. 575</ref> 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>Yau and Nadis 2010, p. 175</ref> The category having these branes as its objects is called the Fukaya category.<ref>Aspinwal et al. 2009, p. 575</ref>
The derived category of coherent sheaves is constructed using tools from complex geometry, a branch of mathematics that describes geometric shapes in algebraic terms and solves geometric problems using algebraic equations.<ref>Yau and Nadis 2010, pp. 180–1</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>Zaslow 2008, p. 531</ref>
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.<ref>Aspinwall et al. 2009, p. 616</ref> This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.<ref>Yau and Nadis 2010, p. 181</ref>
== See also == * Black brane * Brane cosmology * Dirac membrane * Lagrangian submanifold * M2-brane * M5-brane * NS5-brane {{Subfields of physics}} == Citations == {{Reflist|2}}
== General and cited references == * {{Cite book |editor1-first=Paul |editor1-last=Aspinwall |editor2-first=Tom |editor2-last=Bridgeland |editor3-first=Alastair |editor3-last=Craw |editor4-first=Michael |editor4-last=Douglas |editor5-first=Mark |editor5-last=Gross |editor6-first=Anton |editor6-last=Kapustin |editor7-first=Gregory |editor7-last=Moore |editor8-first=Graeme |editor8-last=Segal |editor9-first=Balázs |editor9-last=Szendröi |editor10-first=P.M.H. |editor10-last=Wilson |title=Dirichlet Branes and Mirror Symmetry |year=2009 |publisher=American Mathematical Society | series = Clay Mathematics Monographs | volume = 4 | isbn=978-0-8218-3848-8}} * {{Cite book |last1=Mac Lane |first1=Saunders |author-link=Saunders Mac Lane |title=Categories for the Working Mathematician |year=1998 |isbn=978-0-387-98403-2}} * {{Cite journal| last = Moore |first = Gregory |author-link= Greg Moore (physicist)| title=What is ... a Brane?| journal=Notices of the AMS| year=2005 | url=https://www.ams.org/notices/200502/what-is.pdf | access-date=June 7, 2018 |page=214| volume=52}} * {{Cite book| first1 = Shing-Tung | last1 = Yau |author-link1=Shing-Tung Yau | first2 = Steve | last2 = Nadis | year = 2010 | title = The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions | publisher = Basic Books | isbn = 978-0-465-02023-2 }} * {{Cite book| last1=Zaslow | first1=Eric |author-link=Eric Zaslow | contribution=Mirror Symmetry | year=2008 | title=The Princeton Companion to Mathematics | editor-last=Gowers | editor-first=Timothy | isbn=978-0-691-11880-2 }} {{String theory topics}} {{Authority control}}
Category:String theory