HOME

TheInfoList



OR:

In
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
to
higher dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
. Branes are dynamical objects which can propagate through
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
according to the rules of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
. They have mass and can have other attributes such as
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
. Mathematically, branes can be represented within
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
, and are studied in pure mathematics for insight into
homological mirror symmetry Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address ...
and
noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
.


''p''-branes

A point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. In addition to point particles and strings, it is possible to consider higher-dimensional branes. A ''p''-dimensional brane is generally called "''p''-brane". The term "''p''-brane" was coined by M. J. Duff ''et al.'' in 1988; "brane" comes from the word "membrane" which refers to a two-dimensional brane. A ''p''-brane sweeps out a (''p''+1)-dimensional volume in spacetime called its worldvolume. Physicists often study
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
analogous to the electromagnetic field, which live on the worldvolume of a brane.


D-branes

In
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
, a string may be open (forming a segment with two endpoints) or closed (forming a closed loop).
D-brane In string theory, D-branes, short for ''Dirichlet membrane'', are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Jin Dai, Leigh, and Polch ...
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 the
Dirichlet boundary condition In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
, which the D-brane satisfies. 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 In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter s ...
, a theoretical tool that physicists use to translate difficult problems in gauge theory into more mathematically tractable problems in string theory.


Categorical description

Mathematically, branes can be described using the notion of a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. This is a mathematical structure consisting of ''objects'', and for any pair of objects, a set of ''
morphisms In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
'' between them. In most examples, the objects are mathematical structures (such as sets,
vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, or
topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
) and the morphisms are functions between these structures. One can likewise consider categories where the objects are D-branes and the morphisms between two branes \alpha and \beta are states of open strings stretched between \alpha and \beta. In one version of string theory known as the
topological B-model In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological qu ...
, 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 Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aqu ...
at the endpoints of strings. 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. In mathematical language, the category having these branes as its objects is known as the
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
on the Calabi–Yau. In another version of string theory called the
topological A-model In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological qu ...
, the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold. Roughly speaking, they are what mathematicians call special Lagrangian submanifolds. 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. The category having these branes as its objects is called the Fukaya category. The derived category of coherent sheaves is constructed using tools from
complex geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, a branch of mathematics that describes geometric curves in algebraic terms and solves geometric problems using
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s. 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 In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...
, a mathematical tool that can be used to compute
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
in two-dimensional examples. The
homological mirror symmetry Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address ...
conjecture of
Maxim Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
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. This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.Yau and Nadis 2010, p. 181


See also

*
Black brane In general relativity, a black brane is a solution of the equations that generalizes a black hole solution but it is also extended—and translationally symmetric—in ''p'' additional spatial dimensions. That type of solution would be called a bl ...
*
Brane cosmology Brane cosmology refers to several theories in particle physics and cosmology related to string theory, superstring theory and M-theory. Brane and bulk The central idea is that the visible, three-dimensional universe is restricted to a brane i ...
* Dirac membrane *
Lagrangian submanifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
*
M2-brane In theoretical physics, an M2-brane, is a spatially extended mathematical object (brane) that appears in string theory and in related theories (e.g. M-theory, F-theory). In particular, it is a solution of eleven-dimensional supergravity which pos ...
*
M5-brane In theoretical physics, an M5-brane is a brane which carries magnetic charge, and the dual under electric-magnetic duality is the M2-brane. M5-brane is analogous to the NS5-brane in string theory. In addition, it is a soliton In mathemati ...
*
NS5-brane In theoretical physics, the NS5-brane is a five-dimensional p-brane that carries a magnetic charge under the B-field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, a ...


Citations


General references

* * * * * {{Cite book, last1=Zaslow , first1=Eric , contribution=Mirror Symmetry , year=2008 , title=
The Princeton Companion to Mathematics ''The Princeton Companion to Mathematics'' is a book providing an extensive overview of mathematics that was published in 2008 by Princeton University Press. Edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, it ...
, editor-last=Gowers , editor-first=Timothy , isbn=978-0-691-11880-2 String theory