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In
string theory In , string theory is a in which the of are replaced by objects called . String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just ...

string theory
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 particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, l ...
to higher dimensions. Branes are
dynamical In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. Examples include the mathematical models that describe the s ...
objects which can propagate through
spacetime In , spacetime is any which fuses the and the one of into a single . can be used to visualize effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three ...
according to the rules of
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
. 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'' (David Ford album) * Charge (Machel Montano album), ''Charge'' (Mac ...
. Mathematically, branes can be represented within
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
, and are studied in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
for insight into
homological mirror symmetry Homological mirror symmetry is a mathematical conjecture In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
and
noncommutative geometryNoncommutative geometry (NCG) is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chang ...
.


''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 File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
analogous to the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the in ...
, which live on the worldvolume of a brane.


D-branes

In
string theory In , string theory is a in which the of are replaced by objects called . String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just ...

string theory
, a
string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * Strings (1991 film), ''Strings'' (1991 fil ...
may be open (forming a segment with two endpoints) or closed (forming a closed loop).
D-brane In string theory In , string theory is a in which the of are replaced by objects called . String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a ...
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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
, 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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
, 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 The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis ...
. This connection has led to important insights into
gauge theory In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. For example, it led to the discovery of the
AdS/CFT correspondence In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in c ...
, 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 Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
. This is a mathematical structure consisting of ''objects'', and for any pair of objects, a set of ''
morphisms In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

morphisms
'' between them. In most examples, the objects are mathematical structures (such as sets,
vector spaces In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, or
topological spaces In mathematics, a topological space is, roughly speaking, a geometry, geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a Set (mathematics), set of ...
) and the morphisms are
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, Un ...
of open strings stretched between \alpha and \beta. 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. 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 categoryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of
coherent sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of Sheaf (mathematics), sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves ...
on the Calabi–Yau. 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. 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 categoryIn symplectic topology, a Fukaya category of a symplectic manifold (M, \omega) is a Category (mathematics), category \mathcal F (M) whose objects are Lagrangian submanifolds of M, and morphisms are Floer homology, Floer chain groups: \mathrm (L_0, L ...
. The derived category of coherent sheaves is constructed using tools from
complex geometry In mathematics, complex geometry is the study of complex manifolds, Complex algebraic variety, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this cate ...
, a branch of mathematics that describes geometric curves in algebraic terms and solves geometric problems using
algebraic equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s. On the other hand, the Fukaya category is constructed using
symplectic geometry Symplectic geometry is a branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study prob ...
, a branch of mathematics that arose from studies of
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
. Symplectic geometry studies spaces equipped with a
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a Field (mathematics), field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping that is ...
, a mathematical tool that can be used to compute
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
in two-dimensional examples. The
homological mirror symmetry Homological mirror symmetry is a mathematical conjecture In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
conjecture of
Maxim Kontsevich Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and France, French mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished pro ...

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. 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 *
Brane cosmology Brane cosmology refers to several theories in particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of na ...
* Dirac membrane *
Eric Weinstein The given name Eric, Erich, Erikk, Erik, Erick, or Eirik is derived from the Old Norse name ''Eiríkr'' (or ''Eríkr'' in Eastern Old Norse due to monophthongization). The first element, ''ei-'' may be derived from the older Proto-Norse langua ...
’s observerse theory (14 dimensions) * M2-brane * M5-brane * NS5-brane


Notes


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, edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, and published in 2008 by Princeton University Press (). It provides an extensive overview of mathematics, and ...
, editor-last=Gowers , editor-first=Timothy , isbn=978-0-691-11880-2 Quantum field theory String theory