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 interac ...
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
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
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 differen ...
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 chemistry, ...
. 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), ''Categories'' (Aristotle)
*Category (Kant)
...
, and are studied in
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
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 t ...
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 ge ...
.
''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
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
, which live on the worldvolume of a brane.
D-branes
![D3-brane et D2-brane](https://upload.wikimedia.org/wikipedia/commons/8/88/D3-brane_et_D2-brane.PNG)
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 interac ...
, 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 Polchi ...
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
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
, 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 is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It wa ...
. This connection has led to important insights into
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
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, 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 points ...
) 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
and
are
states of open strings stretched between
and
.
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 proce ...
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 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 sympl ...
s. 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
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
, 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 ''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
A shape or figure is a graphics, graphical representation of an obje ...
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 t ...
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 In quantum mechanics, a Dirac membrane is a model of a charged membrane introduced by Paul Dirac in 1962. Dirac's original motivation was to explain the mass of the muon as an excitation of the ground state corresponding to an electron. Anticipatin ...
*
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 ha ...
, editor-last=Gowers , editor-first=Timothy , isbn=978-0-691-11880-2
String theory