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.[1] The corresponding object of arbitrary dimension p is called a p-brane, a term coined by M. J. Duffet al. in 1988.[2]
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.[3]
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.[4]
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.[5]
Categorical descriptionedit
Mathematically, branes can be described using the notion of a category.[6] 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.[7] 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 .[8]
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.[9] 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.[10] In mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi–Yau.[11] 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.[12] 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.[13] The category having these branes as its objects is called the Fukaya category.[14]
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.[15] 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.[16]
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.[17] This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.[18]
Mac Lane, Saunders (1998). Categories for the Working Mathematician. ISBN978-0-387-98403-2.
Moore, Gregory (2005). "What is ... a Brane?" (PDF). Notices of the AMS. 52: 214. Retrieved June 7, 2018.
Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. ISBN978-0-465-02023-2.
brane, other, uses, disambiguation, organ, brain, string, theory, related, theories, such, supergravity, theories, brane, physical, object, that, generalizes, notion, zero, dimensional, point, particle, dimensional, string, dimensional, membrane, higher, dimen. For other uses see Brane disambiguation For the organ see brain In string theory and related theories such as supergravity theories 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 Contents 1 p branes 2 D branes 3 Categorical description 4 See also 5 Citations 6 General and cited referencesp branes editA 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 1 The corresponding object of arbitrary dimension p is called a p brane a term coined by M J Duff et al in 1988 2 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 3 D branes editMain article D brane nbsp Open strings attached to a pair of D branesIn 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 4 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 5 Categorical description editMathematically branes can be described using the notion of a category 6 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 7 One can likewise consider categories where the objects are D branes and the morphisms between two branes a displaystyle alpha nbsp and b displaystyle beta nbsp are states of open strings stretched between a displaystyle alpha nbsp and b displaystyle beta nbsp 8 nbsp A cross section of a Calabi Yau manifoldIn 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 9 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 10 In mathematical language the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi Yau 11 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 12 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 13 The category having these branes as its objects is called the Fukaya category 14 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 15 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 16 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 17 This equivalence provides an unexpected bridge between two branches of geometry namely complex and symplectic geometry 18 See also editBlack brane Brane cosmology Dirac membrane Lagrangian submanifold M2 brane M5 brane NS5 braneField Subfields Major theories ConceptsNuclear and particle physics Nuclear physics Nuclear astrophysics Particle physics Astroparticle physics Particle physics phenomenology Standard Model Quantum field theory Quantum electrodynamics Quantum chromodynamics Electroweak theory Effective field theory Lattice field theory Gauge theory Supersymmetry Grand Unified Theory Superstring theory M theory AdS CFT correspondence Fundamental interaction gravitational electromagnetic weak strong Elementary particle Spin Antimatter Spontaneous symmetry breaking Neutrino oscillation Seesaw mechanism Brane String Quantum gravity Theory of everything Vacuum energyAtomic molecular and optical physics Atomic physics Molecular physics Atomic and molecular astrophysics Chemical physics Optics Photonics Quantum optics Quantum chemistry Quantum information science Photon Atom Molecule Diffraction Electromagnetic radiation Laser Polarization waves Spectral line Casimir effectCondensed matter physics Solid state physics High pressure physics Low temperature physics Surface physics Nanoscale and mesoscopic physics Polymer physics BCS theory Bloch s theorem Density functional theory Fermi gas Fermi liquid theory Many body theory Statistical mechanics Phases gas liquid solid Bose Einstein condensate Electrical conduction Phonon Magnetism Self organization Semiconductor superconductor superfluidity Spin Astrophysics Astronomy Astrometry Cosmology Gravitation physics High energy astrophysics Planetary astrophysics Plasma physics Solar physics Space physics Stellar astrophysics Big Bang Cosmic inflation General relativity Newton s law of universal gravitation Lambda CDM model Magnetohydrodynamics Black hole Cosmic background radiation Cosmic string Cosmos Dark energy Dark matter Galaxy Gravity Gravitational radiation Gravitational singularity Planet Solar System Star Supernova UniverseApplied physics Accelerator physics Acoustics Agrophysics Atmospheric physics Biophysics Chemical physics Communication physics Econophysics Engineering physics Fluid dynamics Geophysics Laser physics Materials physics Medical physics Nanotechnology Optics Optoelectronics Photonics Photovoltaics Physical chemistry Physical oceanography Physics of computation Plasma physics Solid state devices Quantum chemistry Quantum electronics Quantum information science Vehicle dynamicsCitations edit Moore 2005 p 214 M J Duff T Inami C N Pope E Sezgin de and K S Stelle Semiclassical quantization of the supermembrane Nucl Phys B297 1988 515 Moore 2005 p 214 Moore 2005 p 215 Moore 2005 p 215 Aspinwall et al 2009 A basic reference on category theory is Mac Lane 1998 Zaslow 2008 p 536 Zaslow 2008 p 536 Yau and Nadis 2010 p 165 Aspinwal et al 2009 p 575 Aspinwal et al 2009 p 575 Yau and Nadis 2010 p 175 Aspinwal et al 2009 p 575 Yau and Nadis 2010 pp 180 1 Zaslow 2008 p 531 Aspinwall et al 2009 p 616 Yau and Nadis 2010 p 181General and cited references editAspinwall Paul Bridgeland Tom Craw Alastair Douglas Michael Gross Mark Kapustin Anton Moore Gregory Segal Graeme Szendroi Balazs Wilson P M H eds 2009 Dirichlet Branes and Mirror Symmetry Clay Mathematics Monographs Vol 4 American Mathematical Society ISBN 978 0 8218 3848 8 Mac Lane Saunders 1998 Categories for the Working Mathematician ISBN 978 0 387 98403 2 Moore Gregory 2005 What is a Brane PDF Notices of the AMS 52 214 Retrieved June 7 2018 Yau Shing Tung Nadis Steve 2010 The Shape of Inner Space String Theory and the Geometry of the Universe s Hidden Dimensions Basic Books ISBN 978 0 465 02023 2 Zaslow Eric 2008 Mirror Symmetry In Gowers Timothy ed The Princeton Companion to Mathematics ISBN 978 0 691 11880 2 Retrieved from https en wikipedia org w index php title Brane amp oldid 1184337503, wikipedia, wiki, book, books, library,
