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Brane

For other uses, seeBrane (disambiguation).
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Instring theoryand related theories (such assupergravity theories), abraneis a physical object that generalizes the notion of a zero-dimensionalpoint particle,a one-dimensionalstring,or a two-dimensional membrane to higher-dimensional objects. Branes aredynamicalobjects which can propagate throughspacetimeaccording to the rules ofquantum mechanics.They havemassand can have other attributes such ascharge.

Mathematically, branes can be represented withincategories,and are studied inpure mathematicsfor insight intohomological mirror symmetryandnoncommutative geometry.

The word "brane" originated in 1987 as a contraction of "membrane".[1]

p-branes

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A point particle is a 0-brane, of dimension zero; a string, named after vibratingmusical strings,is a 1-brane; a membrane, named aftervibrating membranessuch asdrumheads,is a 2-brane.[2]The corresponding object of arbitrary dimensionpis called ap-brane, a term coined byM. J. Duffet al.in 1988.[3]

Ap-brane sweeps out a (p+1)-dimensional volume in spacetime called itsworldvolume.Physicists often studyfieldsanalogous to theelectromagnetic field,which live on the worldvolume of a brane.[4]

D-branes

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Main article:D-brane
Open strings attached to a pair ofD-branes

Instring theory,astringmay be open (forming a segment with two endpoints) or closed (forming a closed loop).D-branesare 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 theDirichlet boundary condition,which the D-brane satisfies.[5]

One crucial point about D-branes is that the dynamics on the D-brane worldvolume is described by agauge theory,a kind of highly symmetric physical theory which is also used to describe the behavior of elementary particles in thestandard model of particle physics.This connection has led to important insights intogauge theoryandquantum field theory.For example, it led to the discovery of theAdS/CFT correspondence,a theoretical tool that physicists use to translate difficult problems in gauge theory into more mathematically tractable problems in string theory.[6]

Categorical description

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Mathematically, branes can be described using the notion of acategory.[7]This is a mathematical structure consisting ofobjects,and for any pair of objects, a set ofmorphismsbetween them. In most examples, the objects are mathematical structures (such assets,vector spaces,ortopological spaces) and the morphisms arefunctionsbetween these structures.[8]One can likewise consider categories where the objects are D-branes and the morphisms between two branesandarestatesof open strings stretched betweenand.[9]

A cross section of aCalabi–Yau manifold

In one version of string theory known as thetopological B-model,the D-branes arecomplex submanifoldsof certain six-dimensional shapes calledCalabi–Yau manifolds,together with additional data that arise physically from havingchargesat the endpoints of strings.[10]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.[11]In mathematical language, the category having these branes as its objects is known as thederived categoryofcoherent sheaveson the Calabi–Yau.[12]In another version of string theory called thetopological A-model,the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold. Roughly speaking, they are what mathematicians callspecial Lagrangian submanifolds.[13]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.[14]The category having these branes as its objects is called theFukaya category.[15]

The derived category of coherent sheaves is constructed using tools fromcomplex geometry,a branch of mathematics that describes geometric shapes inalgebraic termsand solves geometric problems usingalgebraic equations.[16]On the other hand, the Fukaya category is constructed usingsymplectic geometry,a branch of mathematics that arose from studies ofclassical physics.Symplectic geometry studies spaces equipped with asymplectic form,a mathematical tool that can be used to computeareain two-dimensional examples.[17]

Thehomological mirror symmetryconjecture ofMaxim Kontsevichstates 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.[18]This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.[19]

See also

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Field Subfields Major theories Concepts
Nuclearandparticle 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 energy
Atomic, 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 effect
Condensed 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,Universe
Applied 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 dynamics

Citations

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  1. ^"brane".Oxford English Dictionary(Online ed.).Oxford University Press.(Subscription orparticipating institution membershiprequired.)
  2. ^Moore 2005, p. 214
  3. ^M. J. Duff,T. Inami,C. N. Pope,E. Sezgin[de],andK. S. Stelle,"Semiclassical quantization of the supermembrane",Nucl. Phys.B297(1988), 515.
  4. ^Moore 2005, p. 214
  5. ^Moore 2005, p. 215
  6. ^Moore 2005, p. 215
  7. ^Aspinwall et al. 2009
  8. ^A basic reference on category theory is Mac Lane 1998.
  9. ^Zaslow 2008, p. 536
  10. ^Zaslow 2008, p. 536
  11. ^Yau and Nadis 2010, p. 165
  12. ^Aspinwal et al. 2009, p. 575
  13. ^Aspinwal et al. 2009, p. 575
  14. ^Yau and Nadis 2010, p. 175
  15. ^Aspinwal et al. 2009, p. 575
  16. ^Yau and Nadis 2010, pp. 180–1
  17. ^Zaslow 2008, p. 531
  18. ^Aspinwall et al. 2009, p. 616
  19. ^Yau and Nadis 2010, p. 181

General and cited references

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