Instring theory and related theories (such as supergravity theories), abrane is 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 are dynamical objects which can propagate throughspacetime according to the rules ofquantum mechanics. They havemass and can have other attributes such ascharge.
Mathematically, branes can be represented withincategories, and are studied inpure mathematics for insight intohomological mirror symmetry andnoncommutative geometry.
The word "brane" originated in 1987 as a contraction of "membrane".[1]
A point particle is a 0-brane, of dimension zero; a string, named after vibratingmusical strings, is a 1-brane; a membrane, named aftervibrating membranes such asdrumheads, is a 2-brane.[2] The corresponding object of arbitrary dimensionp is 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 studyfields analogous to theelectromagnetic field, which live on the worldvolume of a brane.[4]
Instring theory, astring 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 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 into gauge theory andquantum 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]
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 ofmorphisms between them. In most examples, the objects are mathematical structures (such assets,vector spaces, ortopological spaces) and the morphisms arefunctions between these structures.[8] One can likewise consider categories where the objects are D-branes and the morphisms between two branes and arestates of open strings stretched between and.[9]
In one version of string theory known as thetopological B-model, the D-branes arecomplex submanifolds of certain six-dimensional shapes calledCalabi–Yau manifolds, together with additional data that arise physically from havingcharges at 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 category ofcoherent sheaves on 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 terms and 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 computearea in two-dimensional examples.[17]
Thehomological mirror symmetry conjecture ofMaxim 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.[18] This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.[19]
