brane

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.Moore 2005, p. 214 The corresponding object of arbitrary dimension p is called a p-brane, a term coined by M. J. Duff et al. in 1988.M. J. Duff, T. Inami, C. N. Pope, {{Interlanguage link multi|Ergin Sezgin|lt=E. Sezgin|de}}, and K. S. Stelle, "Semiclassical quantization of the supermembrane", Nucl. Phys. B297 (1988), 515.

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.Moore 2005, p. 214

{{Main|D-brane}}

File:D3-brane et D2-brane.PNGs]]

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.Moore 2005, p. 215

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.Moore 2005, p. 215

Mathematically, branes can be described using the notion of a category.Aspinwall et al. 2009 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.A basic reference on category theory is Mac Lane 1998. One can likewise consider categories where the objects are D-branes and the morphisms between two branes $\backslash alpha$ and $\backslash beta$ are states of open strings stretched between $\backslash alpha$ and $\backslash beta$.Zaslow 2008, p. 536

Image:Calabi yau.jpg ]]

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.Zaslow 2008, p. 536 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.Yau and Nadis 2010, p. 165 In mathematical language, the category having these branes as its objects is known as the derived category of coherent sheaves on the Calabi–Yau.Aspinwal et al. 2009, p. 575 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.Aspinwal et al. 2009, p. 575 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.Yau and Nadis 2010, p. 175 The category having these branes as its objects is called the Fukaya category.Aspinwal et al. 2009, p. 575

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.Yau and Nadis 2010, pp. 180–1 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.Zaslow 2008, p. 531

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.Aspinwall et al. 2009, p. 616 This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.Yau and Nadis 2010, p. 181

• Black brane
• Brane cosmology
• Dirac membrane
• Lagrangian submanifold
• M2-brane
• M5-brane
• NS5-brane

{{Subfields of physics}}

{{Reflist|2}}

• {{Cite book |editor1-first=Paul |editor1-last=Aspinwall |editor2-first=Tom |editor2-last=Bridgeland |editor3-first=Alastair |editor3-last=Craw |editor4-first=Michael |editor4-last=Douglas |editor5-first=Mark |editor5-last=Gross |editor6-first=Anton |editor6-last=Kapustin |editor7-first=Gregory |editor7-last=Moore |editor8-first=Graeme |editor8-last=Segal |editor9-first=Balázs |editor9-last=Szendröi |editor10-first=P.M.H. |editor10-last=Wilson |title=Dirichlet Branes and Mirror Symmetry |year=2009 |publisher=American Mathematical Society | series = Clay Mathematics Monographs | volume = 4 | isbn=978-0-8218-3848-8}}
• {{Cite book |last1=Mac Lane |first1=Saunders |title=Categories for the Working Mathematician |year=1998 |isbn=978-0-387-98403-2}}
• {{Cite journal| last = Moore |first = Gregory |author-link= Greg Moore (physicist)| title=What is ... a Brane?| journal=Notices of the AMS| year=2005 | url=https://www.ams.org/notices/200502/what-is.pdf | access-date=June 7, 2018 |page=214| volume=52}}
• {{Cite book| first1 = Shing-Tung | last1 = Yau | first2 = Steve | last2 = Nadis | year = 2010 | title = The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions | publisher = Basic Books | isbn = 978-0-465-02023-2 }}
• {{Cite book| last1=Zaslow | first1=Eric | contribution=Mirror Symmetry | year=2008 | title=The Princeton Companion to Mathematics | editor-last=Gowers | editor-first=Timothy | isbn=978-0-691-11880-2 }}

{{String theory topics}}

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Category:String theory