Brane

<h2 id="p-branes">p-branes</h2>
A point particle is a 0-brane, of dimension zero; a string, named after vibrating <a href="/facts/String_(music)/ITsEwger">musical strings</a>, is a 1-brane; a membrane, named after <a href="/facts/Acoustic_membrane/JXh4XtzK">vibrating membranes</a> such as <a href="/facts/Drumhead/jfoEbSsM">drumheads</a>, is a 2-brane.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> The corresponding object of arbitrary dimension p is called a p-brane, a term coined by <a href="/facts/Michael_Duff_(physicist)/bmbXLn02">M. J. Duff</a> et al. in 1988.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
A p-brane sweeps out a (p+1)-dimensional volume in spacetime called its worldvolume. Physicists often study <a href="/facts/Field_(physics)/FJdAKeRA">fields</a> analogous to the <a href="/facts/Electromagnetic_field/MacraS8y">electromagnetic field</a>, which live on the worldvolume of a brane.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="d-branes">D-branes</h2>
Main article: <a href="/facts/D-brane/hX47HDTk">D-brane</a>

In <a href="/facts/String_theory/HUJFIPYH">string theory</a>, a <a href="/facts/String_(physics)/3EtMw6n2">string</a> may be open (forming a segment with two endpoints) or closed (forming a closed loop). <a href="/facts/D-brane/hX47HDTk">D-branes</a> 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 <a href="/facts/Dirichlet_boundary_condition/m9wsgG89">Dirichlet boundary condition</a>, which the D-brane satisfies.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
One crucial point about D-branes is that the dynamics on the D-brane worldvolume is described by a <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a>, a kind of highly symmetric physical theory which is also used to describe the behavior of elementary particles in the <a href="/facts/Standard_model_of_particle_physics/LhVGf9ow">standard model of particle physics</a>. This connection has led to important insights into gauge theory and <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>. For example, it led to the discovery of the <a href="/facts/AdS/CFT_correspondence/8Ide9C9m">AdS/CFT correspondence</a>, a theoretical tool that physicists use to translate difficult problems in gauge theory into more mathematically tractable problems in string theory.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="categorical-description">Categorical description</h2>
Mathematically, branes can be described using the notion of a <a href="/facts/Category_(mathematics)/7WzxUThf">category</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> This is a mathematical structure consisting of objects, and for any pair of objects, a set of <a href="/facts/Morphisms/Kdpuyz3S">morphisms</a> between them. In most examples, the objects are mathematical structures (such as <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a>, <a href="/facts/Vector_spaces/M1pxMLx2">vector spaces</a>, or <a href="/facts/Topological_spaces/v2ut7CbK">topological spaces</a>) and the morphisms are <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> between these structures.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> One can likewise consider categories where the objects are D-branes and the morphisms between two branes 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 are <a href="/facts/Wavefunction/KgM5ykjY">states</a> of open strings stretched between 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

In one version of string theory known as the <a href="/facts/Topological_string_theory/SAKGh0nV">topological B-model</a>, the D-branes are <a href="/facts/Complex_manifold/DlvPnpzm">complex submanifolds</a> of certain six-dimensional shapes called <a href="/facts/Calabi%E2%80%93Yau_manifolds/LUyEX4kT">Calabi–Yau manifolds</a>, together with additional data that arise physically from having <a href="/facts/Charge_(physics)/jnQEDBI3">charges</a> at the endpoints of strings.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 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.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> In mathematical language, the category having these branes as its objects is known as the <a href="/facts/Derived_category/pBQHnFYo">derived category</a> of <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaves</a> on the Calabi–Yau.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> In another version of string theory called the <a href="/facts/Topological_string_theory/SAKGh0nV">topological A-model</a>, the D-branes can again be viewed as submanifolds of a Calabi–Yau manifold. Roughly speaking, they are what mathematicians call <a href="/facts/Special_Lagrangian_submanifold/JvNrJkC7">special Lagrangian submanifolds</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> 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.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> The category having these branes as its objects is called the <a href="/facts/Fukaya_category/YWXy8Fux">Fukaya category</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
The derived category of coherent sheaves is constructed using tools from <a href="/facts/Complex_geometry/ekCsxAXb">complex geometry</a>, a branch of mathematics that describes geometric shapes in <a href="/facts/Algebra/nMdqczjo">algebraic terms</a> and solves geometric problems using <a href="/facts/Algebraic_equation/1QauIGxs">algebraic equations</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> On the other hand, the Fukaya category is constructed using <a href="/facts/Symplectic_geometry/runXFCea">symplectic geometry</a>, a branch of mathematics that arose from studies of <a href="/facts/Classical_physics/gGnINVPE">classical physics</a>. Symplectic geometry studies spaces equipped with a <a href="/facts/Symplectic_form/o95tWKBc">symplectic form</a>, a mathematical tool that can be used to compute <a href="/facts/Area/KsNG1G9t">area</a> in two-dimensional examples.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
The <a href="/facts/Homological_mirror_symmetry/Wbv35FAn">homological mirror symmetry</a> conjecture of <a href="/facts/Maxim_Kontsevich/7No1RaTQ">Maxim Kontsevich</a> 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.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> This equivalence provides an unexpected bridge between two branches of geometry, namely complex and symplectic geometry.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Black_brane/0lIN7qBx">Black brane</a></li>
<li><a href="/facts/Brane_cosmology/je3aBCDf">Brane cosmology</a></li>
<li><a href="/facts/Dirac_membrane/BLdQmnmU">Dirac membrane</a></li>
<li><a href="/facts/Lagrangian_submanifold/JvNrJkC7">Lagrangian submanifold</a></li>
<li><a href="/facts/M2-brane/pIS57ksc">M2-brane</a></li>
<li><a href="/facts/M5-brane/59sdkeQM">M5-brane</a></li>
<li><a href="/facts/NS5-brane/yDJrL6Ey">NS5-brane</a></li></ul>
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<h2 id="citations">Citations</h2>

<h2 id="general-and-cited-references">General and cited references</h2>
<ul><li>Aspinwall, Paul; Bridgeland, Tom; Craw, Alastair; Douglas, Michael; Gross, Mark; Kapustin, Anton; Moore, Gregory; Segal, Graeme; Szendröi, Balázs; Wilson, P.M.H., eds. (2009). Dirichlet Branes and Mirror Symmetry. <a href="/facts/Clay_Mathematics_Monographs/7fdsuMwQ">Clay Mathematics Monographs </a>. Vol. 4. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3848-8.</li>
<li>Mac Lane, Saunders (1998). Categories for the Working Mathematician. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98403-2.</li>
<li><a href="/facts/Greg_Moore_(physicist)/7WCVjHHt">Moore, Gregory</a> (2005). <a href="https://www.ams.org/notices/200502/what-is.pdf">"What is ... a Brane?"</a> (PDF). Notices of the AMS. 52: 214. Retrieved June 7, 2018.</li>
<li>Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. <a href="/facts/Basic_Books/mfV01W4C">Basic Books</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-465-02023-2.</li>
<li>Zaslow, Eric (2008). "Mirror Symmetry". In Gowers, Timothy (ed.). <a href="/facts/The_Princeton_Companion_to_Mathematics/MzrNYmAz">The Princeton Companion to Mathematics</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-11880-2.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"brane". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.) <a href="https://www.oed.com/search/dictionary/?q=brane" target="_blank">https://www.oed.com/search/dictionary/?q=brane</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Moore 2005, p. 214 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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. <a href="/wiki/Michael_Duff_(physicist)" target="_blank">/wiki/Michael_Duff_(physicist)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Moore 2005, p. 214 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Moore 2005, p. 215 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Moore 2005, p. 215 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Aspinwall et al. 2009 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">A basic reference on category theory is Mac Lane 1998. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Zaslow 2008, p. 536 <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Zaslow 2008, p. 536 <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Yau and Nadis 2010, p. 165 <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Aspinwal et al. 2009, p. 575 <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Aspinwal et al. 2009, p. 575 <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Yau and Nadis 2010, p. 175 <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Aspinwal et al. 2009, p. 575 <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Yau and Nadis 2010, pp. 180–1 <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Zaslow 2008, p. 531 <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Aspinwall et al. 2009, p. 616 <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Yau and Nadis 2010, p. 181 <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
</ol>

Brane open-in-new

Brane