Fontem paginae D-brana inspicere

'''D-brana,''' in [[theoria chordarum]] [[physica theoretica|physicae theoreticae]], est specialis ''p''-branarum classis, honore [[Iohannes Petrus Gustavus Lejeune Dirichlet|Iohannis Petri Gustavi Lejeune Dirichlet]] ([[1805]]–[[1859]]) appellata. <!--

[[Dirichlet limitum conditio|Dirichlet limitum conditiones]] 
have long been used in the study of [[Fluid dynamics|fluids]] and [[potential]] theory, where they involve specifying some quantity all along a boundary. In fluid dynamics, fixing a Dirichlet boundary condition could mean assigning a known fluid velocity to all points on a surface; when studying [[electrostatics]], one may establish Dirichlet boundary conditions by fixing the [[voltage]] to known values at particular locations, like the surfaces of conductors. In either case, the locations at which values are specified is called a D-brane. These constructions take on special importance in [[string theory]], because [[String (physics)|open strings]] must have their endpoints attached to D-branes.

D-branes are typically classified by their [[dimension]], which is indicated by a number written after the ''D.''  A D0-brane is a single point, a D1-brane is a line (sometimes called a "D-string"), a D2-brane is a plane, and a D25-brane fills the highest-dimensional space considered in [[bosonic string theory]].

==Theoretical background==
Most versions of string theory involve two, closely related, types of string: [[String (physics)|''open'' strings]] with unjoined endpoints and [[String (physics)|''closed'' strings]] which connect upon themselves to form closed loops.  Exploring the consequences of the [[Nambu-Goto action]], it becomes clear that energy can flow along a string, slipping off the endpoint and vanishing.  This poses a problem: [[conservation of energy]] dictates that energy should not be able to disappear from the system.  Therefore, a consistent string theory must include places into which energy can flow once it leaves a string; these objects are termed D-branes.  Any version of string theory which allows open strings must necessarily incorporate D-branes, and all open strings must have their endpoints attached to these branes.  To a string theorist, D-branes are ''physical objects,'' just as "real" as strings themselves—not just mathematical surfaces where a value is specified.

Because the endpoints of open strings cannot detach from the D-branes to which they are affixed, the D-branes determine the [[boundary condition]]s for the string's equation of motion.  Each direction of motion is affected independently:  the endpoint is free to move along axes parallel to the brane, but the endpoint's coordinates are "locked" to the brane's position for all dimensions perpendicular to the brane.  Technically speaking, the D-brane determines whether a particular string coordinate obeys [[Neumann boundary condition|Neumann]] or [[Dirichlet boundary condition]]s:  directions parallel to the brane are Neumann, while directions perpendicular to it are Dirichlet.

All [[elementary particle]]s are expected to be vibrational states of quantum strings, and it is natural to ask if D-branes are somehow "made of" strings themselves.  In a sense, this turns out to be true: among the [[spectrum (disambiguation)|spectrum]] of particles which the string vibrations allow, we find a type known as a [[tachyon]], which has some odd properties, like [[imaginary number|imaginary]] mass.  Consider a situation where we have a "space-filling" D-brane, ''i.e.,'' one of infinite extent and the same dimensionality as the universe. (In [[bosonic string theory]], this requires a D25-brane.) The strings attached to this brane lead to a tachyon field "living" on the brane's volume.  Other D-branes, of lower dimensionality, can exist within the space-filling brane's volume — say, D1-branes ("D-strings") or D2-branes.  These lower-dimensional branes can be thought of as large collections of tachyons, coherent in a way reminiscent of the [[photon]]s in a laser beam. Many studies in string theory ignore this viewpoint, for simplicity treating the D-brane as a single object. (In [[thermodynamics]], classroom discussions frequently involve a gas of atoms interacting with a large object, like a piston in a cylinder. Of course, physicists believe that the piston is also made of atoms, but for many problems, it is not necessary to consider all the extra complexity, and they model it as a single, macroscopic object.  The case of D-branes is analogous.)

[[Tachyon condensation]] is a central concept in this field. [[Ashoke Sen]] has argued that in [[Type II string theory|Type IIB string theory]], tachyon condensation allows (in the absence of Neveu-Schwarz 3-[[differential form|form]] flux) an arbitrary D-brane configuration to be obtained from a stack of D9 and anti D9-branes. [[Edward Witten]] has shown that such configurations will be classified by the [[K-theory (physics)|K-theory]] of the [[spacetime]].

==Braneworld cosmology==
This has implications for [[physical cosmology]]. Because string theory implies that the Universe has more dimensions than we expect—26 for [[bosonic string theory|bosonic string theories]] and 10 for [[superstring theory|superstring theories]]—we have to find a reason why the extra dimensions are not apparent. One possibility would be that the visible Universe is in fact a very large D-brane extending over three spatial dimensions. Material objects, made of open strings, are bound to the D-brane, and cannot move "at right angles to reality" to explore the Universe outside the brane. This scenario is called a [[brane cosmology]].  Interestingly, the force of [[gravity]] is ''not'' due to open strings; the [[graviton]]s which carry gravitational forces are vibrational states of ''closed'' strings.  Because closed strings do not have to be attached to D-branes, gravitational effects could depend upon the extra dimensions at right angles to the brane.  (This is a fairly simple braneworld model.  More recent innovations under close study [[2005|as of 2005]] are more intricate, but this discussion reflects some of their spirit.)

==Gauge theories==
The arrangement of D-branes constricts the types of string states which can exist in a system. For example, if we have two parallel D2-branes, we can easily imagine strings stretching from brane 1 to brane 2 or vice versa. (In most theories, strings are ''oriented'' objects:  each one carries an "arrow" defining a direction along its length.) The open strings permissible in this situation then fall into two categories, or "sectors":  those originating on brane 1 and terminating on brane 2, and those originating on brane 2 and terminating on brane 1. Symbolically, we say we have the <math>[1\ 2]</math> and the <math>[2\ 1]</math> sectors.  In addition, a string may begin and end on the same brane, giving <math>[1\ 1]</math> and <math>[2\ 2]</math> sectors.  (The numbers inside the brackets are called ''[[Chan-Paton factor|Chan-Paton indices]],''  but they are really just labels identifying the branes.)  A string in either the <math>[1\ 2]</math> or the <math>[2\ 1]</math> sector has a minimum length: it cannot be shorter than the separation between the branes.  All strings have some tension, against which one must pull to lengthen the object; this pull does work on the string, adding to its energy.  Because string theories are by nature [[special relativity|relativistic]], adding energy to a string is equivalent to adding mass, by Einstein's relation <math>E = mc^2</math>.  Therefore, the separation between D-branes controls the minimum mass open strings may have.

Furthermore, affixing a string's endpoint to a brane influences the way the string can move and vibrate.  Because particle states "emerge" from the string theory as the different vibrational states the string can experience, the arrangement of D-branes controls the types of particles present in the theory. The simplest case is the <math>[1\ 1]</math> sector for a D''p''-brane, that is to say the strings which begin and end on any particular D-brane of ''p'' dimensions.  Examining the consequences of the [[Nambu-Goto action]] (and applying the rules of [[quantum mechanics]] to [[quantization (physics)|quantize]] the string), one finds that among the spectrum of particles is one resembling the [[photon]], the fundamental quantum of the electromagnetic field. The resemblance is precise: a ''p''-dimensional version of the electromagnetic field, obeying a ''p''-dimensional analogue of [[Maxwell's equations]], "lives" on every D''p''-brane.

In this sense, then, one can say that string theory "predicts" electromagnetism: D-branes are a necessary part of the theory if we permit open strings to exist, and all D-branes carry an electromagnetic field on their volume.

Other particle states originate from strings beginning and ending on the same D-brane. Some correspond to massless particles like the photon; also in this group are a set of massless scalar particles. If a D''p''-brane is embedded in a spacetime of ''d'' spatial dimensions, the brane carries (in addition to its Maxwell field) a set of ''d - p'' massless [[Scalar field theory|scalars]] (particles which do not have polarizations like the photons making up light). Intriguingly, there are just as many massless scalars as there are directions perpendicular to the brane; the ''geometry'' of the brane arrangement is closely related to the ''[[quantum field theory]]'' of the particles "living" on it. In fact, these massless scalars are [[Nambu-Goldstone boson|Goldstone excitations]] of the brane, corresponding to the different ways the symmetry of empty space can be broken. Placing a D-brane in a universe breaks the symmetry among locations, because it defines a particular place, assigning a special meaning to a particular location along each of the ''d - p'' directions perpendicular to the brane.

The quantum version of Maxwell's electromagnetism is only one kind of [[gauge theory]], a ''[[unitary group|U(1)]]'' gauge theory where the gauge [[group (mathematics)|group]] is made of unitary [[matrix (mathematics)|matrices]] of order 1.  D-branes can be used to generate gauge theories of higher order, in the following way:

Consider a group of ''N'' separate D''p''-branes, arranged in parallel for simplicity.  The branes are labeled 1,2,...,''N'' for convenience. Open strings in this system exist in one of many sectors: the strings beginning and ending on some brane ''i'' give that brane a Maxwell field and some massless scalar fields on its volume. The strings stretching from brane ''i'' to another brane ''j'' have more intriguing properties. For starters, it is worthwhile to ask which sectors of strings can interact with one another.  One straightforward mechanism for a string interaction is for two strings to join endpoints (or, conversely, for one string to "split down the middle" and make two "daughter" strings). Since endpoints are restricted to lie on D-branes, it is evident that a <math>[1\ 2]</math> string may interact with a <math>[2\ 3]</math> string, but not with a <math>[3\ 4]</math> or a <math>[4\ 17]</math> one.  The masses of these strings will be influenced by the separation between the branes, as discussed above, so for simplicity's sake we can imagine the branes squeezed closer and closer together, until they lie atop one another.  If we regard two overlapping branes as distinct objects, then we still have all the sectors we had before, but without the effects due to the brane separations.

The zero-mass states in the open-string particle spectrum for a system of ''N'' coincident D-branes yields a set of interacting quantum fields which is exactly a ''U(N)'' gauge theory. (The string theory does contain other interactions, but they are only detectable at very high energies.) Gauge theories were not invented starting with bosonic or fermionic strings; they originated from a different area of physics, and have become quite useful in their own right.  If nothing else, the relation between D-brane geometry and gauge theory offers a useful pedagogical tool for explaining gauge interactions, even if string theory fails to be the "theory of everything".

==Black holes==
Another important use of D-branes has been in the study of [[black hole]]s. Since the [[1970s]], scientists have debated the problem of black holes having [[entropy]]. Consider, as a [[thought experiment]], dropping an amount of hot gas into a black hole. Since the gas cannot escape from the hole's gravitational pull, its entropy would seem to have vanished from the universe. To maintain the [[second law of thermodynamics]], one must postulate that the black hole gained whatever entropy the infalling gas originally had.  Attempting to apply [[quantum mechanics]] to the study of black holes, [[Stephen Hawking]] discovered that a hole should emit energy with the characteristic spectrum of thermal radiation. The characteristic temperature of this [[Hawking radiation]] is given by
:<math>T_{\rm H} = \frac{\hbar c^3}{8\pi GM k_B},</math>
where ''G'' is [[Isaac Newton|Newton]]'s [[gravitational constant]], ''M'' is the black hole's mass and ''k<sub>B</sub>'' is [[Boltzmann constant|Boltzmann's constant]].

Using this expression for the Hawking temperature, and assuming that a zero-mass black hole has zero entropy, one can use thermodynamic arguments to derive the "[[Jacob Bekenstein|Bekenstein]] entropy":
:<math>S_{\rm B} = \frac{k_B 4\pi G}{\hbar c} M^2.</math>
The Bekenstein entropy is proportional to the black hole mass squared; because the [[Schwarzschild radius]] is proportional to the mass, the Bekenstein entropy is proportional to the black hole's ''surface area.'' In fact,
:<math>S_{\rm B} = \frac{A k_B}{4 l_{\rm P}^2},</math>
where <math>l_{\rm P}</math> is the [[Planck length]].

The concept of black hole entropy poses some interesting conundra. In an ordinary situation, a system has entropy when a large number of different "microstates" can satisfy the same macroscopic condition. For example, given a box full of gas, many different arrangements of the gas atoms can have the same total energy. However, a black hole was believed to be a featureless object (in [[John Wheeler]]'s catchphrase, "[[No hair theorem|Black holes have no hair]]").  What, then, are the "degrees of freedom" which can give rise to black hole entropy?

String theorists have constructed models in which a black hole is a very long (and hence very massive) string. This model gives rough agreement with the expected entropy of a Schwarzschild black hole, but an exact proof has yet to be found one way or the other. The chief difficulty is that it is relatively easy to count the degrees of freedom quantum strings possess ''if they do not interact with one another.'' This is analogous to the [[ideal gas]] studied in introductory thermodynamics:  the easiest situation to model is when the gas atoms do not have interactions among themselves. Developing the [[kinetic theory]] of gases in the case where the gas atoms or molecules experience inter-particle forces (like the [[Van der Waals force|van der Waals]] force) is more difficult.  However, a world without interactions is an uninteresting place: most significantly for the black hole problem, gravity is an interaction, and so if the "string coupling" is turned off, no black hole could ever arise. Therefore, calculating black hole entropy requires working in a regime where string interactions exist.

Extending the simpler case of non-interacting strings to the regime where a black hole could exist requires [[supersymmetry]]. In certain cases, the entropy calculation done for zero string coupling remains valid when the strings interact. The challenge for a string theorist is to devise a situation in which a black hole can exist which does not "break" supersymmetry. In recent years, this has been done by building black holes out of D-branes.  Calculating the entropies of these hypothetical holes gives results which agree with the expected Bekenstein entropy. Unfortunately, the cases studied so far all involve higher-dimensional spaces — D5-branes in nine-dimensional space, for example.  They do not directly apply to the familiar case, the Schwarzschild black holes observed in our own universe.

==History==

Dirichlet boundary conditions and D-branes had a long `pre-history' before their full significance was recognized.  Mixed Dirichlet/Neumann boundary conditions were first considered by Warren Siegel in 1976 as a means of lowering the critical dimension of open string theory from 26 or 10 to 4 (Siegel also cites unpublished work by Halpern, and a 1974 paper by Chodos and Thorn, but a reading of the latter paper shows that it is actually concerned with linear dilation backgrounds, not Dirichlet boundary conditions).  This paper, though prescient, was little-noted in its time (a 1985 parody by Siegel, `The Super-g String,' contains an almost dead-on description of braneworlds).  Dirichlet conditions for all coordinates including Euclidean time (defining what are now known as D-instantons) were introduced by Michael Green in 1977 as a means of introducing point-like structure into string theory, in an attempt to construct a string theory of the strong interaction.  String compactifications studied by Harvey and Minahan, Ishibashi and Onogi, and Pradisi and Sagnotti in 1987-89 also employed Dirichlet boundary conditions.  

The fact that T-duality interchanges the usual Neumann boundary conditions with Dirichlet boundary conditions was discovered independently by Horava and by Dai, Leigh, and Polchinski in 1989; this result implies that such boundary conditions must necessarily appear in regions of the moduli space of any open string theory. The Dai et al. paper also notes that the locus of the Dirichlet boundary conditions is dynamical, and coins the term Dirichlet-brane (D-brane) for the resulting object (this paper also coins [[orientifold]] for another object that arises under string T-duality). A 1989 paper by Leigh showed that D-brane dynamics are governed by the Dirac-Born-Infeld action. D-instantons were extensively studied by Green in the early 1990's, and were shown by Polchinski in 1994 to produce the e^{-1/g} nonperturbative string effects anticipated by Shenker.  In 1995 Polchinski showed that D-branes are the sources of electric and magnetic Ramond-Ramond fields that are required by string duality, leading to rapid progress in the nonperturbative understanding of string theory.-->

== Vide ==
*[[Bogomol'nyi-Prasad-Sommerfield Limes]]
*[[Brana]]
*[[Brana atra]]
*[[Cyclicalis Universi Theoria]]
*[[Cosmologia branarum]]
*[[M-theoria]]
*[[S-brana]]
*[[Theoria chordarum]]

== Fontes ==
* Bachas, C. P. 1998. "Lectures on D-branes."
* Giveon, A., et D. Kutasov. 1999. "Brane Dynamics and Gauge Theory." ''Rev. Mod. Phys.'' 71:983).
* Johnson, Clifford. 2003. ''D-branes.'' Cantabrigiae: Cambridge University Press. ISBN 0-521-80912-6.
* Polchinski, Joseph. 1996. ''TASI Lectures on D-branes,'' Lectures given at Theoretical Advanced Study Institute.
* Polchinski, Joseph. 1995. [Commentarium quod significationem D-branarum in theoria chordarum primum explicavit.] ''Phys. Rev. Lett.'' 75:4724.
* Zwiebach, Barton. 2004. ''A First Course in String Theory.'' Cantabrigiae: Cambridge University Press. ISBN 0-521-83143-1.

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[[Categoria:Theoria chordarum]]