D-brane

理論物理入面，**D-branes**，有人叫做D-膜，係一類p-branes；渠個名來自渠跟嘅「Dirichlet 邊界條件」，其中Dirichlet 即係數學家Johann Dirichlet。一直以來Dirichlet 邊界條件係流體力學同埋勢論都有用；喺流體力學，固定一 Dirichlet 邊界條件可以即係畀一表示上嘅每一點一支流速；而研究靜電時，我地可以響特定嘅地方定死咗電壓來設定 Dirichlet 條件，好似傳導體嘅表面。喺是但一例，嗰啲有固定值嘅點就組成塊 D-brane。呢啲嘢響弦論特别重要，因開弦嘅每一尾一定要連住塊 D-branes。

一般我地用維度來分類 D-branes。例如， D0-brane 即係一點；D1-brane 係條線（有啲叫 D-弦 (*D-string*)）；D2-brane 係塊平面；D25-brane 就佔曬玻色弦論中最頂龍維度嘅空間。

## 理論背景
多數弦論模型入面有用兩種關係密切嘅弦：有一頭一尾嘅開弦同埋成個圈咁嘅閂弦(*closed strings*)。睇下南部後滕作用(:en:Nambu-Goto action)，就會發現條弦上能量可以流，重會響頭尾消失。咁就出問題：能量守恆律話，響封閉系統入面，能量唔會無咗。  所以，弦論要行得通，就要包括埋啲能量嘅出路；呢的嘢就叫 D-branes。  任何弦論模型，有親開弦嘅，都一定要有 D-branes，而且全部開弦嘅尾都要黏疑住呢啲 branes。弦論學者認為  D-branes 真係實質嘅「物理嘢」，同啲弦本身一樣咁真— 唔只係抽象嘅曲面。因為開弦嘅尾唔甩得開渠黏住嗰幾塊 D-branes，所以 D-branes 定咗條弦嘅運動方程嘅邊界條件。 每個方向嘅運動都各自各被影響：條弦尾可自由沿住平行塊 D-brane  嘅方向郁，但垂直住塊 D-brane 嘅方向就鎖死咗。嚴格講，一塊  D-brane 決定咗條弦嘅任何座標究竟跟Neumann定跟  Dirichlet 邊界條件行：平行住塊 brane 嘅就行 Neumann，垂直住塊 brane 嘅就行 Dirichlet。

弦論預測，全部基本粒子都係量子弦嘅震態(*vibrational state*)，咁我地自然可以問，啲 D-branes 都係唔係由弦「組成」。 某種意義上，真係嘅：嚮弦震態畀嘅粒子譜入面，有種怪嘢叫 tachyon(例如：渠有虚質量)。假設我哋有舊填滿空間 (*space-filling*) 嘅 D-brane，即係話，佢同宇宙有一樣嘅維度，而且無限大。（嚮玻色弦論，咁就要舊 D25-brane。）黐住呢塊 brane 嘅弦會引發「住喺」塊 brane 入面嘅一幅tachyon 場。第啲低維度嘅 D-branes 可以住嚮舊填滿成個空間嘅 brane 裏面 — 例如， D1-branes (「D-弦」) 或者 D2-branes。呢啲低維度 branes 可以當做一拃拃 tachyons， coherent in a way reminiscent of the photons 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 IIB string theory, tachyon condensation allows (in the absence of Neveu-Schwarz 3-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 of the spacetime.

## Braneworld 宇宙學

D-brane 喺物理宇宙學都有影響。 因為弦論學者預測宇宙嘅維數多過我地認為—玻色弦論要 26維，超弦論就要 10維—咁我地就要揾下點解呢啲凸出來嘅維度咁唔明顯。一種可能係見到嘅宇宙其實係舊好大嘅 D-brane，有三支空間嘅維度。 由開弦組成嘅物質嘢，黐住塊 D-brane，唔可以「垂直住現實 (reality)」咁郁，離唔開塊 brane 去外面嘅宇宙玩。呢種景況就叫brane 宇宙學。  Interestingly, the force of gravity is *not* due to open strings; the gravitons 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 as of 2005 are more intricate, but this discussion reflects some of their spirit.)

## 規範場論

啲 D-branes 嘅擺位，限制住個系統入面可以有邊種弦。例如，如我地有兩塊平行嘅 D2-branes，咁我地就可以想象啲弦由 1 號brane 拉去 2 號 brane，或者相反。（嚮多數模型，啲弦都係有向嘅嘢：每條都帶住支箭嘴，定咗沿弦嘅一個方向。）  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 [1\ 2] and the [2\ 1] sectors.  In addition, a string may begin and end on the same brane, giving [1\ 1] and [2\ 2] 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 [1\ 2] or the [2\ 1] 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 relativistic, adding energy to a string is equivalent to adding mass, by Einstein's relation E = mc^2.  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 [1\ 1] 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 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 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 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 is made of unitary 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 [1\ 2] string may interact with a [2\ 3] string, but not with a [3\ 4] or a [4\ 17] 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".

## 黑洞

D-branes 又可用來研究黑洞。 自從1970年代，物理學人辯論緊黑洞有無熵。 諗下一次理想實驗：攞舊熱氣掟落黑洞度。因為舊熱氣出唔到黑洞嘅引力場，咁渠嘅熵就似乎由宇宙消失咗。為咗維持熱力學第二定律， 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
:T_{\rm H} = \frac{\hbar c^3}{8\pi GM k_B},
where *G* is Newton's gravitational constant, *M* is the black hole's mass and *k<sub>B</sub>* is 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 "Bekenstein entropy":
:S_{\rm B} = \frac{k_B 4\pi G}{\hbar c} M^2.
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,
:S_{\rm B} = \frac{A k_B}{4 l_{\rm P}^2},
where l_{\rm P} 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, "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) 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.

## 史

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.

## 睇埋
* Bogomol'nyi-Prasad-Sommerfield bound
* Brane
* M-theory

## 參考資料
* Bachas, C. P. "Lectures on D-branes" (1998). .
* Giveon, A. and Kutasov, D. "Brane dynamics and gauge theory," *Rev. Mod. Phys.* **71**, 983 (1999). .
* 
* Polchinski, Joseph, *TASI Lectures on D-branes,* . Lectures given at TASI '96.
* Polchinski, Joseph, *Phys. Rev. Lett.* **75**, 4724 (1995). An article which established D-branes' significance in string theory.
* Zwiebach, Barton. *A First Course in String Theory.* Cambridge University Press (2004). ISBN 0-521-83143-1.