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In theoretical physics, D-branes are a special class of p-branes, named for the physicist Johann Dirichlet. Dirichlet boundary conditions have long been used in the study of 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 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.


D-branes in string theory

Theoretical background

Most versions of string theory involve two, closely related, types of string: open strings with unjoined endpoints and 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.

All elementary particles 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 of particles which the string vibrations allow, we find a type known as a tachyon, which has some odd properties, like imaginary mass. D-branes can be thought of as large collections of 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.) The conjectures of Ashoke Sen are central to research in this field; see tachyon condensation.

Braneworld cosmology

This has implications for cosmology. Because string theory implies that the Universe has more dimensions than we expect—26 for bosonic string theories and 10 for 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 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.)

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 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 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. (Technically speaking, the D-brane determines whether a particular string coordinate obeys Neumann or Dirichlet boundary conditions: directions parallel to the brane are Neumann, while directions perpendicular to it are Dirichlet.) 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 Dp-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 Dp-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 Dp-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 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 Dp-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 holes. D-brane theory allows one to study the quantum states of black holes.

See also


  • Bachas, C. P. "Lectures on D-branes" (1998). arXiv:hep-th/9806199 (http://arxiv.org/abs/hep-th/9806199).
  • Giveon, A. and Kutasov, D. "Brane dynamics and gauge theory," Rev. Mod. Phys. 71, 983 (1999). arXiv:hep-th/9802067.
  • Johnson, C. V. D-branes. Cambridge University Press (2003).
  • 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.de:D-branes

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