Everything about D-branes totally explained
In
string theory,
D-branes are a class of extended objects upon which open
strings can end with
Dirichlet boundary conditions, after which they're named. D-branes were discovered by Dai, Leigh and
Polchinski, and independently by
Horava in 1989. In 1995, Polchinski identified D-branes with
black p-brane solutions of
supergravity, a discovery that triggered the
Second Superstring Revolution and led to both
holographic and
M-theory dualities.
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. There are also
instantonic D(-1)-branes, which are localized in both space and time.
Theoretical background
The equations of motion of string theory require that the endpoints of an open string (a string with endpoints) satisfy one of two types of boundary conditions: The
Neumann boundary condition, corresponding to free endpoints moving through spacetime at the speed of light, or the
Dirichlet boundary conditions, which pin the string endpoint. Each coordinate of the string must satisfy one or the other of these conditions. If p spatial dimensions satisfy the Neumann boundary condition, then the string endpoint is confined to move within a p-dimensional hyperplane. This hyperplane provides one description of a Dp-brane.
Although rigid in the limit of zero coupling, the spectrum of open strings ending on a D-brane contains modes associated with its fluctuations, implying that D-branes are dynamical objects. When
D-branes are nearly coincident, the spectrum of strings stretching between them becomes very rich. One set of modes produce a non-abelian gauge theory on the world-volume. Another set of modes is an
dimensional matrix for each transverse dimension of the brane. If these matrices commute, they may be diagonalized, and the eigenvalues define the position of the
D-branes in space. More generally, the branes are described by non-commutative geometry, which allows exotic behavior such as the Myers effect, in which a collection of Dp-branes expand into a D(p+2)-brane.
All
elementary particles are expected to be vibrational states of quantum strings, and it's 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. Consider a situation where we've a "space-filling" D-brane,
for example, 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
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 recognize that the piston is also made of atoms, but for many problems, it isn't 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 cosmology
This has implications for
physical 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 can't 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 don't 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've 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've the
and the
sectors. In addition, a string may begin and end on the same brane, giving
and
sectors. (The numbers inside the brackets are called
Chan-Paton indices, but they're really just labels identifying the branes.) A string in either the
or the
sector has a minimum length: it can't 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
. 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
sector for a D
p-brane, that's 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 don't 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 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's 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's evident that a
string may interact with a
string, but not with a
or a
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'd 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're 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. 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 can't escape from the hole's gravitational pull, its entropy would seem to have vanished from the universe. In order 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
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