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In
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open
string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
s can end with
Dirichlet boundary condition In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed. The question of finding solutions to such equat ...
s, after which they are named. D-branes are typically classified by their spatial
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
, 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 Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the context of string theory, and a new ve ...
. There are also
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
ic D(−1)-branes, which are localized in both
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
and
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
.


Discovery

D-branes were discovered by Jin Dai, Robert Leigh, and
Joseph Polchinski Joseph Gerard Polchinski Jr. (; May 16, 1954 – February 2, 2018) was an American theoretical physicist and string theorist. Biography Polchinski was born in White Plains, New York, the elder of two children to Joseph Gerard Polchinski Sr. (19 ...
, and independently by Petr Hořava, in 1989. In 1995, Polchinski identified D-branes with
black p-brane In general relativity, a black brane is a solution of the Einstein field equations that generalizes a black hole solution but it is also extended—and translationally symmetric—in additional spatial dimensions. That type of solution would be ...
solutions of
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
, a discovery that triggered the
second superstring revolution The history of string theory spans several decades of intense research including two superstring revolutions. Through the combined efforts of many researchers, string theory has developed into a broad and varied subject with connections to quantum ...
and led to both
holographic Holography is a technique that allows a wavefront to be recorded and later reconstructed. It is best known as a method of generating three-dimensional images, and has a wide range of other uses, including data storage, microscopy, and interfe ...
and
M-theory In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1 ...
dualities.


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 In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative app ...
, corresponding to free endpoints moving through spacetime at the speed of light, or the
Dirichlet boundary conditions In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed. The question of finding solutions to such equat ...
, which pin the string endpoint. Each coordinate of the string must satisfy one or the other of these conditions. There can also exist strings with mixed boundary conditions, where the two endpoints satisfy NN, DD, ND and DN boundary 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 D''p''-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 N 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 N \times N 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 N D-branes in space. More generally, the branes are described by non-commutative geometry, which allows exotic behavior such as the
Myers effect Myers as a surname has several possible origins, e.g. Old French ("physician"), Old English ("mayor"), and Old Norse ("marsh"). People * Abram F. Myers (1889–after 1960), chair of the Federal Trade Commission and later general counsel and b ...
, in which a collection of D''p''-branes expand into a D(''p''+2)-brane.
Tachyon condensation Tachyon condensation is a process in particle physics in which a system can lower its potential energy by spontaneously producing particles. The end result is a "condensate" of particles that fills the volume of the system. Tachyon condensation is ...
is a central concept in this field.
Ashoke Sen Ashoke Sen FRS (; born 1956) is an Indian theoretical physicist and distinguished professor at the International Centre for Theoretical Sciences (ICTS), Bangalore. A former distinguished professor at the Harish-Chandra Research Institute, Pra ...
has argued that in
Type IIB string theory In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theorie ...
, tachyon condensation allows (in the absence of Neveu-Schwarz 3-
form Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens. Form may also refer to: *Form (document), a document (printed or electronic) with spaces in which to write or enter dat ...
flux) an arbitrary D-brane configuration to be obtained from a stack of D9 and anti D9-branes.
Edward Witten Edward Witten (born August 26, 1951) is an American theoretical physics, theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the sc ...
has shown that such configurations will be classified by the
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
of the
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. Tachyon condensation is still very poorly understood. This is due to the lack of an exact string field theory that would describe the off-shell evolution of the tachyon.


Braneworld cosmology

This has implications for
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...
. 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 Brane cosmology refers to several theories in particle physics and cosmology related to string theory, superstring theory and M-theory. Brane and bulk The central idea is that the visible, four-dimensional spacetime is restricted to a brane i ...
. The force of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
is ''not'' due to open strings; the
graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...
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 orthogonal to the brane.


D-brane scattering

When two D-branes approach each other the interaction is captured by the one loop annulus amplitude of strings between the two branes. The scenario of two parallel branes approaching each other at a constant velocity can be mapped to the problem of two stationary branes that are rotated relative to each other by some angle. The annulus amplitude yields singularities that correspond to the on-shell production of open strings stretched between the two branes. This is true irrespective of the charge of the D-branes. At non-relativistic scattering velocities the open strings may be described by a low-energy effective action that contains two complex scalar fields that are coupled via a term \phi^2\chi^2. Thus, as the field \phi (separation of the branes) changes, the mass of the field \chi changes. This induces open string production and as a result the two scattering branes will be trapped.


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  2and the  1sectors. In addition, a string may begin and end on the same brane, giving  1and  2sectors. (The numbers inside the brackets are called Chan–Paton indices, but they are really just labels identifying the branes.) A string in either the  2or the  1sector 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 Relativity may refer to: Physics * Galilean relativity, Galileo's conception of relativity * Numerical relativity, a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity ...
, adding energy to a string is equivalent to adding mass, by Einstein's relation ''E'' = ''mc2''. 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  1sector 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 The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate string-like objects (for example, cosmic strings). It is the starting point of the analysis of zero-thickness ...
(and applying the rules of
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
to quantize the string), one finds that among the spectrum of particles is one resembling the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
, 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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
, exists 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 Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
(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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
'' of the particles existing 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 In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, a U(1) gauge theory where the gauge
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
is made of unitary
matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...
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  2string may interact with a  3string, but not with a  4or a  17one. 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 A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s. Since the 1970s, scientists have debated the problem of black holes having
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
. Consider, as a
thought experiment A thought experiment is an imaginary scenario that is meant to elucidate or test an argument or theory. It is often an experiment that would be hard, impossible, or unethical to actually perform. It can also be an abstract hypothetical that is ...
, 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. In order to maintain the
second law of thermodynamics The second law of thermodynamics is a physical law based on Universal (metaphysics), universal empirical observation concerning heat and Energy transformation, energy interconversions. A simple statement of the law is that heat always flows spont ...
, one must postulate that the black hole gained whatever entropy the infalling gas originally had. Attempting to apply
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
to the study of black holes,
Stephen Hawking Stephen William Hawking (8January 194214March 2018) was an English theoretical physics, theoretical physicist, cosmologist, and author who was director of research at the Centre for Theoretical Cosmology at the University of Cambridge. Between ...
discovered that a hole should emit energy with the characteristic spectrum of thermal radiation. The characteristic temperature of this
Hawking radiation Hawking radiation is black-body radiation released outside a black hole's event horizon due to quantum effects according to a model developed by Stephen Hawking in 1974. The radiation was not predicted by previous models which assumed that onc ...
is given by T_ = \frac \;\quad(\approx \; K), where G is the
Newtonian constant of gravitation The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's theory of general relativity. It is also known as t ...
, M is the black hole's mass and k_\text is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
. 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_ = \frac M^2. The Bekenstein entropy is proportional to the black hole mass squared; because the
Schwarzschild radius The Schwarzschild radius is a parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius of a sphere in flat space that has the same surface area as that of the event horizon of a Schwarzschild black ho ...
is proportional to the mass, the Bekenstein entropy is proportional to the black hole's ''surface area.'' In fact, S_ = \frac, where l_ is the
Planck length In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: '' c'', '' G'', '' ħ'', and ''k''B (described further below). Expressing one of ...
. 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 An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
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 The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small ...
in the case where the gas atoms or molecules experience inter-particle forces (like the
van der Waals force In molecular physics and chemistry, the van der Waals force (sometimes van der Waals' force) is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical elec ...
) 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 Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
. 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. A series of 1975–76 papers by Bardeen, Bars, Hanson and Peccei dealt with an early concrete proposal of interacting particles at the ends of strings (quarks interacting with QCD flux tubes), with dynamical boundary conditions for string endpoints where the Dirichlet conditions were dynamical rather than static. Mixed Dirichlet/Neumann boundary conditions were first considered by
Warren Siegel Warren Siegel ( ) is a theoretical physicist specializing in supersymmetric quantum field theory and string theory. He was a professor at the C. N. Yang Institute for Theoretical Physics at Stony Brook University. He retired in Fall of 2022. Back ...
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 In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interaction, fundamental interactions. It confines Quark, quarks into proton, protons, n ...
. String compactifications studied by Harvey and Minahan, Ishibashi and Onogi, and Pradisi and Sagnotti in 1987–1989 also employed Dirichlet boundary conditions. In 1989, Dai,
Leigh Leigh may refer to: Places In England Pronounced : * Leigh, Greater Manchester, Borough of Wigan ** Leigh (UK Parliament constituency) * Leigh-on-Sea, Essex Pronounced : * Leigh, Dorset * Leigh, Gloucestershire * Leigh, Kent * Leigh, Staffor ...
, Polchinski, and Hořava independently, discovered that
T-duality T-duality (short for target-space duality) in theoretical physics is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories descr ...
interchanges the usual Neumann boundary conditions with Dirichlet boundary conditions. This result implies that such boundary conditions must necessarily appear in regions of the
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
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 In theoretical physics orientifold is a generalization of the notion of orbifold, proposed by Augusto Sagnotti in 1987. The novelty is that in the case of string theory the non-trivial element(s) of the orbifold group includes the reversal of the ...
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 1990s, and were shown by Polchinski in 1994 to produce the nonperturbative string effects anticipated by Shenker. In 1995 Polchinski showed that D-branes are the sources of electric and magnetic
Ramond–Ramond field In theoretical physics, Ramond–Ramond fields are differential form fields in the 10-dimensional spacetime of type II supergravity theories, which are the classical limits of type II string theory. The ranks of the fields depend on which type II t ...
s that are required by
string duality String duality is a class of symmetries in physics that link different string theories, theories which assume that the fundamental building blocks of the universe are strings instead of point particles. Overview Before the so-called "duality r ...
,Polchinski, J. (1995). "Dirichlet branes and Ramond-Ramond charges". ''Physical Review D'', 50(10): R6041–R6045. leading to rapid progress in the nonperturbative understanding of string theory.


See also

* Bogomol'nyi–Prasad–Sommerfield bound *
M-theory In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1 ...


Notes


References

* * * ; ibid. * Bachas, C. P. "Lectures on D-branes" (1998). . * * Hashimoto, Koji, ''D-Brane: Superstrings and New Perspective of Our World.'' Springer (2012). * * Polchinski, Joseph, ''TASI Lectures on D-branes'', . Lectures given at TASI '96. * . An article which established D-branes' significance in string theory. * Zwiebach, Barton. ''A First Course in String Theory.'' Cambridge University Press (2004). . {{String theory topics , state=collapsed String theory