Origin12 : The Universe On a String

8/7/2019 Origin12 : The Universe On a String

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If the quadverse view of the universe is correct (matterverse / antimatterverse / mirrorverse /antimirrorverse) here is how it would work. Assuming that we live in 3D space and that theantiverse exists on the opposite side of the CY manifold of our universe within threecomplementary dimensional space, we have two CY manifolds, each with 6 dimensions thatmirror each other. Also, using this model, the space near the cone actually represents the centralblack hole of the universe ("singularity") which has the ability to create baby universes andthereby connect all the CY manifolds (universes) throughout the omniverse. If our universe isinside a black hole inside a larger universe, this can be seen "from the outside" as a single point(the dimensions cancel each other out -- 3 regular spatial dimensions, 3 complementary spatialdimensions, and time and its complement), therefore rather than expanding into the black hole, itwould expand into its own space and the net affect to the outside would be no change.

The big bangs (which are actually big bounces) are coordinated, between the universe and itsmirroverse (which are a quantumly entangled pair) and the antiverse and the mirror antiversewhich are similarly entangled.

Where does the 2-sphere and 3-sphere model come in? Enter type 2 string theory.... in type 2Athe topology smooths out to a 2-sphere (actually a torus) and in type 2B it becomes the 3-sphereor the glome mentioned earlier. So really, type 2A and 2B are related via T duality. Mirrorsymmetry can also be understood as T-duality applied to three dimensional toroidal CY space. F-

theory also compactifies the background to a torus (and obtains type 2B superstring theory.) 6+2dimensional space is preserved as "large." (all the spatial and temporal dimensions plus theircomplements.) To include the entire omniverse in M-theory or F-theory, to explain the presenceof exotic matter, hyperspace and imaginary time, as well as dark matter and dark energy, weneed to complete the description of it as an infinite but bounded analogy to the hypersphere with4+1 dimensions (Einstein was on the right track with Relativity). The extra temporal dimension isImaginary Time, which now gives us a total of three time dimensions with which to navigatethrough time in any direction we wish (this explains how time can go both ways at the quantumlevel.) So now we have a theory of 10+3 dimensions, our three spacial dimensions plus time(3+1), the complementary spatial dimensions plus complementary time (3'+1') and, finally, theomniverse's own (4+1) dimensions which can be fractally reduced through string theory (just likewith the universe) from a hypersphere to a torus to a mobius strip, which is not only emblematicof the infinite but bounded structure present on every level, but the cyclical time nature of the

omniverse and all its component universes. It is in this 4+1 that other universes get created andwhere exotic matter and ZPE exist, and what wormholes tunnel through. So the total structure ofthe omniverse is a 10+3 cosmology with the 6+2 being emergent from the original 4+1. The 4original dimensions of the omniverse make up hyperspace (it should have been namedhypersphere) and the original temporal dimension is imaginary time. In addition, T duality actingon D-branes changes their dimension by 1, thereby validating the Holographic Principle. Thisagrees well with the AdS/CFT correspondence which explains superconductivity, quantumentanglement, quark soup BEC and black holes. Another descendant of string theory, M-theory,is used to explain why gravity is so much weaker than the other forces-- its closed loop geometryattenuates it across the extra dimensions at right angles to reality (and parallel time lines acrossthe second temporal dimension) while the other forces are not so affected (they are local to eachtimeline and universe.) M-theory was used to explain how 2A and 2B are really two differentways of looking at the same thing and how the universe can be toroidal and hyperspherical at the

same time (the volumetric equations for both are the same.) The fact that one is the inverse ofthe other identifies both the "normal" side of the universe as well as the antiverse side. Heteroticstring theory (which defines the universe as a line segment) can be looped using M-theory tocreate a mobius strip (the twist is the vertex of the cone) and thus the mobius, torus and glomeare all united under the umbrella of M-theory. In addition, they show why larger black holes warpspace-time more and lead to more distances across space and time because larger black holesare composed of larger strings which means longer worm holes. In M theory, the torus hasmultiple holes, and each of these holes can correspond to a black hole which births a babyuniverse (the CYM connection across the omniverse through the conifolds is preserved.) SO andSE heterotic string theory flip the same way as type 2A and type 2B do, inversely, thus together


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describing the universe/antiverse pair. Another duality, called S-duality, occurs with type 1 stringtheory, where weakly interacting particles become strongly interacting ones, thus each ordinaryparticle is seen to have a supersymmetric dark matter partner. Even bosonic string theory with its26 dimensions predicts particles with ghost states that do not interact with ordinary matter andspacetime curls up to form a torus. It also predicts the existence of tachyons, which can beresolved by considering that a luxon wall separates our universe from the antiverse; along theluxon wall photons exist that travel at the speed of light, while "on the other side" we haveparticles that appear to be going faster than light and backward in time. For an observer on theother side, they would think we were undergoing the same phenomena. Thus the arrow of time isconserved because, in actuality, all possible temporal states exist at once, which explains suchthings as precognition, which occurs through the quantum foam of consciousness (informationaltunneling) and retrocausal post selection of the present from the future. This is why the bigbounces are reversed between the universe and the antiverse and the mirrorverse and theantimirrorverse. This is mathematically outlined in K-theory, which states that tachyoncondensation occurs when universe/antiverse pairs stack together to form a CY manifold. Thus,conservation laws are preserved and this works well with Type 2B superstring theory using 9dimensional D branes and Type 2A superstring theory using 8 dimensional D branes. Anti Dbranes define the antiverse. Not only that, twisted K-theory also identifies SU (3) a majorcomponent of the weak interaction. Twisted K-theory also works with the 2x2 matrix of theuniverse/antiverse arrangement as the mobius twists of the pair add up to the same number.

This also works with the mirrorverse and its partner to create mirror symmetry. Furthermore CYmanifolds are created as supersymmetry breaks to create 3+1 spacetime , as the otherdimensions appear to compactify as they become the torus of the antiverse. Mirror symmetryalso predicts the torus. Topological string theory presents a 2D holographic surface that allowsfor quantum bubbles in space time that would show up as quantum foam (like the holes in thetorus earlier mentioned under M theory membranes). This quantum foam would fulfill theobjective of connecting all the CY Manifold universes together through micro wormholes. Thisquantizes space-time and this model. Just like the other models of string theory presented, eachCY Manifold consists of 6 dimensions, 3 spatial dimensions for each side (universe andantiverse). Mirror symmetry also exists in this model, as described by T-duality on a three torus.And so we have two manifolds each with 6 dimensions. They also observe S-duality. Wittenargues that the dimensions quantize, which basically is the same thing as saying the CY Manifoldhalves, creating the universe and the antiverse, with the luxon wall on the boundary. Another

duality is the conversion of the 3 torus to the 3 sphere. This is wrapped around the conifold andthe two vertices coincide (universal black hole), which either expands the universe with darkenergy entering through the wormhole part or shrinks it on the black hole end-- depending onwhether you're in the universe or the antiverse. It never shrinks to zero because gravity becomesrepellant at very short distances, thus there is a big bounce at 10 planck lengths while the otheruniverse is at maximum extent and starts to collapse as gravity increases on that side as the bigbounce causes a gravity reversal as black hole turns to white hole in the universe and white holeturns to black hole in the antiverse and the collapse accelerates ("deflation") as the time linesconverge and gravity increases, while inflation at greater than light speeds causes the paralleltime lines to emerge and separate in the second temporal dimension in the universe (and theseparation causes gravity to weaken further increasing expansion.) This separation of time linesactually occurs in the original temporal dimension of the omniverse, imaginary time (which is whythese time lines are navigable by worm hole time machines, with the fact that all moments of all

times exist simultaneously making that possible. The reason these wormholes exist is becausethe expansion occurred so rapidly that what was once connected spacetime remains so throughthe higher 4+1 dimensions of the omniverse.) Another important thing to note is that there is aflip from black hole to white hole and it is simultaneous in the quadverse (universe andmirrorverse together, antiverse and antimirrorverse together) as the force of gravity flips theunified gravimagnetic fields of each universe (analogous to the earth's own magnetic fieldswitching and also to the way negative absolute temperatures occur.) These also obey theholographic principle of dimensional reduction (glome to torus to mobius with the hole and thetwist the black hole / white hole.) As a matter of fact, the hole neatly reduces to the twist and sothe geometry is the same. The holographic principle links together seemingly unconnected


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regions of space through dimension reduction and thus performs a function analogous toquantum entanglement, teleportation and tunneling. They also obey the 2x2 matrix (universe +antiverse), with 1x1 being identified on each surface. Instantons describe quantum tunneling inone dimension and mirror symmetry in two. It also results in the monopole, whose mass willeventually help the universe collapse (as part of dark matter.) As a matter of fact, the monopolehas recently been discovered in spin ice (see below.) The central black hole is quite possibly aBTZ black hole, a much larger one of the type found in quantum foam (2+1 spacetime), whichmight indicate that our own universe exists in the quantum foam of a larger universe (and that wehave myriads "under" us also, as everything loops back, infinite yet bounded) and rotates just likea 3+1 Kerr Black Hole (thus imparting angular momentum to the quadverse) and thereforecreates the same wormhole tunneling and linkages to other universes. This also follows theAdS/CFT correspondence. Quantum topology can also link entanglement to geometric linkingand braiding. This is just like the double helix structure of DNA, which is why the quadverse canbe described by that model, another example of fractality.

http://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory)

In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yaumanifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may lookvery different geometrically, but nevertheless they are equivalent if they are employed as hiddendimensions of string theory. The classical formulation of mirror symmetry relates two Calabi-Yauthreefolds M and W whose Hodge numbers h1,1 and h1,2 are swapped; string theorycompactified on these two manifolds lead to identical effective field theories.Contents [hide]1 History2 Applications3 Generalizations3.1 Mirror symmetry in 2-dimensional gauged sigma models

3.2 Mirror symmetry in 3-dimensional gauge theories4 Notes5 References

[edit]History

The discovery of mirror symmetry is connected with names such as Lance Dixon, WolfgangLerche, Cumrun Vafa, Nicholas Warner, Brian Greene, Ronen Plesser, Philip Candelas, MonikaLynker, Rolf Schimmrigk and others. Andrew Strominger, Shing-Tung Yau, and Eric Zaslow haveshowed that mirror symmetry is a special example of T-duality: the Calabi-Yau manifold may bewritten as a fiber bundle whose fiber is a three-dimensional torus. The simultaneous action of T-duality on all three dimensions of this torus is equivalent to mirror symmetry.

Mathematicians became interested in mirror symmetry in 1990, after Candelas-de la Ossa-Green-Parkes gave predictions for numbers of rational curves in a quintic threefold via datacoming from variation of Hodge structure on the mirror family. These predictions weremathematically proven a few years later by Givental and Lian-Liu-Yau.[edit]Applications

Mirror symmetry allowed the physicists to calculate many quantities that seemed virtuallyincalculable before, by invoking the "mirror" description of a given physical situation, which can be


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often much easier. Mirror symmetry has also become a very powerful tool in mathematics, andalthough mathematicians have proved many rigorous theorems based on the physicists' intuition,a full mathematical understanding of the phenomenon of mirror symmetry is still being developed.Most of the physicist's examples could be conceptualized by the Batyrev-Borisov mirrorconstruction which uses the duality of reflexive polytopes and nef partitions. In their constructionthe mirror partners appear as anticanonically embedded hypersurfaces or certain completeintersections in Fano toric varieties. The Gross-Siebert mirror construction generalizes this tonon-embedded cases by looking at degenerating families of Calabi-Yau manifolds. This point ofview also includes the T-duality. Another mathematical framework is provided by the homologicalmirror symmetry conjecture.[edit]Generalizations

There are two different, but closely related, string theory statements of mirror symmetry[1].1. Type IIA string theory on a Calabi-Yau M is mirror dual to Type IIB on W.2. Type IIB string theory on a Calabi-Yau M is mirror dual to Type IIA on W.

This follows from the fact that Calabi-Yau hodge numbers satisfy h1,1 1 but h2,1 0. If the Hodgenumbers of M are such that h2,1=0 then by definition its mirror dual W is not Calabi-Yau. As aresult mirror symmetry allows for the definition of an extended space of compact spaces, which

are defined by the W of the above two mirror symmetries.

Mirror symmetry has also been generalized to a duality between supersymmetric gauge theoriesin various numbers of dimensions. In this generalized context the original mirror symmetry, whichrelates pairs of toric Calabi-Yau manifolds, relates the moduli spaces of 2-dimensional abeliansupersymmetric gauge theories when the sums of the electric charges of the matter are equal tozero.

In all manifestations of mirror symmetry found so far a central role is played by the fact that in a d-dimensional quantum field theory a differential p-form potential admits a dual formulation as a (d-p-2)-form potential. In 4-dimensions this relates the electric and magnetic vector potentials and iscalled electric-magnetic duality. In 3-dimensions this duality relates a vector and a scalar, whichin an abelian gauge theory correspond to a photon and a squark. In 2-dimensions it relates two

scalars, but while one carries an electric charge, the dual scalar is an uncharged Fayet-Iliopoulosterm. In the process of this duality topological solitons called Abrikosov-Nielsen-Oleson vorticesare intercharged with elementary quark fields in the 3-dimensional case and play the role ininstantons in the 2-dimensional case.

The derivations of 2-dimensional mirror symmetry and 3-dimensional mirror symmetry are bothinspired by Alexander Polyakov's instanton calculation in non-supersymmetric quantumelectrodynamics with a scalar Higgs field. In a 1977 article[2] he demonstrated that instantoneffects give the photon a mass, where the instanton is a 't Hooft-Polyakov monopole embedded inan ultraviolet nonabelian gauge group.[edit]Mirror symmetry in 2-dimensional gauged sigma models

Mirror symmetries in 2-dimensional sigma models are usually considered in cases with N=(2,2)supersymmetry, which means that the fermionic supersymmetry generators are the four realcomponents of a single Dirac spinor. This is the case which is relevant, for example, totopological string theories and type II superstring theory. Generalizations to N=(2,0)supersymmetry have also appeared.[3]

The matter content of N=(2,2) gauged linear sigma models consists of three kinds ofsupermultiplet. The gauge bosons occur in vector multiplets, the charged matter occurs in chiralmultiplets and the Fayet-Ilipolous (FI) terms of the various abelian gauge symmetries occur intwisted chiral multiplets. Mirror symmetry exchanges chiral and twisted chiral multiplets.


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Mirror symmetry, in a class of models of toric varieties with zero first Chern class Calabi-Yaumanifolds and positive first Chern class (Fano varieties) was proven by Kentaro Hori and CumrunVafa.[4] Their approach is as follows. A sigma model whose target space is a toric variety may bedescribed by an abelian gauge theory with charged chiral multiplets. Mirror symmetry thenreplaces these charged chiral multiplets with uncharged twisted chiral multiplets whose vacuumexpectation values are FI terms. Instantons in the dual theory are now vortices whose action isgiven by the exponential of the FI term. These vortices each have precisely 2 fermion zeromodes,and so the sole correction to the superpotential is given by a single vortex. The nonperturbativecorrections to the dual superpotential may then be found by simply summing the exponentials ofthe FI terms. Therefore mirror symmetry allows one to find the full nonperturbative solutions to thetheory.

In addition to finding many new dualities, this allowed them to demonstrate many dualities thathad been conjectured in the literature. For example, beginning with a sigma model whose targetspace is the 2-sphere they found an exactly solvable Sine-Gordon model. More generally, whenthe original sigma model's target space is the n-complex dimensional projective space they foundthat the dual theory is the exactly solvable affine Toda model.[edit]Mirror symmetry in 3-dimensional gauge theories

Mirror symmetry in 3-dimensional gauge theories with N=4 supersymmetry, or 8 supercharges,was first proposed by Kenneth Intriligator and Nathan Seiberg in their 1996 paper[5] as a relationbetween pairs of 3-dimensional gauge theories such that the Coulomb branch of the modulispace of one is the Higgs branch of the moduli space of the other. It was demonstrated using D-brane cartoons by Amihay Hanany and Edward Witten 4 months later,[6] where they found that itis a consequence of S-duality in type IIB string theory.

Four months later it was extended to N=2 gauge theories resulting from supersymmetry breakingin N=4 theories.[7] Here it was given a physical interpretation in terms of vortices. Vortices in 3-dimensional gauge theories are particles. BPS vortices, which are those vortices that preservesome supersymmetry, have masses which are given by the FI term of the gauge theory. Inparticular, on the Higgs branch, where the squarks are massless and condense yielding nontrivial

vacuum expectation values (VEVs), the vortices are massive. On the other hand they interpretthe Coulomb branch of the gauge theory, where the scalar in the vector multiplet has a VEV, asbeing the regime where massless vortices condense. Thus the duality between the Coulumbbranch in one theory and the Higgs branch in the dual theory is the duality between squarks andvortices.

In this theory the instantons are 't Hooft-Polyakov magnetic monopoles, whose actions areproportional to the VEV of the scalar in the vector multiplet. In this case instanton calculationsagain reproduce the nonperturbative superpotential. In particular, in the N=4 case with SU(2)gauge symmetry, the metric on the moduli space was found by Nathan Seiberg and EdwardWitten[8] using holomorphy and supersymmetric nonrenormalization theorems several daysbefore Intriligator and Seiberg's 3-dimensional mirror symmetry paper appeared. Their resultswere reproduced using standard instanton techniques.[9]

The description of conifolds here reminds me of spaces connected with worm holes.

http://en.wikipedia.org/wiki/Conifold

In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds,


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conifolds can contain conical singularities i.e. points whose neighbourhoods look like cones overa certain base. In physics, in particular in flux compactifications of string theory, the base isusually a five-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.

Conifolds are important objects in string theory: Brian Greene explains the physics of conifolds inChapter 13 of his book The Elegant Universe - including the fact that the space can tear near thecone, and its topology can change. This possibility was first noticed by Candelas et al. (1988) andemployed by Green & Hbsch (1988) to prove that conifolds provide a connection between all(then) known Calabi-Yau compactifications in string theory; this partially supports a conjecture byReid (1987) whereby conifolds connect all possible Calabi-Yau complex 3-dimensional spaces.

A well-known example of a conifold is obtained as a deformation limit of a quintic - i.e. a quintichypersurface in the projective space . The space has complex dimension equal to four, andtherefore the space defined by the quintic (degree five) equations

in terms of homogeneous coordinates zi on , for any fixed complex , has complex dimensionthree. This family of quintic hypersurfaces is the most famous example of Calabi-Yaumanifolds. If the complex structure parameter is chosen to become equal to one,

the manifold described above becomes singular since the derivatives of the quinticpolynomial in the equation vanish when all coordinates zi are equal or their ratios arecertain fifth roots of unity. The neighbourhood of this singular point looks like a conewhose base is topologically just .

In the context of string theory, the geometrically singular conifolds can be shown tolead to completely smooth physics of strings. The divergences are "smeared out" byD3-branes wrapped on the shrinking three-sphere in Type IIB string theory and byD2-branes wrapped on the shrinking two-sphere in Type IIA string theory, asoriginally pointed out by Strominger (1995). As shown by Greene, Morrison &Strominger (1995), this provides the string-theoretic description of the topology-change via the conifold transition originally described by Candelas, Green & Hbsch(1990), who also invented the term "conifold" and the diagram

for the purpose. The two topologically distinct ways of smoothing a conifold are thus shown toinvolve replacing the singular vertex (node) by either a 3-sphere (by way of deforming thecomplex structure) or a 2-sphere (by way of a "small resolution"). It is believed that nearly allCalabi-Yau manifolds can be connected via these "critical transitions", resonating with Reid'sconjecture.

http://en.wikipedia.org/wiki/Type_IIB_string_theory

In theoretical physics, type II string theory is a unified term that includes both type IIA strings andtype IIB strings. These account for two of the five consistent superstring theories in tendimensions. Both theories have the maximal amount of supersymmetry namely 32

supercharges in ten dimensions. Both theories are based on oriented closed strings. On theworldsheet, they differ only in the choice of GSO projection.Contents [hide]1 Type IIA string2 Type IIB string3 Relationship between the type II theories4 See also

Type IIA string

At low energies, type IIA string theory is described by type IIA supergravity in ten dimensions


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which is a non-chiral theory (i.e. left-right symmetric) with (1,1) d=10 supersymmetry; the fact thatthe anomalies in this theory cancel is therefore trivial.

In the 1990s it was realized by Edward Witten (building on previous insights by Michael Duff, PaulTownsend, and others) that the limit of type IIA string theory in which the string coupling goes toinfinity becomes a new 11-dimensional theory called M-theory.

The mathematical treatment of type IIA string theory belongs to symplectic topology and algebraicgeometry, particularly Gromov-Witten invariants.Type IIB string

At low energies, type IIB string theory is described by type IIB supergravity in ten dimensionswhich is a chiral theory (left-right asymmetric) with (2,0) d=10 supersymmetry; the fact that theanomalies in this theory cancel is therefore nontrivial.

In the 1990s it was realized that type II string theory with the string coupling constant g isequivalent to the same theory with the coupling 1 / g. This equivalence is known as S-duality.

Orientifold of type IIB string theory leads to type I string theory.

The mathematical treatment of type IIB string theory belongs to algebraic geometry, specificallythe deformation theory of complex structures originally studied by Kunihiko Kodaira and DonaldC. Spencer.Relationship between the type II theories

In the late 1980s, it was realized that type IIA string theory is related to type IIB string theory by T-duality.

http://en.wikipedia.org/wiki/T-duality

Open Strings and D-branes

T-duality acting on D-branes changes their dimension by +1 or -1.

Andrew Strominger, Shing-Tung Yau, and Eric Zaslow have showed that mirror symmetry can beunderstood as T-duality applied to three-dimensional toroidal fibres of the Calabi-Yau space.

http://en.wikipedia.org/wiki/Calabi-Yau_space#Applications_in_superstring_theory

Applications in superstring theory

CalabiYau manifolds are important in superstring theory. In the most conventional superstringmodels, ten conjectural dimensions in string theory are supposed to come as four of which we areaware, carrying some kind of fibration with fiber dimension six. Compactification on CalabiYau n-folds are important because they leave some of the original supersymmetry unbroken. Moreprecisely, in the absence of fluxes, compactification on a CalabiYau 3-fold (real dimension 6)

leaves one quarter of the original supersymmetry unbroken if the holonomy is the full SU(3).

More generally, a flux-free compactification on an n-manifold with holonomy SU(n) leaves 21n ofthe original supersymmetry unbroken, corresponding to 26n supercharges in a compactificationof type II supergravity or 25n supercharges in a compactification of type I. When fluxes areincluded the supersymmetry condition instead implies that the compactification manifold be ageneralized CalabiYau, a notion introduced by Hitchin (2003). These models are known as fluxcompactifications.

Essentially, CalabiYau manifolds are shapes that satisfy the requirement of space for the six


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"unseen" spatial dimensions of string theory, which may be smaller than our currently observablelengths as they have not yet been detected. A popular alternative known as large extradimensions, which often occurs in braneworld models, is that the CalabiYau is large but we areconfined to a small subset on which it intersects a D-brane.

F-theory compactifications on various CalabiYau four-folds provide physicists with a method tofind a large number of classical solution in the so-called string theory landscape.

http://en.wikipedia.org/wiki/F-theory

F-theory is a branch of string theory developed by Cumrun Vafa. The new vacua described as F-theory were discovered by Vafa, and it also allowed string theorists to construct new realisticvacua in the form of F-theory compactified on elliptically fibered Calabi-Yau four-folds. Theletter "F" supposedly stands for "Father".[citation needed]Contents [hide]1 Compactifications2 Phenomenology3 Extra time dimensions4 References

Compactifications

F-theory is formally a 12-dimensional theory, but the only way to obtain an acceptablebackground is to compactify this theory on a two-torus. By doing so, one obtains type IIBsuperstring theory in 10 dimensions. The SL(2,Z) S-duality symmetry of the resulting type IIBstring theory is manifest because it arises as the group of large diffeomorphisms of the two-dimensional torus.

More generally, one can compactify F-theory on an elliptically fibered manifold (elliptic fibration),i.e. a fiber bundle whose fiber is a two-dimensional torus (also called an elliptic curve). Forexample, a subclass of the K3 manifolds is elliptically fibered, and F-theory on a K3 manifold isdual to heterotic string theory on a two-torus. (Eight dimensions are large.)

The well-known large number of semirealistic solutions to string theory referred to as the string

theory landscape, with 10500 elements or so, is dominated by F-theory compactifications onCalabi-Yau four-folds.Phenomenology

New models of GUT unification of the fundamental forces have recently been developed using F-theory.[1]Extra time dimensions

F-theory, as it has metric signature (11,1), as needed for the Euclidean interpretation of thecompactification spaces (e.g. the four-folds), is not a "two-time" theory of physics.

However, the signature of the two additional dimensions is somewhat ambiguous due to theirinfinitesimal character. For example, the supersymmetry of F-theory on a flat background

corresponds to type IIB (i.e. (2,0)) supersymmetry with 32 real supercharges which may beinterpreted as the dimensional reduction of the chiral real 12-dimensional supersymmetry if itsspacetime signature is (10,2). In (11,1) dimensions, the minimum number of components wouldbe 64.

http://en.wikipedia.org/wiki/D-branes

In string theory, D-branes are a class of extended objects upon which open strings can end withDirichlet boundary conditions, after which they are named. D-branes were discovered by Dai,Leigh and Polchinski, and independently by Hoava in 1989. In 1995, Polchinski identified


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D-branes with black p-brane solutions of supergravity, a discovery that triggered theSecond Superstring Revolution and led to both holographic and M-theory dualities.

D-branes are typically classified by their spatial dimension, which is indicated by anumber 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 alsoinstantonic D(-1)-branes, which are localized in both space and time.Contents [hide]1 Theoretical background2 Braneworld cosmology3 D-brane scattering4 Gauge theories5 Black holes6 History7 See also8 References

[edit]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: TheNeumann boundary condition, corresponding to free endpoints moving throughspacetime at the speed of light, or the Dirichlet boundary conditions, which pin thestring endpoint. Each coordinate of the string must satisfy one or the other of theseconditions. There can also exist strings with mixed boundary conditions, where thetwo endpoints satisfy NN, DD, ND and DN boundary conditions. If p spatialdimensions satisfy the Neumann boundary condition, then the string endpoint isconfined to move within a p-dimensional hyperplane. This hyperplane provides onedescription of a Dp-brane.

Although rigid in the limit of zero coupling, the spectrum of open strings ending on aD-brane contains modes associated with its fluctuations, implying that D-branes aredynamical objects. When N D-branes are nearly coincident, the spectrum of stringsstretching 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 dimensionalmatrix for each transverse dimension of the brane. If these matrices commute, theymay be diagonalized, and the eigenvalues define the position of the N D-branes inspace. 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.

Tachyon condensation is a central concept in this field. Ashoke Sen has argued thatin 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 stackof D9 and anti D9-branes. Edward Witten has shown that such configurations will be

classified by the K-theory of the spacetime. Tachyon condensation is still very poorlyunderstood. This is due to the lack of an exact string field theory that would describethe off-shell evolution of the tachyon.[edit]Braneworld cosmology

This has implications for physical cosmology. Because string theory implies that theUniverse has more dimensions than we expect26 for bosonic string theories and 10for superstring theorieswe have to find a reason why the extra dimensions are not


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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 openstrings, are bound to the D-brane, and cannot move "at right angles to reality" toexplore the Universe outside the brane. This scenario is called a brane cosmology.The force of gravity is not due to open strings; the gravitons which carry gravitationalforces are vibrational states of closed strings. Because closed strings do not have to

be attached to D-branes, gravitational effects could depend upon the extradimensions at right angles to the brane.[edit]D-brane scattering

When two D-branes approach each other the interaction is captured by the one loopannulus amplitude of strings between the two branes. The scenario of two parallelbranes approaching each other at a constant velocity can be mapped to the problemof two stationary branes that are rotated relative to each other by some angle. Theannulus amplitude yields singularities that correspond to the on-shell production ofopen strings stretched between the two branes. This is true irrespective of the chargeof the D-branes. At non-relativistic scattering velocities the open strings may bedescribed by a low-energy effective action that contains two complex scalar fieldsthat are coupled via a term 22. Thus, as the field (separation of the branes)changes, the mass of the field changes. This induces open string production and asa result the two scattering branes will be trapped.[edit]Gauge theories

The arrangement of D-branes constricts the types of string states which can exist in asystem. For example, if we have two parallel D2-branes, we can easily imaginestrings stretching from brane 1 to brane 2 or vice versa. (In most theories, strings areoriented 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 thoseoriginating on brane 2 and terminating on brane 1. Symbolically, we say we have theand the sectors. In addition, a string may begin and end on the same brane, givingand sectors. (The numbers inside the brackets are called Chan-Paton indices, butthey are really just labels identifying the branes.) A string in either the or the sectorhas 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 bynature relativistic, adding energy to a string is equivalent to adding mass, byEinstein's relation E = mc2. Therefore, the separation between D-branes controls theminimum mass open strings may have.

Furthermore, affixing a string's endpoint to a brane influences the way the string canmove and vibrate. Because particle states "emerge" from the string theory as thedifferent vibrational states the string can experience, the arrangement of D-branescontrols the types of particles present in the theory. The simplest case is the 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 (andapplying the rules of quantum mechanics to quantize the string), one finds thatamong the spectrum of particles is one resembling the photon, the fundamentalquantum of the electromagnetic field. The resemblance is precise: a p-dimensionalversion of the electromagnetic field, obeying a p-dimensional analogue of Maxwell'sequations, exists on every Dp-brane.

In this sense, then, one can say that string theory "predicts" electromagnetism: D-


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branes are a necessary part of the theory if we permit open strings to exist, and allD-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 aset 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 masslessscalars (particles which do not have polarizations like the photons making up light).Intriguingly, there are just as many massless scalars as there are directionsperpendicular to the brane; the geometry of the brane arrangement is closely relatedto the quantum field theory of the particles existing on it. In fact, these masslessscalars are Goldstone excitations of the brane, corresponding to the different waysthe symmetry of empty space can be broken. Placing a D-brane in a universe breaksthe symmetry among locations, because it defines a particular place, assigning aspecial meaning to a particular location along each of the d - p directionsperpendicular to the brane.

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

Consider a group of N separate Dp-branes, arranged in parallel for simplicity. Thebranes are labeled 1,2,...,N for convenience. Open strings in this system exist in oneof many sectors: the strings beginning and ending on some brane i give that brane aMaxwell field and some massless scalar fields on its volume. The strings stretchingfrom brane i to another brane j have more intriguing properties. For starters, it isworthwhile to ask which sectors of strings can interact with one another. Onestraightforward 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 stringmay interact with a string, but not with a or a one. The masses of these strings willbe influenced by the separation between the branes, as discussed above, so forsimplicity'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 distinctobjects, then we still have all the sectors we had before, but without the effects dueto the brane separations.

The zero-mass states in the open-string particle spectrum for a system of Ncoincident 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 onlydetectable at very high energies.) Gauge theories were not invented starting withbosonic or fermionic strings; they originated from a different area of physics, andhave 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 explaininggauge interactions, even if string theory fails to be the "theory of everything".

[edit]Black holes

Another important use of D-branes has been in the study of black holes. Since the1970s, 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 thegas cannot escape from the hole's gravitational pull, its entropy would seem to havevanished 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


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originally had. Attempting to apply quantum mechanics to the study of black holes,Stephen Hawking discovered that a hole should emit energy with the characteristicspectrum of thermal radiation. The characteristic temperature of this Hawkingradiation is given by,

where G is Newton's gravitational constant, M is the black hole's mass and kB isBoltzmann's constant.

Using this expression for the Hawking temperature, and assuming that a zero-massblack hole has zero entropy, one can use thermodynamic arguments to derive the"Bekenstein entropy":

The Bekenstein entropy is proportional to the black hole mass squared; because theSchwarzschild radius is proportional to the mass, the Bekenstein entropy isproportional to the black hole's surface area. In fact,

where lP is the Planck length.

The concept of black hole entropy poses some interesting conundra. In an ordinarysituation, a system has entropy when a large number of different "microstates" cansatisfy the same macroscopic condition. For example, given a box full of gas, manydifferent arrangements of the gas atoms can have the same total energy. However, ablack 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 giverise to black hole entropy?

String theorists have constructed models in which a black hole is a very long (andhence very massive) string. This model gives rough agreement with the expectedentropy of a Schwarzschild black hole, but an exact proof has yet to be found oneway or the other. The chief difficulty is that it is relatively easy to count the degreesof freedom quantum strings possess if they do not interact with one another. This isanalogous to the ideal gas studied in introductory thermodynamics: the easiestsituation to model is when the gas atoms do not have interactions amongthemselves. Developing the kinetic theory of gases in the case where the gas atomsor molecules experience inter-particle forces (like the van der Waals force) is moredifficult. However, a world without interactions is an uninteresting place: mostsignificantly for the black hole problem, gravity is an interaction, and so if the "stringcoupling" is turned off, no black hole could ever arise. Therefore, calculating blackhole entropy requires working in a regime where string interactions exist.

Extending the simpler case of non-interacting strings to the regime where a blackhole could exist requires supersymmetry. In certain cases, the entropy calculationdone 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 doesnot "break" supersymmetry. In recent years, this has been done by building blackholes out of D-branes. Calculating the entropies of these hypothetical holes givesresults which agree with the expected Bekenstein entropy. Unfortunately, the casesstudied so far all involve higher-dimensional spaces D5-branes in nine-dimensionalspace, for example. They do not directly apply to the familiar case, the Schwarzschildblack holes observed in our own universe.[edit]History


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Dirichlet boundary conditions and D-branes had a long pre-history' before their fullsignificance was recognized. Mixed Dirichlet/Neumann boundary conditions were firstconsidered by Warren Siegel in 1976 as a means of lowering the critical dimension ofopen 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 boundaryconditions). This paper, though prescient, was little-noted in its time (a 1985 parodyby Siegel, `The Super-g String,' contains an almost dead-on description ofbraneworlds). Dirichlet conditions for all coordinates including Euclidean time(defining what are now known as D-instantons) were introduced by Michael Green in1977 as a means of introducing point-like structure into string theory, in an attemptto construct a string theory of the strong interaction. String compactifications studiedby Harvey and Minahan, Ishibashi and Onogi, and Pradisi and Sagnotti in 1987-89also employed Dirichlet boundary conditions.

The fact that T-duality interchanges the usual Neumann boundary conditions withDirichlet boundary conditions was discovered independently by Horava and by Dai,Leigh, and Polchinski in 1989; this result implies that such boundary conditions mustnecessarily appear in regions of the moduli space of any open string theory. The Daiet al. paper also notes that the locus of the Dirichlet boundary conditions isdynamical, and coins the term Dirichlet-brane (D-brane) for the resulting object (thispaper also coins orientifold for another object that arises under string T-duality). A1989 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, andwere shown by Polchinski in 1994 to produce the e^{-1/g} nonperturbative stringeffects anticipated by Shenker. In 1995 Polchinski showed that D-branes are thesources of electric and magnetic Ramond-Ramond fields that are required by stringduality, leading to rapid progress in the nonperturbative understanding of stringtheory.

http://en.wikipedia.org/wiki/M-theory#P-branes

In theoretical physics, M-theory is an extension of string theory in which 11dimensions are identified. Because the dimensionality exceeds the dimensionality ofsuperstring theories in 10 dimensions, it is believed that the 11-dimensional theoryunites all five string theories (and supersedes them). Though a full description of thetheory is not known, the low-entropy dynamics are known to be supergravityinteracting with 2- and 5-dimensional membranes.

This idea is the unique supersymmetric theory in eleven dimensions, with its low-entropy matter content and interactions fully determined, and can be obtained as thestrong coupling limit of type IIA string theory because a new dimension of spaceemerges as the coupling constant increases.

Drawing on the work of a number of string theorists (including Ashoke Sen, Chris Hull,

Paul Townsend, Michael Duff and John Schwarz), Edward Witten of the Institute forAdvanced Study suggested its existence at a conference at USC in 1995, and used M-theory to explain a number of previously observed dualities, sparking a flurry of newresearch in string theory called the second superstring revolution.

In the early 1990s, it was shown that the various superstring theories were related bydualities, which allow physicists to relate the description of an object in one superstring theory to the description of a different object in another super string theory.These relationships imply that each of the super string theories is a different aspect


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of a single underlying theory, proposed by Witten, and named "M-theory".

Originally the letter M in M-theory was taken from membrane, a construct designedto generalize the strings of string theory. However, as Witten was more skepticalabout membranes than his colleagues, he opted for "M-theory" rather than"Membrane theory". Witten has since stated that the interpretation of the M can be a

matter of taste for the user of the name.[1]

M-theory (and string theory) has been criticized (e.g. by Lawrence Krauss) for lackingpredictive power or being untestable. Further work continues to find mathematicalconstructs that join various surrounding theories. New formulations are proposed tojoin many theoretic situations (usually by exploiting string theoreticdualities[clarification needed]). Witten has suggested that a general formulation of M-theory will probably require the development of new mathematical language.[citationneeded] However, the tangible success of M-theory can be questioned, given itscurrent incompleteness and limited predictive power, even after so many years ofintense research.Contents [hide]1 History and development1.1 Prior to May 19951.2 Type I string theory and supplements1.3 String vibrational patterns1.4 M-theory1.5 Type IIA and Type IIB1.6 Other dualities1.7 Only two string theories1.8 Last step1.9 Supergravity theories1.10 Same underlying theory1.11 Recent developments2 Nomenclature3 M-theory and membranes3.1 P-branes3.2 Strings with "loose ends"3.3 Strings with closed loops4 Membrane interactions5 Matrix theory5.1 Analogy with water5.2 BFSS model5.3 IKKT model6 Mysterious duality7 See also8 Notes9 References10 Further reading11 External links

[edit]History and development[edit]Prior to May 1995

Antecedent to 1995 there were five (known) consistent superstring theories(henceforth referred to as string theories), which were given the names Type I stringtheory, Type IIA string theory, Type IIB string theory, heterotic SO(32) (the HO string)theory, and heterotic E8E8 (the HE string) theory. The five theories all share essential


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features that relate them to the name of string theory. Each theory is fundamentally based onvibrating, one dimensional strings at approximately the length of the Planck length. Calculationshave also shown that each theory requires more than the normal four spacetime dimensions(although all extra dimensions are in fact spatial). Yet, when the theories are analyzed in detail,significant differences appear.[edit]Type I string theory and supplements

The Type I string theory has vibrating strings like the rest of the string theories. These stringsvibrate both in closed loops, so that the strings have no ends, and as open strings with two looseends. The open loose strings are what separates the Type I string theory from the other fourstring theories. This was a feature that the other string theories did not contain.[edit]String vibrational patterns

The calculations of the String Vibrational Patterns show that the list of string vibrational patternsand the way each pattern interacts and influences others vary from one theory to another. Theseand other differences hindered the development of the string theory as being the theory thatunited quantum mechanics and general relativity successfully. Attempts by the physicscommunity to eliminate four of the theories, leaving only one string theory, have not been

successful.[edit]M-theory

M-theory attempts to unify the five string theories by examining certain identifications anddualities. Thus each of the five string theories become special cases of M-theory.

As the names suggest, some of these string theories were thought to be related to each other. Inthe early 1990s, string theorists discovered that some relations were so strong that they could bethought of as an identification.[edit]Type IIA and Type IIB

The Type IIA string theory and the Type IIB string theory were known to be connected by T-duality; this essentially meant that the IIA string theory description of a circle of radius R is exactlythe same as the IIB description of a circle of radius 1/R, where distances are measured in units ofthe Planck length.

This was a profound result. First, this was an intrinsically quantum mechanical result; theidentification did not hold in the realm of classical physics. Second, because it is possible to buildup any space by gluing circles together in various ways,[dubious discuss] it would seem thatany space described by the IIA string theory can also be seen as a different space described bythe IIB theory. This implies that the IIA string theory can identify with the IIB string theory: anyobject which can be described with the IIA theory has an equivalent, although seemingly different,description in terms of the IIB theory. This suggests that the IIA string theory and the IIB stringtheory are really aspects of the same underlying theory.

[edit]Other dualities

There are other dualities between the other string theories. The heterotic SO(32) and theheterotic E8E8 theories[2][3] are also related by T-duality; the heterotic SO(32) description of acircle of radius R is exactly the same as the heterotic E8E8 description of a circle of radius 1/R.This implies that there are really only three superstring theories, which might be called (fordiscussion) the Type I theory, the Type II theory, and the heterotic theory.

There are still more dualities, however. The Type I string theory is related to the heterotic SO(32)


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theory by S-duality; this means that the Type I description of weakly interacting particles can alsobe seen as the heterotic SO(32) description of very strongly interacting particles. Thisidentification is somewhat more subtle, in that it identifies only extreme limits of the respectivetheories. String theorists have found strong evidence that the two theories are really the same,even away from the extremely strong and extremely weak limits, but they do not yet have a proofstrong enough to satisfy mathematicians. However, it has become clear that the two theories arerelated in some fashion; they appear as different limits of a single underlying theory.[edit]Only two string theories

Given the above commonalities there appear to be only two string theories: the heterotic stringtheory (which is also the type I string theory) and the type II theory. There are relations betweenthese two theories as well, and these relations are in fact strong enough to allow them to beidentified.[edit]Last step

This last step is best explained first in a certain limit. In order to describe our world, strings mustbe extremely tiny objects. So when one studies string theory at low energies, it becomes difficultto see that strings are extended objects they become effectively zero-dimensional (pointlike).

Consequently, the quantum theory describing the low energy limit is a theory that describes thedynamics of these points moving in spacetime, rather than strings. Such theories are calledquantum field theories. However, since string theory also describes gravitational interactions, oneexpects the low-energy theory to describe particles moving in gravitational backgrounds. Finally,since superstring string theories are supersymmetric, one expects to see supersymmetryappearing in the low-energy approximation. These three facts imply that the low-energyapproximation to a superstring theory is a supergravity theory.[edit]Supergravity theories

The possible supergravity theories were classified by Werner Nahm in the 1970s. In 10dimensions, there are only two supergravity theories, which are denoted Type IIA and Type IIB.This similar denomination is not a coincidence; the Type IIA string theory has the Type IIA

supergravity theory as its low-energy limit and the Type IIB string theory gives rise to Type IIBsupergravity. The heterotic SO(32) and heterotic E8E8 string theories also reduce to Type IIAand Type IIB supergravity in the low-energy limit. This suggests that there may indeed be arelation between the heterotic/Type I theories and the Type II theories.

In 1994, Edward Witten outlined the following relationship: The Type IIA supergravity(corresponding to the heterotic SO(32) and Type IIA string theories) can be obtained bydimensional reduction from the single unique eleven-dimensional supergravity theory. Thismeans that if one studied supergravity on an eleven-dimensional spacetime that looks like theproduct of a ten-dimensional spacetime with another very small one-dimensional manifold, onegets the Type IIA supergravity theory. (And the Type IIB supergravity theory can be obtained byusing T-duality.) However, eleven-dimensional supergravity is not consistent on its own it doesnot make sense at extremely high energy, and likely requires some form of completion. It seems

plausible, then, that there is some quantum theory which Witten dubbed M-theory in eleven-dimensions which gives rise at low energies to eleven-dimensional supergravity, and is related toten-dimensional string theory by dimensional reduction. Dimensional reduction to a circle yieldsthe Type IIA string theory, and dimensional reduction to a line segment yields the heteroticSO(32) string theory.[edit]Same underlying theory

M-theory would implement the notion that all of the different string theories are different specialcases.


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[edit]Recent developments

In late 2007, Bagger and Lambert set off renewed interest in M-theory with the discovery of acandidate Lagrangian description of coincident M2-branes, based on a non-associativegeneralization of Lie Algebra, Nambu 3-algebra or Filippov 3-algebra. Practitioners hope theBaggerLambertGustavsson action will provide the long-sought microscopic description of M-theory.[edit]Nomenclature

When Edward Witten named M-theory, he did not specify what the M stood forperhapsbecause the nascent theory wasn't fully defined. Some[who?] speculate that Witten chose theletter because it resembles an inverted W. According to Witten, "M can stand variously for'magic', 'mystery', or 'matrix', according to one's taste."[4]

Faced with this ambiguous initial, countless scientists and commentators have offered their ownexpansions of the Msome sincere, others facetious. M should stand for membrane, say some.[who?] Meanwhile, Michio Kaku, Michael Duff, Neil Turok, and others suggest mother or master(i.e., the "mother of all theories" or the "master theory").[5]

Although Witten coined the term M-theory to refer to his model of an eleven-dimensionaluniverse, other scientists have generalized the moniker for application to any of various meta-theories involving string theory and brane cosmology. (Ashoke Sen proposed u-theory (ur, 'ber','ultimate', 'underlying', or perhaps 'unified') as a more distinctive appellation.)[citation needed]When unqualified, M-theory now usually denotes this more general definition, rather than the oneWitten originally advanced.[edit]M-theory and membranes

In the standard string theories, strings are assumed to be the single fundamental constituent ofthe universe. M-theory adds another fundamental constituent - membranes. Like the tenth spatialdimension, the approximate equations in the original five superstring models proved too weak to

reveal membranes.[edit]P-branes

A membrane, or brane, is a multidimensional object, usually called a P-brane, with P referring tothe number of dimensions in which it exists. The value of 'P' can range from zero to nine, thusgiving branes dimensions from zero (0-brane point particle) to nine - five more than the worldwe are accustomed to inhabiting. The inclusion of p-branes does not render previous work instring theory wrong on account of not taking note of these P-branes. P-branes are much moremassive ("heavier") than strings, and when all higher-dimensional P-branes are much moremassive than strings, they can be ignored, as researchers had done unknowingly in the 1970s.[edit]Strings with "loose ends"

Shortly after Witten's breakthrough in 1995, Joseph Polchinski of the University of California,Santa Barbara discovered a fairly obscure feature of string theory. He found that in certainsituations the endpoints of strings (strings with "loose ends") would not be able to move withcomplete freedom as they were attached, or stuck within certain regions of space. Polchinski thenreasoned that if the endpoints of open strings are restricted to move within some p-dimensionalregion of space, then that region of space must be occupied by a p-brane. These type of "sticky"branes are called Dirichlet-P-branes, or D-p-branes. His calculations showed that the newlydiscovered D-P-branes had exactly the right properties to be the objects that exert a tight grip onthe open string endpoints, thus holding down these strings within the p-dimensional region of


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space they fill.[edit]Strings with closed loops

Not all strings are confined to p-branes. Strings with closed loops, like the graviton, arecompletely free to move from membrane to membrane. Of the four force carrier particles, thegraviton is unique in this way. Researchers speculate that this is the reason why investigationthrough the weak force, the strong force, and the electromagnetic force have not hinted at thepossibility of extra dimensions. These force carrier particles are strings with endpoints thatconfine them to their p-branes. Further testing is needed in order to show that extra spatialdimensions indeed exist through experimentation with gravity.[edit]Membrane interactions

One of the reasons M-theory is so difficult to formulate is that the numbers of different types ofmembranes in the various dimensions increases exponentially. For example once one gets to 3dimensional surfaces, one has to deal with solid objects with knot-shaped holes, and then oneneeds the whole of knot theory just to classify them. Since M-theory is thought to operate in 11dimensions this problem then becomes very difficult. But just like string theory, in order for thetheory to satisfy causality, the theory must be local, and so the topology changing must occur at a

single point. The basic orientable 2-brane interactions are easy to show. Orientable 2-branes aretori with multiple holes cut out of them.[edit]Matrix theory

The original formulation of M-theory was in terms of a (relatively) low-energy effective field theory,called 11-dimensional Supergravity. Though this formulation provided a key link to the low-energylimits of string theories, it was recognized that a full high-energy formulation (or "UV-completion")of M-theory was needed.[edit]Analogy with water

For an analogy, the Super gravity description is like treating water as a continuous,

incompressible fluid. This is effective for describing long-distance effects such as waves andcurrents, but inadequate to understand short-distance/high-energy phenomena such asevaporation, for which a description of the underlying molecules is needed. What, then, are theunderlying degrees of freedom of M-theory?[edit]BFSS model

Banks, Fischler, Shenker and Susskind (BFSS) conjectured that Matrix theory could provide theanswer. They demonstrated that a theory of 9 very large matrices, evolving in time, couldreproduce the Super gravity description at low energy, but take over for it as it breaks down athigh energy. While the Super gravity description assumes a continuous space-time, Matrix theorypredicts that, at short distances, non-commutative geometry takes over, somewhat similar to theway the continuum of water breaks down at short distances in favor of the graininess of

molecules.[edit]IKKT model

Another matrix string theory equivalent to Type IIB string theory was constructed in 1996 byIshibashi, Kawai, Kitazawa, and Tsuchiya.[edit]Mysterious dualityMain article: Mysterious duality


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There is a duality theory called mysterious duality relating M-theory and the geometry of delPezzo surfaces.[edit]See alsoAdS/CFT correspondence

http://en.wikipedia.org/wiki/AdS/CFT_correspondence

In physics, the AdS/CFT correspondence (anti-de-Sitter space/conformal field theorycorrespondence), sometimes called the Maldacena duality, is the conjectured equivalencebetween a string theory and gravity defined on one space, and a quantum field theory withoutgravity defined on the conformal boundary of this space, whose dimension is lower by one ormore. The name suggests that the first space is the product of anti de Sitter space (AdS) withsome closed manifold like sphere, orbifold, or noncommutative space, and that the quantum fieldtheory is a conformal field theory (CFT).[1]

An example is the duality between Type IIB string theory on AdS5 S5 space (a product of fivedimensional AdS space with a five dimensional sphere) and a supersymmetric N=4 Yang-Millsgauge theory (which is a conformal field theory) on the 4-dimensional boundary of AdS5. It is themost successful realization of the holographic principle, a speculative idea about quantum gravity

originally proposed by Gerard 't Hooft and improved and promoted by Leonard Susskind.

The AdS/CFT correspondence was originally proposed by Juan Maldacena in late 1997.[2]Important aspects of the correspondence were given in articles by Steven Gubser, Igor Klebanovand Alexander Markovich Polyakov,[3] and by Edward Witten.[4] The correspondence has alsobeen generalized to many other (non-AdS) backgrounds and their dual (non-conformal) theories.In about five years, Maldacena's article had 3000 citations and became one of the most importantconceptual breakthroughs in theoretical physics of the 1990s, providing many new lines ofresearch into both quantum gravity and quantum chromodynamics (QCD).

http://en.wikipedia.org/wiki/Matrix_theory_(physics)

In physics, Matrix string theory is a set of equations that describe superstring theory in a non-

perturbative framework. Type IIA string theory can be shown to be equivalent to a maximallysupersymmetric two-dimensional gauge theory, the gauge group of which is U(N) for a largevalue of N. This Matrix string theory was first proposed by Lubo Motl in 1997 [1] and laterindependently in a more complete paper by Robbert Dijkgraaf, Erik Verlinde, and HermanVerlinde [2]. Another matrix string theory equivalent to Type IIB string theory was constructed in1996 by Ishibashi, Kawai, Kitazawa and Tsuchiya [3]. This version is known as the IKKT matrixmodel.M(atrix) Theory

M(atrix) theory (also known as BFSS-Matrix theory) is a fundamental formulation of M-theory as aRandom matrix model. Matrix string theory is related to M(atrix) theory in the same sense thatsuperstring theory is related to M-theory.

M(atrix) theory is written in terms of interacting D0-branes (zero-dimensional Dirichlet branes) ininfinite momentum frame. It was proposed by Banks, Fischler, Shenker, and Susskind in 1996 [4].See also the discussion in M-theory.

http://en.wikipedia.org/wiki/Bosonic_string_theory

Bosonic string theory is the original version of string theory, developed in the late 1960s.

In the early 1970s, supersymmetry was discovered in the context of string theory, and a newversion of string theory called superstring theory (supersymmetric string theory) became the real


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focus. Nevertheless, bosonic string theory remains a very useful "toy model" to understand manygeneral features of perturbative string theory, and string theory textbooks usually start with thebosonic string. The first volume of Polchinski's String Theory and Zwiebach's A First Course inString Theory are good examples.Contents [hide]1 Problems2 See also3 References4 External links

Problems

Although bosonic string theory has many attractive features, it has a pair of features that render itunattractive as a physical model.

Firstly it predicts only the existence of bosons whereas many physical particles are fermions.Secondly, it predicts the existence of a particle whose mass is imaginary implying that it travelsfaster than light. The existence of such a particle, commonly known as a tachyon, would conflictwith much of what is known about physics, and such particles have never been observed.

Another feature of bosonic string theory is that in general the theory displays inconsistencies due

to the conformal anomaly. But, as was first noticed by Claud Lovelace, in a spacetime of 26dimensions, with 25 dimensions of space and one of time, the inconsistencies cancel. Anotherway to look at this is that in general bosonic string theory predicts unphysical particle states called'ghosts'. In 26 dimensions the no-ghost theorem predicts that these ghost states have nointeraction whatsoever with any other states and hence that they can be ignored leaving aconsistent theory. So bosonic string theory predicts a 26 dimensional spacetime. This highdimensionality is not a problem for bosonic string theory because it can be formulated in such away that along the 22 excess dimensions, spacetime is folded up to form a small torus. Thiswould leave only the familiar four dimensions of spacetime visible.

http://en.wikipedia.org/wiki/K-theory_(physics)#Reconciling_twisted_K-theory_and_S-duality

In string theory, the K-theory classification refers to a conjectured application of K-theory (in

abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond-Ramondfield strengths as well as the charges of stable D-branes.

In condensed matter physics K-theory has also found important applications, specially in thetopological classification of topological insulators, superconductors and stable Fermi Surfaces(Kitaev (2009), Horava (2005)).Contents [hide]1 History2 Applications2.1 Open problems3 K-theory classification of RR fluxes4 K-theory classification of D-branes4.1 K-theory charge versus BPS charge4.2 K-theory from tachyon condensation

4.3 Twisted K-theory from MMS instantons5 Reconciling twisted K-theory and S-duality6 Researchers7 See also8 References9 References (Condensed Matter Physics)10 Further reading11 External links

History


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This conjecture, applied to D-brane charges, was first proposed by Minasian & Moore (1997). Itwas popularized by Witten (1998) who demonstrated that in type IIB string theory arises naturallyfrom Ashoke Sen's realization of arbitrary D-brane configurations as stacks of D9 and anti-D9-branes after tachyon condensation.

Such stacks of branes are inconsistent in a non-torsion Neveu-Schwarz (NS) 3-form background,which, as was highlighted by Kapustin (2000), complicates the extension of the K-theoryclassification to such cases. Bouwknegt & Varghese (2000) suggested a solution to this problem:D-branes are in general classified by a twisted K-theory, that had earlier been defined byRosenberg (1989).ApplicationsString theory

Superstring theory [show]Theory[show]Concepts[show]Related Topics[show]

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The K-theory classification of D-branes has had numerous applications. For example, Hanany &Kol (2000) used it to argue that there are eight species of orientifold one-plane. Uranga (2001)applied the K-theory classification to derive new consistency conditions for flux compactifications.K-theory has also been used to conjecture a formula for the topologies of T-dual manifolds byBouwknegt, Evslin & Varghese (2004). Recently K-theory has been conjectured to classify thespinors in compactifications on generalized complex manifolds.Open problems

Despite these successes, RR fluxes are not quite classified by K-theory. Diaconescu, Moore &Witten (2003) argued that the K-theory classification is incompatible with S-duality in IIB stringtheory.

In addition, if one attempts to classify fluxes on a compact ten-dimensional spacetime, then acomplication arises due to the self-duality of the RR fluxes. The duality uses the Hodge star,which depends on the metric and so is continuously valued and in particular is genericallyirrational. Thus not all of the RR fluxes, which are interpreted as the Chern characters in K-theory,can be rational. However Chern characters are always rational, and so the K-theory classificationmust be replaced. One needs to choose a half of the fluxes to quantize, or apolarization[disambiguation needed] in the geometric quantization-inspired language ofDiaconescu, Moore, and Witten and later of Varghese & Sati (2004). Alternately one may use theK-theory of a 9-dimensional time slice as has been done by Maldacena, Moore & Seiberg (2001).

K-theory classification of RR fluxes

In the classical limit of type II string theory, which is type II supergravity, the Ramond-Ramondfield strengths are differential forms. In the quantum theory the well-definedness of the partitionfunctions of D-branes implies that the RR field strengths obey Dirac quantization conditions whenspacetime is compact, or when a spatial slice is compact and one considers only the (magnetic)components of the field strength which lie along the spatial directions. This led twentieth centuryphysicists to classify RR field strengths using cohomology with integral coefficients.

However some authors have argued that the cohomology of spacetime with integral coefficients


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is too big. For example, in the presence of Neveu-Schwarz H-flux or non-spin cycles some RRfluxes dictate the presence of D-branes. In the former case this is a consequence of thesupergravity equation of motion which states that the product of a RR flux with the NS 3-form is aD-brane charge density. Thus the set of topologically distinct RR field strengths that can exist inbrane-free configurations is only a subset of the cohomology with integral coefficients.

This subset is still too big, because some of these classes are related by large gaugetransformations. In QED there are large gauge transformations which add integral multiples oftwo pi to Wilson loops. The p-form potentials in type II supergravity theories also enjoy theselarge gauge transformations, but due to the presence of Chern-Simons terms in the supergravityactions these large gauge transformations transform not only the p-form potentials but alsosimultaneously the (p+3)-form field strengths. Thus to obtain the space of inequivalent fieldstrengths from the forementioned subset of integral cohomology we must quotient by these largegauge transformations.

The Atiyah-Hirzebruch spectral sequence constructs twisted K-theory, with a twist given by theNS 3-form field strength, as a quotient of a subset of the cohomology with integral coefficients. Inthe classical limit, which corresponds to working with rational coefficients, this is precisely thequotient of a subset described above in supergravity. The quantum corrections come from torsionclasses and contain mod 2 torsion corrections due to the Freed-Witten anomaly.

Thus twisted K-theory classifies the subset of RR field strengths that can exist in the absence ofD-branes quotiented by large gauge transformations. Daniel Freed has attempted to extend thisclassification to include also the RR potentials using differential K-theory.K-theory classification of D-branes

K-theory classifies D-branes in noncompact spacetimes, intuitively in spacetimes in which we arenot concerned about the flux sourced by the brane having nowhere to go. While the K-theory of a10d spacetime classifies D-branes as subsets of that spacetime, if the spacetime is the product oftime and a fixed 9-manifold then K-theory also classifies the conserved D-brane charges on each9-dimensional spatial slice. While we were required to forget about RR potentials to obtain the K-theory classification of RR field strengths, we are required to forget about RR field strengths toobtain the K-theory classification of D-branes.

K-theory charge versus BPS charge

As has been stressed by Petr Hoava, the K-theory classification of D-branes isindependent of, and in some ways stronger than, the classification of BPS states. K-theory appears to classify stable D-branes missed by supersymmetry basedclassifications.

For example, D-branes with torsion charges, that is with charges in the order N cyclicgroup , attract each other and so can never be BPS. In fact, N such branes can decay,whereas no superposition of branes that satisfy a Bogomolny bound may ever decay.However the charge of such branes is conserved modulo N, and this is captured bythe K-theory classification but not by a BPS classification. Such torsion branes havebeen applied, for example, to model Douglas-Shenker strings in supersymmetric U(N)

gauge theories.K-theory from tachyon condensation

Ashoke Sen has conjectured that, in the absence of a topologically nontrivial NS 3-form flux, all IIB brane configurations can be obtained from stacks of spacefilling D9and anti D9 branes via tachyon condensation. The topology of the resulting branes isencoded in the topology of the gauge bundle on the stack of the spacefilling branes.The topology of the gauge bundle of a stack of D9s and anti D9s can be decomposedinto a gauge bundle on the D9's and another bundle on the anti D9's. Tachyoncondensation transforms such a pair of bundles to another pair in which the same


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bundle is direct summed with each component in the pair. Thus the tachyoncondensation invariant quantity, that is, the charge which is conserved by thetachyon condensation process, is not a pair of bundles but rather the equivalenceclass of a pair of bundles under direct sums of the same bundle on both sides of thepair. This is precisely the usual construction of topological K-theory. Thus the gaugebundles on stacks of D9's and anti-D9's are classified by topological K-theory. If Sen's

conjecture is right, all D-brane configurations in type IIB are then classified by K-theory. Petr Horava has extended this conjecture to type IIA using D8-branes.Twisted K-theory from MMS instantons

While the tachyon condensation picture of the K-theory classification classifies D-branes as subsets of a 10-dimensional spacetime with no NS 3-form flux, theMaldacena, Moore, Seiberg picture classifies stable D-branes with finite mass assubsets of a 9-dimensional spatial slice of spacetime.

The central observation is that D-branes are not classified by integral homologybecause Dp-branes wrapping certain cycles suffer from a Freed-Witten anomaly,which is cancelled by the insertion of D(p-2)-branes and sometimes D(p-4)-branesthat end on the afflicted Dp-brane. These inserted branes may either continue toinfinity, in which case the composite object has an infinite mass, or else they mayend on an anti-Dp-brane, in which case the total Dp-brane charge is zero. In eithercase, one may wish to remove the anomalous Dp-branes from the spectrum, leavingonly a subset of the original integral cohomology.

The inserted branes are unstable. To see this, imagine that they extend in time away(into the past) from the anomalous brane. This corresponds to a process in which theinserted branes decay via a Dp-brane that forms, wraps the forementioned cycle andthen disappears. MMS refer to this process as an instanton, although really it neednot be instantonic.

The conserved charges are thus the nonanomolous subset quotiented by theunstable insertions. This is precisely the Atiyah-Hirzebruch spectral sequenceconstruction of twisted K-theory as a set.Reconciling twisted K-theory and S-duality

Diaconescu, Moore, and Witten have pointed out that the twisted K-theoryclassification is not compatible with the S-duality covariance of type IIB string theory.For example, consider the constraint on the Ramond-Ramond 3-form field strengthG3 in the Atiyah-Hirzebruch spectral sequence (AHSS):

where d3=Sq3+H is the first nontrivial differential in the AHSS, Sq3 is the thirdSteenrod square and the last equality follows from the fact that the nth Steenrodsquare acting on any n-form x is xx.

The above equation is not invariant under S-duality, which exchanges G3 and H.

Instead Diaconescu, Moore, and Witten have proposed the following S-dualitycovariant extension

where P is an unknown characteristic class that depends only on the topology, and inparticular not on the fluxes. Diaconescu, Freed & Moore (2007) have found aconstraint on P using the E8 gauge theory approach to M-theory pioneered byDiaconescu, Moore, and Witten.


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Thus D-branes in IIB are not classified by twisted K-theory after all, but someunknown S-duality-covariant object that inevitably also classifies both fundamentalstrings and NS5-branes.

However the MMS prescription for calculating twisted K-theory is easily S-covariantized, as the Freed-Witten anomalies respect S-duality. Thus the S-

covariantized form of the MMS construction may be applied to construct the S-covariantized twisted K-theory, as a set, without knowing having any geometricdescription for just what this strange covariant object is. This program has beencarried out in a number of papers, such as Evslin & Varadarajan (2003) and Evslin(2003a), and was also applied to the classification of fluxes by Evslin (2003b).Bouwknegt et al. (2006) use this approach to prove Diaconescu, Moore, and Witten'sconjectured constraint on the 3-fluxes, and they show that there is an additional termequal to the D3-brane charge. Evslin (2006) shows that the Klebanov-Strasslercascade of Seiberg dualities consists of a series of S-dual MMS instantons, one foreach Seiberg duality. The group, of universality classes of the supersymmetricgauge theory is then shown to agree with the S-dual twisted K-theory and not withthe original twisted K-theory.

Some authors have proposed radically different solutions to this puzzle. For example,Kriz & Sati (2005) propose that instead of twisted K-theory, II string theoryconfigurations should be classified by elliptic cohomology.Researchers

Prominent researchers in this area include Peter Bouwknegt, Angel Uranga, EmanuelDiaconescu, Gregory Moore, Anton Kapustin, Jonathan Rosenberg, Ruben Minasian,Amihay Hanany, Hisham Sati, Nathan Seiberg, Juan Maldacena, Daniel Freed, andIgor Kriz.See alsoTwisted K-theoryRamond-Ramond fieldKalb-Ramond fieldD-brane

http://en.wikipedia.org/wiki/Twisted_K-theory

More specifically, twisted K-theory with twist H is a particular variant of K-theory, inwhich the twist is given by an integral 3-dimensional cohomology class. It is specialamong the various twists that K-theory admits for two reasons. First, it admits ageometric formulation. This was provided in two steps ; the first one was done in1970 (Publ. Math. de l'IHES) by Peter Donovan and Max Karoubi [1]; the second onein 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the BundleTheoretic Point of View.

In physics, it has been conjectured to classify D-branes, Ramond-Ramond fieldstrengths and in some cases even spinors in type II string theory. For more

information on twisted K-theory in string theory, see K-theory (physics).

In the broader context of K-theory, in each subject it has numerous isomorphicformulations and, in many cases, isomorphisms relating definitions in varioussubjects have been proven. It also has numerous deformations, for example, inabstract algebra K-theory may be twisted by any integral cohomology class.Contents[hide]1 The definition2 What is it?


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3 How to calculate it3.1 Example: the three-sphere4 See also5 References6 External links

[edit]The definition

To motivate Rosenberg's geometric formulation of twisted K-theory, start from theAtiyah-Jnich theorem, stating that,

the Fredholm operators on Hilbert space , is a classifying space for ordinary,untwisted K-theory. This means that the K-theory of the space M consists of thehomotopy classes of maps

from M to .

A slightly more complicated way of saying the same thing is as follows. Consider thetrivial bundle of over M, that is, the Cartesian product of M and . Then the K-theory ofM consists of the homotopy classes of sections of this bundle.

We can make this yet more complicated by introducing a trivial

bundle P over M, where is the group of projective unitary operators on the Hilbertspace . Then the group of maps

from P to which are equivariant under an action of is equivalent to the originalgroups of maps

This more complicated construction of ordinary K-theory is naturally generalized tothe twisted case. To see this, note that bundles on M are classified by elements H ofthe third integral cohomology group of M. This is a consequence of the fact thattopologically is a representative Eilenberg-MacLane space.

The generalization is then straightforward. Rosenberg has definedKH(M),

the twisted K-theory of M with twist given by the 3-class H, to be the space ofhomotopy classes of sections of the trivial bundle over M that are covariant with

respect to a bundle PH fibered over M with 3-class H, that is

Equivalently, it is the space of homotopy classes of sections of the bundlesassociated to a bundle with class H.[edit]What is it?

When H is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring.


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However when H is nontrivial this theory is no longer a ring. It has an addition, but itis no longer closed under multiplication.

However, the direct sum of the twisted K-theories of M with all possible twists is aring. In particular, the product of an element of K-theory with twist H with an elementof K-theory with twist H' is an element of K-theory twisted by H+H'. This element can

be constructed directly from the above definition by using adjoints of Fredholmoperators and construct a specific 2 x 2 matrix out of them (see the reference 1,where a more natural and general Z/2-graded version is also presented). In particulartwisted K-theory is a module over classical K-theory.[edit]How to calculate it

Physicist typically want to calculate twisted K-theory using the Atiyah-Hirzebruchspectral sequence.[1] The idea is that one begins with all of the even or all of the oddintegral cohomology, depending on whether one wishes to calculate the twisted K0 orthe twisted K1, and then one takes the cohomology with respect to a series ofdifferential operators. The first operator, d3, for example, is the sum of the three-class H, which in string theory corresponds to the Neveu-Schwarz 3-form, and thethird Steenrod square. [2] No elementary form for the next operator, d5, has beenfound, although several conjectured forms exist. Higher operators do not contributeto the K-theory of a 10-manifold, which is the dimension of interest in criticalsuperstring theory. Over the rationals Michael Atiyah and Graeme Segal have shownthat all of the differentials reduce to Massey products of H.[3]

After taking the cohomology with respect to the full series of differentials one obtainstwisted K-theory as a set, but to obtain the full group structure one in general needsto solve an extension problem.[edit]Example: the three-sphere

The three-sphere, S3, has trivial cohomology except for H0(S3) and H3(S3) which areboth isomorphic to the integers. Thus the eve

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