Indonesian Journal of PhysicsKontribusi Fisika Indonesia

Vol. 14 No. 4, October 2003

D-Branes and M-Theory in Supersting Theories

Freddy P Zen∗ and Jusak S Kosasih†

Theoretical Physics Laboratory, Department of PhysicsInstitut Teknologi Bandung, Jl Ganesa 10, Bandung, INDONESIA

∗fpzen@fi.itb.ac.id†jusak@fi.itb.ac.id

Abstract

D-Branes in (type IIA and IIB) superstring theories are investigated in spacetime of dimension d = 10. TypeIIA and IIB superstrings are derived using GSO (Gliozzi, Scherk, Olive) projection on Ramond-Ramond (RR)sector. The degrees of freedom in that theory contains extended stable objects D-branes with field strengthsF (1), F (3), F (5) for type IIB and F (2), F (4) for type IIA, in addition to graviton, dilaton and antisymmetrictensor of rank 2. From the T-duality for open strings, we also show the existence of D-branes from the Dirichletboundary conditions. At the end we review the dualities that relate all five of the 10 dimensional superstringtheories and a quantum extension of 11 dimensional supergravity called M theory as well as the recently foundcorrespondence between string theory and gauge theory, the AdS/CFT conjecture.

Keywords: String Theories, D-Branes, M-Theory

1. Bosonic String at Low Energy

The classical action of a bosonic string can becast in a polynomial form known as Polyakov actionwhich is given by the following formula

SB = −T2

∫

d2σ√−hhαβ∂αX

µ∂βXνGµν(X) (1)

where Gµν(X) is the spacetime metric and hαβ isthe metric of the two-dimensional string world-sheet,Xµ(τ, σ). Here 0 ≤ σ ≤ π is the spatial coordiatealong the string, while τ ∈ R describes its propaga-tion in time. T plays the role of the string tension.The action (1) has two local “gauge symmetries”,that is reparametrization invariance and the Weylor conformal invariance. Because of these two lo-cal symmetries, there only left one degree of freedom(from the original four degrees of freedom) and there-fore we can select a gauge in which the three func-tions residing in the symmetric world-sheet metricare expressed in terms of just a single function. Ifthe string world-sheet does not contain non-trivialtopology (in certain gauge) then in light-cone coor-dinate σ± = τ ± σ, the action (1) can be writtenas

SB = 2T

∫

d2σ ∂+Xµ∂−X

νGµν(X). (2)

The consistency of the above gauge choice is guaran-teed if

δSB

δhαβ= Tαβ = 0, (3)

which is also a constraint in the derivation of theequation of motion from the action (2).

The equation of motion for bosonic stringcan be derived by applying the variational princi-ple, δXµ, to the 1+1-dimensional action (2) and in-tegrating by parts. For closed strings, the periodicboundary conditions are imposed as follows

Xµ(σ = π) = Xµ(σ = 0). (4)

Then the equation of motion is, using the boundarycondition, the two dimensional wave equation

∂+∂−Xµ = 0. (5)

The solution to this equation of motion can be writ-ten in Fourier-mode series expansions, which can begrouped into left-moving and right-moving terms

Xµ = XµL +Xµ

R,

XµL =

1

2xµ +

α′

2pµ

L(τ + σ) +

+i

√

α′

2

∑

n6=0

1

nαµ

ne−in(τ+σ), (6)

XµR =

1

2xµ +

α′

2pµ

R(τ − σ) +

+i

√

α′

2

∑

n6=0

1

nαµ

ne−in(τ−σ). (7)

The operators in the above equation satisfy the fol-

121

122 IJP Vol. 14 No. 4, 2003

lowing commutation relations

[xµ, pν ] = iηµν ,

[αµm, α

νn] = [αµ

m, ανn] = mδm+n,0η

µν (8)

and the rest are all zeros. From the creation op-erators αµ

−n(n > 0) and the annihilation operatorsαµ−n(n < 0), one can construct Fock space for the

left-moving part

|N〉 =r∏

p=1

(α−np)ip |0〉, (9)

on which the left-moving and right-moving numberoperators, NL and NR, act as follows

NL|N〉 =

∞∑

n=1

αµ−nα

νnηµν |N〉 =

(

r∑

p=1

ipnp

)

|N〉,

NR|N〉 = 0. (10)

Similarly, one can also build Fock space for the right-moving part.

Using the above oscillator algebra one can nowfind the physical string spectrum explicitly. For thatpurpose, we define the “physical states” |phys〉 of thefull Hilbert space to be those which obey the Vira-soro constraints (3). In Fourier-mode and at quan-tum level (after normal ordering of the creation andannihilation operators), the constraints become

(L0 − 1)|phys〉 = 0,

(L0 − 1)|phys〉 = 0, (11)

where

L0 = −α′

4M2 +NL,

L0 = −α′

4M2 +NR, (12)

and M2 = −pµpµ. The physical spectrum is derived

from the above conditions and it is given by the fol-lowing equation

α′M2 = 4NL − 4, NL = NR. (13)

For example, the lowest (ground) stateNL = NR = 0describes a tachyon α′M2 = −4. The first exitedstate NL = NR = 1 describes massless state givenby the combinations of αµ

−1αν−1. This state can be

decomposed into irreducible representations of thespacetime little group SO(24), which is the residualLorentz symmetry group that remains after the Vira-soro constraints have been taken into account. Thisdecomposition is given by

24⊗ 24 = S⊕A⊕ 1. (14)

where the symmetric traceless tensor correspondsto the spin-2 graviton with Ricci tensor Rµν , the

Kalb-Ramon antisymmetric spin-2 tensor is calledthe Neveu-Schwarz B-field with the field strengthHµνρ, while the scalar field φ is the spin-0 dilaton.Using similar procedures, one can find the next ex-cited levels. Now the effective low energy action (in-cluding up to NL = NR = 1 state) in 26-dimensionalbosonic string theory is given by

S26 = − 1

2K2

∫

d26X√−Ge−2φ(R− 4∇µφ∇µφ+

+1

12HµνρH

µνρ). (15)

2. Type II Superstring

We will now add fermions to the bosonic stringto produce quite naturally supersymmetric stringtheory, called superstring for short. The introduc-tion of supersymmetry in string theory will be carriedout using Ramond-Neveu-Schwarz (RNS) formalism.Starting with the gauge-fixed Polyakov action, theworldsheet action now takes the form

S = −T2

∫

d2σ (∂αXµ∂αXµ − iψµρα∂αψµ). (16)

Here ψµ(µ = 0, 1, . . . , 9) are two-component Majo-rana spinors in the worldsheet and ρα, (α = 0, 1) areDirac matrices in 2-dimension where

ψµ = ψ†µρ0 (17)

Besides the bosonic string symmetries, the action(16) is also invariant under the global, infinitesimalworldsheet supersymmetric transformations

δXµ = εψµ

δψµ = −iρα∂αXµε (18)

where the constant parameter ε is a Majorana spinor.The mode decomposition for Xµ(τ, σ) are exactlyas before, but now we need to consider boundarycondition for the free fermionic fields ψµ

A(τ, σ) whereA = ± is the light-cone index. There are two possibleboundary conditions consistent with the Dirac equa-tion of motion and Lorentz invariance, the periodicboundary condition, called Ramond (R) boundarycondition

ψµA(σ = π) = ψµ

A(σ = 0) (19)

and the antiperiodic boundary condition, calledNeveu-Schwarz (NS) boundary condition

ψµA(σ = π) = −ψµ

A(σ = 0). (20)

The R sector will give particles which are space-time fermions while the NS sector will yield space-time bosons. Corresponding to the different pair-ings between left-moving and right-moving modes|left〉 ⊗ |right〉, there are now four distinct closed

IJP Vol. 14 No. 4, 2003 123

string sectors, namely, NS-NS and R-R (bosons), NS-R and R-NS (fermions).

Using the boundary conditions (19) and (20),the mode decompositions of the general solutions ofthe fermionic fields can be expressed as the Fourierseries

ψµ− =

∑

r∈Z+ 1

2

bµr e−ir(τ−σ), (21)

ψµ+ =

∑

r∈Z+ 1

2

bµr e−ir(τ+σ), (22)

for the NS sector and

ψµ− =

∑

m∈Z

dµme

−im(τ−σ), (23)

ψµ+ =

∑

m∈Z

dµme

−im(τ+σ), (24)

for the R sector. Then, the anticommutation rela-tions of the operators in the NS sector are

bµr , bνr = ηµνδr+s,0 (25)

bµr , bνr = ηµνδr+s,0 (26)

and for the R sector are given by

dµm, d

νn = ηµνδm+n,0 (27)

bµm, bνn = ηµνδm+n,0. (28)

In the R sector, taking m = n = 0, the anticommu-tation relation now becomes (e.g. for right-moving)

dµ0 , d

ν0 = ηµν , (29)

which define a Clifford algebra in (9+1)-dimension.This can be seen explicitly by defining Γµ = i

√2dµ

0 ,such that (29) can now be written as

Γµ,Γν = −2ηµν . (30)

It is now clear that the vacuum state for the R sec-tor, i.e. |0〉aR(a = 1, . . . , 32) is a spinor with 32components (in 10-dimension) transforming underSO(9, 1). The chirality operator can now be definedas

Γ11 = Γ0Γ1 · · ·Γ9 (31)

so

Γ11|0〉+R = +|0〉+R (32)

Γ11|0〉−R = −|0〉−R. (33)

Here |0〉+R and |0〉−R are Majorana-Weyl spinors with16 components which are the decomposition of |0〉aR.

Eventhough we have included supersymmetryinto the theory, there are still left over problems, e.g.the tachyon still appears in the superstring theory.Gliozzi, Scherk and Olive (GSO) proposed an idea

in projecting (truncating) the superstring spectrumconsistently into a subspace without tachyons. Forthe NS sector, GSO projection truncates the spec-trum by removing all states with an even number offermions, such that

(−1)fb |phys〉NS = −|phys〉NS, (34)

where fb is the fermion number operator. For theR-R sector, since the vacuum state has different chi-rality, then the consistent GSO projection is

Γ11(−1)fd |phys〉R = ±|phys〉R. (35)

For the vacuum state fd = 0, the GSO projection on|0〉R is

Γ11|0〉rmR = ±|0〉R, (36)

so the massless fermions are Majorana-Weyl fermionswith a given spacetime chirality. When left-moversand right-movers are combined together with par-ticular choices of chiralities, there are four possiblechoices. Of these, one can indentify two inequiva-lent possibilities corresponding to the relative chiral-ity between the surviving R sector Majorana-Weylspinors, which are of opposite chirality (called TypeIIA, a non-chiral theory) and of the same chirality(called Type IIB, a chiral theory).

Both Type IIA and Type IIB of the R-R sec-tors have a profound implications to the solutions ofthe superstring theory. Consider the GSO projectionon the vertex operator which is defined as follows

VRR = Fab(p)Sa(iΓ0S)beip·x. (37)

Here Fab is the field-strength tensor

Fab =

10∑

k=0

ik

k!Fµ1···µk

(Γµ1···µk )ab, (38)

Sa is the spin operator and Fµ1 ···µkis a differential

k-form. For fd = 0, the GSO projection on Fab andSa in (37) gives

Type IIB : Γ11F = −FΓ11 = F, (39)

Type IIA : Γ11F = +FΓ11 = F. (40)

The above equation limits the number of differentialforms that can be contained in Type IIA and TypeIIB, namely

Type IIB : Fµ1 ···µkwith odd k, (41)

Type IIA : Fµ1 ···µkwith even k. (42)

Now the general form of action for the low energylimit is given by

S10 = − 1

2K2

∫

d10X√−G

[

e−2φ(R − 4∇µφ∇µφ+

+1

12HµνρH

µνρ) +

5∑

k=0

1

2k!F 2

(k)

]

+fermions + gauge fields. (43)

124 IJP Vol. 14 No. 4, 2003

From this action one can deduce that for the R-Rsector in Type II superstring theory, besides gravi-ton, Kalb-Ramond and dilaton fields, there are otherfields contained in field-strength tensor F . Usingthe analogy with gauge particles with 2-form field-strength tensor and 1-form charge, the F in (43) canbe intrepreted as “particle” or stable object. Sincethe field-strength tensor is not only 2-form, then thisobject is extended to what is called Dp-brane. Thisobject has a charge as follows

qp

∫

p−branes

C(p+1) (44)

where C(p+1) is the (p+ 1)-form potential field with(p+ 2)-form field-strength F(p+2).

For completeness, we will give a brief descrip-tion of heterotic string theories. The independenceof the left-movers and right-movers in closed stringtheories enables us to construct a closed string the-ory in which the left-moving sector is of one type andthe right-moving sector is of another type. The socalled heterotic string theory comprises a heterosis oftwo string theories. The theory acquires spacetimesupersymmetry by assuming that the right-movingsector is 10-dimensional supertring. On the otherhand, one use the 26-dimensional bosonic string forthe left-movers, which gives ‘internal’ (gauge) degreeof freedom. The following action gives the heteroticstring1)

S = − 1

2π

∫

d2σ

9∑

µ=0

(∂αXµ∂αXµ − 2iψµ

−∂+ψµ−)+

−2i32∑

A=1

λA+∂−λ

A+

, (45)

where ψµ and λA are Majorana-Weyl fermions. Thefirst term is the bosonic part and the second term isfor the fermions. The remaining gauge scalars are 16right-moving degrees of freedom required by N = 1supersymmetry which can be expressed as 32 realfermionic fields or 16 complex bosonic fields. TheXµ and ψµ have total central charges (c, c) = (10, 10)and the central charges of ghost fields can be addeduntil (cg , cg) = (−26,−10), so the remaining centralcharges are (c, c) = (16, 0). The simplest possibilityis to take 32 left-moving Majorana-Weyl fermions,λA(z), A = 1 . . . , 32.

The additional internal degree of freedomcomes from a 16-dimensional self-dual lattice. Thereare two constraints on the lattice as required by mod-ular invariance. Therefore there are only two suchlattices, corresponding to the weight lattices of theLie groups, E8 × E8 and SO(32). If all λA satisfythe same boundary conditions, then we have SO(32)symmetry, if not all of λA satisfy the same boundaryconditions, then we have E8 ×E8 symmetry.

3. T-Duality and D-Branes

T-duality symmetry is an important conceptand indispensible tool in string theory, similar to S-duality in gauge theory. To systematically demon-strate the existence of D-branes in superstring the-ory, we will describe individually the T-duality sym-metries of closed strings, open strings, and super-strings.

We already know that superstring theoriescan exist only in 10 dimensional spacetime and thiscontradicts our daily experiences and observations.Therefore we need to compactify the extra dimen-sions in order to get a physically acceptable theory.Here we will compactify the extra dimensions follow-ing the Kaluza-Klein procedure. For this, we com-pactify x9 on a circle S1 of radiusR. This means thatthe spacetime coordinate is periodically identified as

X9 = X9 + 2mπR ,m ∈ Z. (46)

For closed strings, the coordinate X9 is nolonger a periodic function and the condition

X9(τ, σ + 2π) = X9(τ, σ) + 2πmR (47)

is an allowed transformation for any integerm. From(47) we can see that m is the winding number of theclosed string, i.e. the number of times the string canwind around the compactified spacetime circle. Fora fixed winding number m ∈ Z, this is representedby adding the term mR/σ to the mode expansion

Xµ(τ, σ) = XµL (τ + σ) +Xµ

R(τ − σ)

= xµ0 + xµ

0 +

√

α′

2

(

αµ0 + αµ

0

)

τ +

+

√

α′

2

(

αµ0 − αµ

0

)

σ + (oscillators) ,

(48)

for X9(τ, σ) and this yields a constraint

α90 − α9

0 = mR

√

2

α′. (49)

On the other hand, single-valuedness of the basiswavefunction eip9

0X9

is satisfied if the momentum p90

is quantized according to

p90 =

n

R, n ∈ Z (50)

which is a requirement for any quantum system.Therefore we also have another constraint

α90 + α9

0 =2n

R

√

α′

2. (51)

From (49) and (51) we get

α90 = pL

√

α′

2, pL =

n

R+mR

α′, (52)

α90 = pR

√

α′

2, pR =

n

R− mR

α′, (53)

IJP Vol. 14 No. 4, 2003 125

where pL and pR are called the left-moving and right-moving momenta.

Let us now consider the mass spectrum in theremaining uncompactified 1+8 dimensions, which isgiven by

m2 = −pµ pµ , µ = 0, 1, . . . , 8

=2

α′

(

α90

)2

+4

α′

(

N − 1)

=2

α′

(

α90

)2

+4

α′

(

N − 1)

, (54)

where the second and third equalities come from theL0 = 1 and L0 = 1 constraints, respectively.

The constructed compactified string theoryhas an interesting feature. The mass spectrum (54)is invariant under the simultaneous exchanges

n ←→ w , R ←→ R′ =α′

R, (55)

which by (53) is equivalent to the transformations

α90 7−→ α9

0 , α90 7−→ − α9

0 . (56)

This symmetry of the compactified string theory iscalled T-duality symmetry. We can also generalizethis symmetry at the full interacting level of massivestates by a spacetime parity transformation of theworldsheet right-movers as

T : X9L(τ + σ) 7−→ X9

L(τ + σ) ,

X9R(τ − σ) 7−→ −X9

R(τ − σ) . (57)

For open strings, we have no quantum num-ber comparable to m since open strings do not windaround the periodic direction of spacetime. There-fore one cannot exchange m ↔ n and we can con-clude that an open strings theory cannot be T-dualto itself. In order to find out what happen if we T-dualize an open strings theory, write the open stringmode expansion

Xµ(τ ± σ) =xµ

0

2± x′ µ0

2+

√

α′

2αµ

0 (τ ± σ) +

+i

√

α′

2

∑

n6=0

αµn

ne−i n(τ±σ) ,

(58)

withαµ

0 =√

2α′ pµ0 (59)

and the total embedding coordinates

Xµ(τ, σ) = Xµ(τ + σ) +Xµ(τ − σ)

= xµ0 + α′pµ

0 τ + i√

2α′∑

n6=0

αµn

ne−i nτ cos(nσ) . (60)

The T-dual open string coordinate can be found fromthe closed ones by reflecting the right-movers, i.e. set

X9(τ+σ)→ X9(τ+σ) andX9(τ−σ)→ −X9(τ−σ).The resulting embedding function is given by

X ′9(τ + σ) = X9(τ, σ) −X9(τ − σ)

= x′9 + 2α′ n

Rσ +√

2α′∑

n6=0

α9n

ne−inτ sinnσ (61)

The exchange of the cosine and the sine factors in theoscillator term indicates that T-duality exchangesNeumann boundary condition and Dirichlet bound-ary condition

∂σX9|σ=0,π = 0 −→ ∂τX

′9|σ=0,π = 0. (62)

This Dirichlet boundary condition shows that theopen string endpoints are at a fixed place in space-time and are equal up to the periodicity of the T-dualdimension (open string of winding number n ∈ Z)

X ′ 9(τ, π) −X ′ 9(τ, 0) =2πα′ n

R= 2πnR′ . (63)

The open string ends are still free to move in theother 1+8 directions that are not T-dualized, whichconstitute a hyperplane called a D-brane, or morespecifically a D8-brane. In general, T-dualizing mdirections of the spacetime gives Dirichlet boundaryconditions in the m directions, and hence a hyper-plance with p = 9 −m spatial dimensions which wecall a Dp-brane.2)

Generalizing the previous results to super-strings, let us consider the effects of T-duality on theclosed, oriented Type II theories. As we have seen,T-duality can be seen as a right-handed parity trans-formation for the right-movers. In other words, T-duality works on the right-movers of the closed stringtheories and changes sign on all oscillation terms inthat right-moving mode and hence T : Γ11 7→ −Γ11.Thus the relative chirality between left-movers andright-movers is flipped, i.e. T-duality reverses thesign of the GSO projection on right-movers

T : P±GSO 7−→ P∓

GSO . (64)

In conclusion, T-duality interchanges theType IIA and Type IIB superstring theories. It isonly a symmetry of the closed string sector, since inthe open string sector it relates two different typesof theories.

4. M-Theory

Supersymmetry representation theory showedthat ten is the largest spacetime dimension in whichthere can be a matter theory with spins ≤ 1. Thisis realized as 10 dimensional super Yang-Mills the-ory which is a very symmetrical classical field theory,but at the quantum level it is both nonrenormaliz-able and anomalous for any nonabelian gauge group.

126 IJP Vol. 14 No. 4, 2003

These problems can be overcome for suitable gaugegroups, i.e. SO(32) or E8 × E8, when the Yang-Mills theory is embedded in a type I or heteroticstring theory. On the other hand, the largest pos-sible spacetime dimension for a supergravity theorywith spins ≤ 2 is eleven. The the 11 dimensional su-pergravity theory is a non-chiral classical field theoryso it has no anomaly problems, but it is also non-renormalizable which also renders it unqualified fora fundamental theory.

In short, there are five consistent superstringtheories which appears to be different from the eachothers and one supergravity theory in 11 dimen-sional. The 11 dimensional supergravity theory isnow believed to be a low-energy effective descriptionof a non-perturbative theory called M-theory. More-over, Witten put forward a convincing case that thedistinction between string theories is just an arti-fact of perturbation theory and hence they are non-perturbatively equivalent. In other words, these fivetheories are just different corners of a deeper theory,or technically speaking, the five string theories andM-theory represent six different special points in themoduli space of the underlying theory. These theo-ries are related by dualities as shown in the picturebelow.

M-theory describes supersymmetric extendedobject with two spatial dimensions (supermem-branes) and its dual object with five spatial dimen-sions (superfivebranes). The relation between themembrane and the fivebrane in 11 dimensions is anal-ogous to the relation beween electric and magneticcharges in 4 dimensions. This is more than just ananalogy because now we can see the importance ofduality in this picture. Electric-magnetic duality in4 dimensional string theory follows as a consequenceof string-string duality in 6 dimensions, which fol-lows, in its turn, as a consequence of membrane-fivebrane duality in 11 dimensions. In particular,heterotic-heterotic duality, Type IIA-heterotic dual-ity, heterotic-Type IIA duality, and Type IIA-TypeIIA duality follow from membrane-fivebrane dualityby compactifying M-theory on certain manifolds. Inaddition, the more realistic kinds of electric-magneticduality envisioned by Seiberg and Witten4, 5) can alsobe explained in terms of string-string duality andhence M-theory.

M-theory will also have important conse-quences in particle physics phenomenology and cos-mology. As we know, in the supersymmetric exten-sions of the standard model, the coupling constantsα3, α2, α1 associated with the SU(3) × SU(2)L ×U(1)Y all meet at about 1016 GeV, entirely consis-tent with the idea of grand unification. The strengthof the dimensionless number αG related to gravita-tion also almost meets the other three, but not quite,which is something that frustrate the theorists. How-

ever, Witten has proposed that spacetime is approx-imately a narrow five dimensional layer bounded byfour dimensional walls. The particle of the standardmodel live on the walls but gravity lives in the fivedimensional bulk, which explains why gravity cou-ples so weakly to matter. As a result, it is possibleto choose the size ot this fifth dimension so that allfour forces meet at this common scale. This scaleis much less than the Planck scale of 1019 GeV andhence would have all kinds of cosmological conse-quences. Other impact of M-theory and D-branes inhigh energy physics and cosmology is a microscopicexplanation of black hole entropy and the rate ofemission of thermal (Hawking) radiation for blackholes in string theory.6, 7)

5. AdS/CFT Correspondence

Finally, the story is not complete withoutmentioning the recently found correspondence ofgauge theory and gravity, popularly known asAdS/CFT correspondence.8, 9) It started from thestudy of quantum gravity, especially the informationparadox of black hole which is related to the prob-lem in quantizing gravity. On the other hand, stringtheory has a large class of black hole solutions whichwere called black p-brane (the 0-brane is the stan-dard black hole), so to address the question of the in-formation paradox one needs to know what happensif a string is thrown into the black p-brane. Thereare two ways to approach this problem, i.e. to studythe propagation of the string in the black p-branegeometry, or alternatively, to use the collective coor-dinates of the p-brane in which one try to formulatethe problem as interactions between the string andthe collective coordinates of the p-brane (D-brane).At low energy, the intrinsic degrees of freedom of thisD-brane is described by a gauge theory in (p+1) di-mensions. The equivalence of these two descriptionsis the origin of the connection between gauge theoryand string theory and its evidence have crystallizedon the work of Maldacena. The conjecture is thatsuperstring theory in a curved 10-dimensional space,which is 5-dimensional anti-de Sitter space times 5-dimensional sphere (AdS5 × S5), is equivalent to agauge theory in 4 dimensions with N = 4 supersym-metry (super Yang-Mills theory) which is a confor-mal field theory (CFT). Thus the conjecture is calledthe AdS/CFT correspondence. More precisely, theconjecture states that Type IIB superstring theoryon AdS5 × S5 is equivalent to the N = 4 super-symmetric gauge theory in 4 dimensions with gaugegroup SU(N). More surprisingly, the string theoryis fully equivalent to the gauge theory although theylive in different spacetime dimensions (this is differ-ent from Kaluza-Klein compactification). Anothersurprise is that string theory contains gravity andthe gauge theory doesn’t so it is suggested that in

IJP Vol. 14 No. 4, 2003 127

M−Theory

I IIA IIB SO(32) E x E8 8

11D

10D

9D

S−Duality

T−Duality onT−Duality on

T−Duality onT−Duality on

1 1

1

9 9

10101

g gs s

o ooo

S R S Rx x

S RxI Rx

Figure 1. The various duality transformations that relate the superstring theories in nine and ten dimensions.T-Duality inverts the radius R of the circle S1, or the length of the finite interval I1, along which a singledirection of the spacetime is compactified, i.e. R 7→ `2P/R. S-duality inverts the (dimensionless) string couplingconstant gs, gs 7→ 1/gs, and is the analog of electric-magnetic duality (or strong-weak coupling duality) in four-dimensional gauge theories. M-Theory originates as the strong coupling limit of either the Type IIA or E8×E8

heterotic string theories.3)

quantum gravity the information of the theory can bestored in lower dimensions, i.e. the idea of hologra-phy of quantum gravity. Thus, the quantum theoryof gravity should be described by a sort of topolog-ical quantum theory in the sense that all degrees offreedom could be projected onto the boundary and,moreover, one can hope to learn much about quan-tum gravity using the armory of gauge theory.

In the original formulation of the conjecture,Maldacena looked at the black D3-brane in super-gravity and superstring theory. For N parallel sep-arated D3-branes, the end points of an open stringmay be attached to the same brane which induce amasless U(1)N gauge theory with N = 4 supersym-metry in the low energy limit. However, an openstring can also have its ends attached to differentbranes and the mass of such string cannot get arbi-trarily small. For N coincident branes, the U(1)N

gauge symmetry is enhanced to the full U(N) gaugesymmetry. In the low energy limit, N coincidentbranes support an N = 4 supersymmetric Yang-Mills in 4 dimensions with gauge group SU(N) inthe low energy limit.

The D3-brane solution is chosen because of itsspecial and interesting properties:

• Its worldbrane has 4 dimensional Poincare in-variance.

• It has constant axion and dilaton fields.

• It is regular at y = 0, where yµ is the coordinatesin the transverse dimension.

• It is self-dual.

The spacetime metric of N coincident D3-branes can be written in the following form,

ds2 =

(

1+R4

y4

)− 1

2

ηijdxidxj+

(

1+R4

y4

)1

2

(dy2+y2dΩ25)

(65)where i, j, · · · are for the 4 dimensional Minkowskispace with metric ηij = diag(−+++) and the radius

R of the D3-brane is given by

R4 = 4πgsNα′2. (66)

The flat space-time R10 limit is reached when

y R. For y < R, the geometry is often referred toas the throat. As y R the geometry would appearto be singular, but a redefinition of the coordinate

u ≡ R2/y (67)

and using the u → ∞ limit, transform the metricinto the following asymptotic form

ds2 = R2

[

1

u2ηijdx

idxj +du2

u2+ dΩ2

5

]

(68)

which corresponds to a product geometry AdS5×S5

with identical radius R for both components. This

128 IJP Vol. 14 No. 4, 2003

Table 1. The three forms of the AdS/CFT conjecturein order of decreasing strength

N = 4 conformal SYM Quantum Type IIB stringall N , gY M ⇔ theory on AdS5 × S5

(gs = g2

Y M ) (R4 = 4πgsNα′2)

‘t Hooft limit of N = 4 SYM Classical Type IIB stringλ = g2

Y M N fixed, N → ∞ ⇔ theory on AdS5 × S5

(1/N expansion) (gs string loop expansion)

Large λ limit of N = 4 SYM Classical Type IIB SUGRAfor N → ∞ ⇔ on AdS5 × S5

(λ−1/2 expansion) (α′ expansion)

shows that the geometry close to the brane (y ∼ 0)is regular and highly symmetrical.

The Maldacena limit8) corresponds to keepingfixed gs and N as well as all physical length scales,while letting α′ → 0. Remarkably, this limit of stringtheory exists and is interesting. In the Maldacenalimit, only the AdS5 × S5 region of the D3-branegeometry survives the limit and contributes to thestring dynamics of physical processes, while the dy-namics in the asymptotically flat region decouplesfrom the theory.

The AdS/CFT or Maldacena conjecture statesthe equivalence (also referred to as duality) betweenthe following theories8)

• Type IIB superstring theory on AdS5×S5 whereboth AdS5 and S5 have the same radius R,where the 5-form F+

5 has integer flux N =∫

S5 F+5 and where the string coupling is gs;

• N = 4 super Yang-Mills theory in 4-dimensions,with gauge group SU(N) and Yang-Mills cou-pling gY M in its (super)conformal phase;

with the following identifications between the param-eters of both theories,

gs = g2Y M R4 = 4πgsN(α′)2 (69)

and the axion expectation value equals the SYM in-stanton angle 〈C〉 = θI . The equivalence includes aprecise map between the states (and fields) on the su-perstring side and the local gauge invariant operatorson the N = 4 SYM side, as well as a correspondencebetween the correlators in both theories.

In the strongest form of the conjecture, thecorrespondence is to hold for all values N and allregimes of coupling gs = g2

Y M . Since string theoryquantization on a general curved manifold appearsto be out of reach at present, one has to seek certainlimits of the conjecture which are more tractable but

still remains highly non-trivial, e.g. the ’t Hooft limitand the large λ limit. These variations of the con-jecture are summed up in the table above.

There is a continual and rapid pace progressin the works of the AdS/CFT duality and its impli-cations. We believe that the conjecture will guide usto a better understanding of nonperturbative stringtheory. We have mentioned that the AdS/CFT cor-respondence realizes the idea that the ’t Hooft largeN limit of the SU(N) gauge theory is string the-ory and also the idea that quantum gravity is holo-graphic. Since string theory and gauge theory areequivalent, we may try to use string theory to dodifficult calculations in gauge theory or vice versa.So, at least, string theory is useful for studying var-ious gauge theories in search for the unified theory.

References

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3. Richard J Szabo, “BUSSTEPP Lectures onString Theory,” hep-th/0207142.

4. N Seiberg and E Witten, “MonopoleCondensation, And Confinement In N=2Supersymmetric Yang-Mills Theory,” Nucl.Phys. B426 (1994) 19, hep-th/9407087.

5. N Seiberg and E Witten, “Monopoles, Dualityand Chiral Symmetry Breaking in N=2Supersymmetric QCD,” Nucl. Phys. B431(1994) 484, hep-th/9408099.

6. C G Callan and J M Maldacena, “D-BraneApproach to Black Hole Quantum Mechanics,”Nucl. Phys. B472 (1996) 591–610,hep-th/9602043.

7. A Strominger and C Vafa, “Microscopic Originof the Bekenstein-Hawking Entropy,” Phys. Lett.B379 (1996) 99–104, hep-th/9601029.

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