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MIT OpenCourseWarehttpocwmitedu

8821 String TheoryFall 2008

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

8102019 Aula 7 - Teoria de cordas MIT

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8821 F2008 Lecture 07

Preview of AdSndashCFT Correspondence String Theory from Gauge Fields and rsquot Hooft Counting

Lecturer

McGreevy

September 25 2008

References

bull Maldacena hepth0309246 sect2 30

bull MAGOO sect12

1 Preview of AdSndashCFT Correspondence

Ideology

fields in AdS larrrarr

local operators of CFT

spin spin

mass scaling dimension ∆

In particular for scalars m2L2 ∆(∆ minus 4) [recall that λ = L4 αprime2]1AdS = AdS

Last time we discussed the KaluzaKlein (KK) harmonics of 10d supergravity (lsquoSUGRArsquo) fields(= massless string modes) These correspond to superconformal primary operators of the N

= 4theory according to

Φ(x y) = φℓ(x)Y ℓ(y)

ℓ

Y ℓ = T i1iℓ

yi1 middot middot middot yiℓ larrrarr T i1iℓ

Tr (X i1 middot middot middot X iℓ)

In which case the massscaling dimension correspondence reads

m(φℓ) = ℓL larrrarr ∆ sim

ℓ

1Wersquoll show this soon

1

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111410911141091114109

11141091114109

On the left hand side this is just the mass of a KK harmonic on a space of size L ie an eigenvalueof the laplacian on the 5sphere On the right hand side we have an operator made from a productof l scalar fields A 4d scalar has engineering dimension 1 so the dimension of the product is l inthe free theory at least (It is actually true for all λ by the BPS property)

The above can be considered as the ldquohighest weightrdquo states Then using the fact that the supersymmetry algebras match on the two sides we can fill out multiplets on both sides by acting withthe generators In this way the correspondence of all SUGRA modes in the N

= 4 theory can befound (an intimidating table can be found in EDrsquoHndashDZF hepth0201253 p50)2

Now consider some observables of a QFT (wersquoll assume Euclidean spacetime for the now)

O1(x1)O2(x2) middot middot middot On(xn)

We can write down a generating functional Z [J ]

L(x) rarr L(x) + J A(x)OA(x) equiv L(x) + LJ (x)

A R

Z [J ] = eminus LJ CFT

where J A(x) are arbitrary functions (sources) and OA(x) is some basis of local operators

The npoint function is then given by

δ On(xn)

= lnZδJ n(xn) J =0

n n

Since LJ is a UV perturbation (∵ it is a perturbation on bare

Lagrangian by local operators) in

AdS it corresponds to perturbation near the boundary (ds2 = dx2r2 +dr2r2 boundary at r = 0)(Recall from the counting of degree of freedom that QFT with UV cutoff E lt 1δ larrrarr

AdS cutoffr gt δ ) The perturbation J will be encoded in the boundary condition on bulk fields

Incidentally this resolves a huge confusion in GR literature dating back to 1970rsquos which claimsthat ldquoCauchy problem3 in AdS is not wellposedrdquo What happens is that signals can reach from thesurface where initial data is supplied to the boundary in finite time4 and hence boundary conditionis needed for the problem to be wellposed

The idea (GKPW) for computing Z [J ] is then

R

Z [J ] = eminus LJ CFT = Z strings[bc depends on J ] sim

eS SUGRA EOM bc depend on J gsrarr0L2αprime rarrinfin=

Note that the limit gs rarr 0 L2αprime rarr infin in AdS corresponds to the limit N rarr infin λrarr infin in QFT

2Note that Tr (X i1 middot middot middotX iℓ) are chiral (QO = 0) primaries (K O = 0) and hence are short (BPS) multipletsThe correspondences of long multiplets is another story which wersquoll get to soon

3A Cauchy problem is one in which initial data is specified in some space-like slice and one is asked to determinethe future evolution of the system

4We will study the geometry of AdS in detail next week

2

8102019 Aula 7 - Teoria de cordas MIT

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As an example of the application of this formula one can compute the 3point function of supergravity states It turns out that the results agree exactly with N = 4 SYM (at leading order in1N ) A priori this is a surprising statement since the field theory calculation is done by pertur-bation theory at small λ = g2 N while the gravity calculation is meant to work at large λ ThisY M

happens because all the supergravity states correspond to BPS (= rdquochiral primaryrdquo) operators

and hence their 3point functions actually do not depend on λ This specialness of the states wersquovebeen able to map so far may worry you how do we describe the rest of the field theory operatorsusing the bulk theory To approach this question we take a small detour

2 Strings from Gauge Fields

There are several longsuspected connection between strings and gauge fields

21

Strings

as

model

of

hadrons

(late

60rsquos)

For hadron with largest spin J for a given mass

J = αprimem2(J ) + const ldquoRegge trajectoryrdquo

which is precisely the form of the spectrum of the vibration modes of a spinning quantum relativisticstring in RD

22

Flux

tubes

in

QCD

We know that the gauge field in QCD is in the confined phase (as opposed to the Coulomb phasesee Fig 1(a)(b) for illustration) Consequently the field lines bunch together and behave like atensionful flux tube

More concretely consider the area law for Wilson loopR

Tr P ei

A =eminusArea()middottension

Here denotes the worldsheet as shown in Fig 1(c) This is an order parameter for confinementand indicates that flux tube has a finite tension if and only if the gauge theory is in the confinedphase

The above suggests that we may try to use flux tubes as microscopic variables5 which shouldbehave as strings in R31 However a result from Polyakov suggests that this is problematic

Polyakovrsquos result is this in quantizing the string worldsheet with metric γ αβ the transformation

γ αβ rarr eφ(σ)γ αβ is a gauge symmetry and hence δΓ

δφeff = 0 There is however a conformal anomaly

5It will turn out that a string description does

not

require confinement The loophole is that one may considerfunny shaped worldsheets

3

8102019 Aula 7 - Teoria de cordas MIT

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917501

3

worldsheet

loop

ldquordquo

t

x

(a) (b) (c)

Figure 1 Flux tubes in QCD (a) meson in confined phase (b) lsquomesonrsquo in Coulomb phase ie adipole (c) the world sheet of flux tube for computing Wilson loop

which generates kinetic term of φ in Γeff Hence φ should be treated as an extra dimension For QFTin d le

1 the procedure for quantizing φ is known (ldquoLiouvillerdquo) and this is really the correct idea(this leads to what is sometimes called lsquoold matrix modelsrsquo including the c = 1 matrix quantummechanics) In d = 4 this result (so far) serves as more inspiration for our extradimensionalpicture

rsquot Hooft Counting

The most explicit evidence that gauge theory leads to string theory comes from rsquot Hooft counting

b=1N

Consider a (any) quantum field theory with matrix

fields6

Φa=1 N macr (for concreteness wersquoll take thematrix group to be U (N )) and consider a Lagrangian of the form

L sim 2

1Tr

(part Φ)2 + Φ2 + Φ3 + Φ4 +

1048573gYM

˜Rescale Φ = gYMΦ

L sim Tr (part + ˜ Φ3 + g2 Φ4 + Φ)2 Φ2 + gYM YM

Now consider propagators It is convenient to adopt the double

line notation in which oriented

index lines follow conserved color flow so that for propagator7

Φab Φ

dc prop

δ caδ b

d =a

c d

c

By matrix field we mean that their products appear in the Lagrangian only in the form of matrix multiplicationeg (Φ2)ca

= ΦbaΦ

c

b

Had we been considering SU (N ) the result would be Φb

aΦcd prop δ c

aδ bd

minus δ baδ d

cN 2 =

4

6

7

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 67

And similarly for vertices

a

b

c

a

b

c

prop gYM

a a

b

b

cc

d

d

prop g2YM

Now consider the rsquot Hooft limit in which N rarr infin and gYM rarr

0 but which λ = g2 N = constYM

Is this limit classicalfree The answer turns out to be no The loophole is that even though thecoupling goes to zero the number of modes diverges

(a)

prop λ3N 2prop

N 2

prop λN 2

(b)

(c)

Figure 2 Planer graphs that contribute to the vacuumrarrvacuum amplitude

2prop gYMN = λN 0

Figure 3 Nonplaner graph that contributes to the vacuumrarrvacuum amplitude

To see this more concretely consider vacuumrarrvacuum diagrams (see Fig 2 and 3 for illustration)In Fig 2 we have a set of planer graphs whose contributions take the general form λnN 2 Howeverthere are also contributing nonplaner graphs such as the one in Fig 3 whose contribution doesthe take that general form

Every doubleline graph specifics a triangulation of a 2dimensional surface Σ There are two waysto construct the explicit mapping

Method 1 (ldquodirect surfacerdquo) Fill in index loops with little plaquettes

Method

2

(ldquodual

surfacerdquo)

(1) draw a vertex in every index loop and (2) draw an edge acrossevery propagator

These constructions are illustrated in Fig 4 and 5

Comparing the resulting surface and the corresponding contribution it can be seen that the contri-bution of a graph with h handles is proportional to N 2minus2hλn Thus the partition function (whichis also the vacuumrarrvacuum amplitude) must take the form

5

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 77

sim S 2 sim

T 2

(a) (b)

Figure 4 Direct surfaces constructed from the vacuum diagram in (a) Fig 2a and (b) Fig 3

infin

sim S 2

Figure 5 Dual surface constructed from the vaccum diagram in Fig 2c Note that points at infinityare identified

infin

lnZ =N 2minus2hf h(λ)

h=0

which is exactly like the result from string perturbation expansion

6

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 57

917501

3

worldsheet

loop

ldquordquo

t

x

(a) (b) (c)

Figure 1 Flux tubes in QCD (a) meson in confined phase (b) lsquomesonrsquo in Coulomb phase ie adipole (c) the world sheet of flux tube for computing Wilson loop

which generates kinetic term of φ in Γeff Hence φ should be treated as an extra dimension For QFTin d le

1 the procedure for quantizing φ is known (ldquoLiouvillerdquo) and this is really the correct idea(this leads to what is sometimes called lsquoold matrix modelsrsquo including the c = 1 matrix quantummechanics) In d = 4 this result (so far) serves as more inspiration for our extradimensionalpicture

rsquot Hooft Counting

The most explicit evidence that gauge theory leads to string theory comes from rsquot Hooft counting

b=1N

Consider a (any) quantum field theory with matrix

fields6

Φa=1 N macr (for concreteness wersquoll take thematrix group to be U (N )) and consider a Lagrangian of the form

L sim 2

1Tr

(part Φ)2 + Φ2 + Φ3 + Φ4 +

1048573gYM

˜Rescale Φ = gYMΦ

L sim Tr (part + ˜ Φ3 + g2 Φ4 + Φ)2 Φ2 + gYM YM

Now consider propagators It is convenient to adopt the double

line notation in which oriented

index lines follow conserved color flow so that for propagator7

Φab Φ

dc prop

δ caδ b

d =a

c d

c

By matrix field we mean that their products appear in the Lagrangian only in the form of matrix multiplicationeg (Φ2)ca

= ΦbaΦ

c

b

Had we been considering SU (N ) the result would be Φb

aΦcd prop δ c

aδ bd

minus δ baδ d

cN 2 =

4

6

7

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 67

And similarly for vertices

a

b

c

a

b

c

prop gYM

a a

b

b

cc

d

d

prop g2YM

Now consider the rsquot Hooft limit in which N rarr infin and gYM rarr

0 but which λ = g2 N = constYM

Is this limit classicalfree The answer turns out to be no The loophole is that even though thecoupling goes to zero the number of modes diverges

(a)

prop λ3N 2prop

N 2

prop λN 2

(b)

(c)

Figure 2 Planer graphs that contribute to the vacuumrarrvacuum amplitude

2prop gYMN = λN 0

Figure 3 Nonplaner graph that contributes to the vacuumrarrvacuum amplitude

To see this more concretely consider vacuumrarrvacuum diagrams (see Fig 2 and 3 for illustration)In Fig 2 we have a set of planer graphs whose contributions take the general form λnN 2 Howeverthere are also contributing nonplaner graphs such as the one in Fig 3 whose contribution doesthe take that general form

Every doubleline graph specifics a triangulation of a 2dimensional surface Σ There are two waysto construct the explicit mapping

Method 1 (ldquodirect surfacerdquo) Fill in index loops with little plaquettes

Method

2

(ldquodual

surfacerdquo)

(1) draw a vertex in every index loop and (2) draw an edge acrossevery propagator

These constructions are illustrated in Fig 4 and 5

Comparing the resulting surface and the corresponding contribution it can be seen that the contri-bution of a graph with h handles is proportional to N 2minus2hλn Thus the partition function (whichis also the vacuumrarrvacuum amplitude) must take the form

5

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 77

sim S 2 sim

T 2

(a) (b)

Figure 4 Direct surfaces constructed from the vacuum diagram in (a) Fig 2a and (b) Fig 3

infin

sim S 2

Figure 5 Dual surface constructed from the vaccum diagram in Fig 2c Note that points at infinityare identified

infin

lnZ =N 2minus2hf h(λ)

h=0

which is exactly like the result from string perturbation expansion

6

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 67

And similarly for vertices

a

b

c

a

b

c

prop gYM

a a

b

b

cc

d

d

prop g2YM

Now consider the rsquot Hooft limit in which N rarr infin and gYM rarr

0 but which λ = g2 N = constYM

Is this limit classicalfree The answer turns out to be no The loophole is that even though thecoupling goes to zero the number of modes diverges

(a)

prop λ3N 2prop

N 2

prop λN 2

(b)

(c)

Figure 2 Planer graphs that contribute to the vacuumrarrvacuum amplitude

2prop gYMN = λN 0

Figure 3 Nonplaner graph that contributes to the vacuumrarrvacuum amplitude

To see this more concretely consider vacuumrarrvacuum diagrams (see Fig 2 and 3 for illustration)In Fig 2 we have a set of planer graphs whose contributions take the general form λnN 2 Howeverthere are also contributing nonplaner graphs such as the one in Fig 3 whose contribution doesthe take that general form

Every doubleline graph specifics a triangulation of a 2dimensional surface Σ There are two waysto construct the explicit mapping

Method 1 (ldquodirect surfacerdquo) Fill in index loops with little plaquettes

Method

2

(ldquodual

surfacerdquo)

(1) draw a vertex in every index loop and (2) draw an edge acrossevery propagator

These constructions are illustrated in Fig 4 and 5

Comparing the resulting surface and the corresponding contribution it can be seen that the contri-bution of a graph with h handles is proportional to N 2minus2hλn Thus the partition function (whichis also the vacuumrarrvacuum amplitude) must take the form

5

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 77

sim S 2 sim

T 2

(a) (b)

Figure 4 Direct surfaces constructed from the vacuum diagram in (a) Fig 2a and (b) Fig 3

infin

sim S 2

Figure 5 Dual surface constructed from the vaccum diagram in Fig 2c Note that points at infinityare identified

infin

lnZ =N 2minus2hf h(λ)

h=0

which is exactly like the result from string perturbation expansion

6

8102019 Aula 7 - Teoria de cordas MIT

httpslidepdfcomreaderfullaula-7-teoria-de-cordas-mit 77

sim S 2 sim

T 2

(a) (b)

Figure 4 Direct surfaces constructed from the vacuum diagram in (a) Fig 2a and (b) Fig 3

infin

sim S 2

Figure 5 Dual surface constructed from the vaccum diagram in Fig 2c Note that points at infinityare identified

infin

lnZ =N 2minus2hf h(λ)

h=0

which is exactly like the result from string perturbation expansion

6