From free gauge theories to strings

Carmen NúñezI.A.F.E. – Physics Dept.-UBA

Buenos Aires

10 Years of AdS/CFT

strings@ar, December 19, 2007

Based on

Work in progress in collaboration with M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)

R. Gopakumar, Phys.Rev.D70(2004)025009, 025010, Phys.Rev.D72 (2005) 066008

O. Aharony, Z. Komargodski and S. Razamat, JHEP 0701 (2007) 063

J. David and R. Gopakumar, JHEP 0701 (2007) 063

O. Aharony, J. David, R. Gopakumar, Z. Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006

Outline

Brief review of a proposal by R. Gopakumar to obtain the string theory dual of large N free gauge theories.

Resulting integrand on moduli space has the right properties to be that of a string theory.

Worldsheet vs. spacetime OPE in several examples

Future work

After 10 years…

Many examples known how to find closed string dual of gauge theories which can be realized as world-volume theories of D-branes in some decoupling limit.

Dual string theory is a standard closed string theory, living in a warped higher dimensional space.

Strongly coupled gauge theory weakly curved string background gravity approx. may be used.

In general, (weakly coupled gauge theories) dual string theory is complicated, and not necessarily has geometrical interpretation.

It is interesting to ask what is the string theory dual of the simplest large N gauge theory: free gauge theory

Free large N gauge theories as a laboratory for understanding the gauge/string correspondence (making this picture precise is essential to obtain a string dual to realistic gauge theories.)

As limit of interacting gauge theories (not just N2 copies of a free U(1) theory). Have topological expansion in powers of 1/N2. In this limit gs ~ 1/N.

Useful starting point for perturbation theory in (perturbative Feynman amplitudes are given in terms of free field diagrams).

Free gauge theories?

At least in the context of string theory on AdS5 S5 , free field theory related to tensionless limit.

For 4D free conformal gauge theories one expects that any geometrical intepretation should have an AdS5 factor.

Peculiar properties needed of w-sh theory: free correlators terminate at finite order of 1/N expansion dual w-sh correlators get contributions upto given maximal genus

General expectations

What exactly is the string dual?

How exactly does a large N field theory reorganize itself into a dual closed string theory?

Can we systematically construct the closed string theory starting from the field theory?

Various proposals: R. Gopakumar, C. Thorn, H. Verlinde, M. Kruczenski, B. Sundborg, G. Bonelli …

Gauge-string duality

General expectation is

wsnnngnn kkkk ng,M

),(),()()( 11111 VVOO

Oi: Gauge invariant operators

Vi : Vertex operators of dual string theory

Can we recast the left hand side

into the form we expect from

the right hand side?

Simple way to organize different Feynman diagram contributions to given n-point function so that the net sum can be written as an integral over the moduli space of an n-punctured Riemann surface.

1. Skeleton diagram

Write gauge theory

amplitudes in Schwinger

parametrised form gluing

together homotopically

equivalent propagators

0

||12

2

||1 ji xx

ji

edxx

Gopakumar’s proposal I

Gopakumar’s prescription II

2. Map Schwinger parameters to the moduli space of a Riemann surface with holes Mg,n R+

n

ijijl CONCRETE PROPOSAL: Identify the Schwinger

parameters with Strebel lengths: Line integrals between the zeroes of certain meromorphic

quadratic differentials (Strebel differentials)

j

i

z

z

ij dzzl )(

# independent for maximally connected Feynman graph of genus g for n-point function (6g 6 + 3n = 6g 6 + 2n + n) =

= # real moduli for genus g Riemann surface with n punctures +

additional n moduli parametrize R+n = # Strebel lengths lij

3. Integrate over the parameters of the holes.

Integral over (with sum over different graphs) can be converted into integral over Mg,n R+

n

Thus potentially a world-sheet n-point correlation function.

This procedure translates any Feynman diagram to a correlation function on the string world-sheet.

Gopakumar’s prescription III

The dictionary

For every Strebel differential there is a critical graph whose vertices are the zeroes of the differential and along whose edges

is real

For generically simple zeroes the vertices of critical graph are cubic.

Each of the n faces of critical graph contains only one double pole

Critical graph can be identified with dual of reduced Feynman graph

ijijl

j

i

z

z

ij dzzl )(

How can we check this hypothesis?

We don’t know how to quantize string theory in the highly curved AdS backgrounds that would presumably

be dual to the free limit of conformal field theory.

Few modest checks

1. Two and three point functions give expected correlators in AdS.

E. g. Planar three point function

can be recast as a product of three bulk-boundary propagators for scalars in AdS

0

3

1321

}{ )(),,(

gi

iJJ

g xTrxxxG ii

])([);,( 2

2/

zxtt

tzxK = J (d-2)/2 x1

x2 x3

0

3

10 1

2

321}{0 );,(),,(

ii

dd

Jg tzxKzd

t

dtxxxG

i

i

Probably special to 2- and 3- point functions

The Y four point function

g

JJJJJg xTrxTrxTrxTrxxxxG i )()()()(),,,( 43214321

}{ 321

),(),,,(}{

)(

24321

}{)4( i

i

iJ

x

J GdxxxxG

2. Consider 4-point correlation functions of the form

with J = J1 + J2 + J3. Mapping gives

with = (l1, l2, l3).

Explicit expression for the candidate worldsheet correlator J. David and R. Gopakumar, JHEP 0701 (2007) 063

Prediction for string dual

The dependence on || and |1- | is what one expects of a correlation function of local operators inserted at 0, 1, and .

J

JJJ

iJx

xxx

JCG i

i

|)]1|||1(|)1|||1(|)1|||1([|)1|||1(|)1|||1(|)1|||1(

|1||||)1|||1(

)(),(

23

22

21

2/12/12/1

2/1}{}{

321

WS

Jx

Jx

Jx

Jx

Jx VVVVG i

i),()()1()0(),(

4

3

3

2

2

1

1

}{}{

Obeys crossing symmetry:),()1,1( }{

}{}{}{ i

i

i

i

Jx

Jx GG

),(||)1

,1

( }{}{

4}{}{

i

i

i

i

Jx

Jx GG

Consistent with locality: all

terms in OPE (when 0)hhxJ

hhhh

Jx

iii

iCG },{

},{,

}{}{ ),(

with hh

Worldsheet vs. spacetime OPE

Consider four point function of single trace operators

As x1 x2 , OPE contains other gauge invariant operators

UV in bdary spacetime IR in bulk spacetime UV on worldsheet

EXPECTATION: As x1 x2 , worldsheet correlator gets dominant

contribution from z 0

)()()()( 44332211 xxxx OOOO

)()()()( 221122211 xxxCxx kk

k OOO

: when two ST positions collide, corresponding ij .

This corresponds to region of moduli space where vertices collide.

ijijl

Worldsheet vs. spacetime OPE (continued)

In free field theory, often correlators in which two operators do not have any Wick contractions with each other, e.g.

has contribution only from

Absence of ST OPE should be reflected in corresponding WS OPE

EXPECTATION: The strongest way in which this could happen is if the corresponding vertex operators also do not have a WS OPE

))(())(())(())(( 4333222111 xTrxTrxTrxTr

x1

x2 x3

x4

Consider correlator in free field theory with three adjoint scalar fields X, Y, Z

The string theory amplitude has support only for negative real values of the modular parameter.

The four point function

))(())(())(())(( 42

32

22

12 xZTrxYZTrxYXTrxXTr

x1

x2 x3

x4

The square and the whale diagrams

Consider the field theory amplitudes

There are no solutions for large The solution can be obtained numerically, and it is always

real and 0< <1 for the square and localizes on small region of complex plane for the whale.

)())(())(()(

))(()())(()(

42

322

222

12

432

212

xZTrxZYTrxYXTrxXTr

xXZTrxZTrxXZTrxXTr

x2

x1

x3

x4

x1x2 x3

x4

LOCALIZATION

The region of moduli space that these diagrams cover precisely excludes the possibility of taking a worldsheet OPE b/corresponding vertex operators (e.g. 1 when localized on the negative real axis).

Pattern behind localization (or absence) in free field diagrams is such that localization occurs only in those diagrams in which there is no contraction between two pairs of vertices.

There is no worldsheet OPE exactly when there is no spacetime OPE.

Realization of EXPECTATION

LOCALIZATION (continued)

Localization on the worldsheet is compatible with properties of a local worldsheet CFT (O. Aharony, J. David, R. Gopakumar, Z. Komargodski and S. Razamat, Phys.Rev.D75 (2007) 106006)

It has contribution from the “broom” diagram.

In the limit j 0, reduces to Pi diagram which

shows localization. ADGKR showed localized

worldsheet correlators correspond to a limit of

the field theory correlation functions which is

governed by saddle point in Schwinger

parameter space

))(())(())(())(( 414334223211133221 xTrxTrxTrxTr JJjJJjJJ

x1

x2 x3

x4

GENERAL LESSONS

The expansion in the position of the saddle point corresponds to an expansion in the length of one or more small edges in the critical graph of the corresponding Strebel differential.

Confirmation of expectation: localization of worldsheet correlators appears to be correlated with absence of non-trivial ST OPE

QUESTIONS:

What is the criterion for localization of general free field diagram?

What is the subspace on which it localizes?

What does this tell us about the WS theory?

The square and the whale from the

The square with a small edge.

Strebel differential

c c = c(0) /2 , 1

Graphical deformation of Strebel graph allows to determine phase of and thus allows to identify potential delocalized diagrams.

2222

22

2

22

)()1()()(

4)( dz

zzzczczp

dzz

2

222

4

2

22

)()1()(

4)( dz

zzzczp

dzz

0

1

= (0) + a(li) 2

Constructing Mg,n

There is a systematic way of constructing Mg,n from the ribbon graph (familiar from open SFT):

When k edges meet at a vertex they form angles 2/k with each other.

one face one zero two bivalent vertices two faces with two edges two single valued vertices two faces with one edge

1

0

Deformation of the

might delocalize

cannot delocalize

0

1

0

0

1

1

The square with one diagonal

Deforming the to get the square with one diagonal

might delocalize

2

The diagram with two diagonals

1 = k 2 , k

0

1

Blow up n-fold zero moving appropriate number of lines along their

central direction allows to identify potentially delocalized diagrams

Conclusions

WS duals to free large N gauge theories exhibit interesting behavior

Adding few contractions to field theory diagram or small edges to dual graph, delocalizes correlators and allows to relate ST with WS OPE. Fruitful approach to extract general features of WS theory.

We obtained graphical method to identify potential delocalization.

Future Work

More diagrams have to be studied in order to extract general properties of the worldsheet duals to

free large N gauge theories.

Allows to obtain new worldsheet correlators which can be studied and lead to better understanding of

the worldsheet CFT.