From free gauge theories to strings
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Transcript of From free gauge theories to strings
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.