From free gauge theories to strings Carmen Núñez I.A.F.E. – Physics Dept.-UBA Buenos Aires 10 Years of AdS/CFT December 19,

Based on  Work in progress in collaboration with M. Bonini (Parma Univ.) and F. Pezzella (Napoli Univ.)  R. Gopakumar, Phys.Rev.D70(2004)025009, , Phys.Rev.D72 (2005)  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)

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 N 2 copies of a free U(1) theory). Have topological expansion in powers of 1/N 2. In this limit g s ~ 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 AdS 5  S 5, free field theory related to tensionless limit.  For 4D free conformal gauge theories one expects that any geometrical intepretation should have an AdS 5 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 O i : Gauge invariant operators V i : 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 Gopakumar’s proposal I

Gopakumar’s prescription II 2.Map Schwinger parameters to the moduli space of a Riemann surface with holes M g, n  R + n  CONCRETE PROPOSAL: Identify the Schwinger parameters with Strebel lengths: Line integrals between the zeroes of certain meromorphic quadratic differentials (Strebel differentials) # 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 l ij

3.Integrate over the parameters of the holes. Integral over  (with sum over different graphs) can be converted into integral over M g, 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

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  = J ( d-2 )/ 2 x 1 x 2 x 3 Probably special to 2- and 3- point functions

The Y four point function 2. Consider 4-point correlation functions of the form with J = J 1 + J 2 + J 3. Mapping gives with  =  ( l 1, l 2, l 3 ).  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 . Obeys crossing symmetry: Consistent with locality: all terms in OPE (when  0) with h  h  

Worldsheet vs. spacetime OPE  Consider four point function of single trace operators  As x 1  x 2, OPE contains other gauge invariant operators  UV in bdary spacetime IR in bulk spacetime UV on worldsheet  EXPECTATION: As x 1  x 2, worldsheet correlator gets dominant contribution from z  0  : when two ST positions collide, corresponding  ij . This corresponds to region of moduli space where vertices collide.

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 x1x1 x2x2 x3x3 x4x4

 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 x1x1 x2x2 x3x3 x4x4

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. x2x2 x1x1 x3x3 x4x4 x1x1 x2x2 x3x3 x4x4

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) ) 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 x1x1 x2x2 x3x3 x4x4

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. 0 1    =  (0) + a ( l i )  2

Constructing M g, n  There is a systematic way of constructing M g, 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     

The square with one diagonal  Deforming the  to get the square with one diagonal might delocalize 2  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.