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Testing AdS/CFT at string loops

Testing AdS/CFT at string loops. Gianluca Grignani. Perugia University & INFN. 2006 PRIN MEETING Alessandria, December 15-16. References. G. Grignani, M. Orselli, B. Ramadanovich, G.W. Semenoff and D. Young, “ Divergence cancellation and loop corrections in string field theory on a

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Testing AdS/CFT at string loops

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  1. Testing AdS/CFT at string loops Gianluca Grignani Perugia University & INFN 2006 PRIN MEETING Alessandria, December 15-16

  2. References G. Grignani, M. Orselli, B. Ramadanovich, G.W. Semenoff and D. Young, “Divergence cancellation and loop corrections in string field theory on a plane wave background”,JHEP 0512:017,2005 e-Print Archive:hep-th/0508126 G. Grignani, M. Orselli, B. Ramadanovich, G.W. Semenoff and D. Young, “AdS/CFT versus string loops”, JHEP 06:040,2006, e-Print Archive:hep-th/0605080 G. De Risi, G. Grignani, M. Orselli, G.W. Semenoff, “DLCQ string spectrum from N=2 SYM theory”,JHEP411:053,2004 e-Print Archive:hep-th/0409315 D. Astolfi, V. Forini, G. Grignani, G.W. Semenoff, “Finite size corrections and integrability of N =2 SYM and DLCQ strings on a pp-wave”,e-Print Archive:hep-th/0606193

  3. AdS/CFT Maximally supersymmetric Yang-Mills theory on 4D flat space with gauge group SU(N) and coupling constant gYM is exactly equivalent to type IIB superstring theory on background AdS5XS5 with N units of 5-form flux through S5 . The radius of curvature is ‘t Hooft coupling and gauge invariant composite operator String state conformal dimension of composite operator Energy This is a weak coupling – strong coupling duality

  4. The Penrose limit of AdS5XS5 geometry gives a pp-wave space transverse SO(4)xSO(4) symmetry Free strings are exactly solvable in the light-cone gauge light-cone Hamiltonian level matching condition • The analogous limit of Super Yang-Mills can be taken (BMN) detailed matching of BMN operators in the large N, planar limit and free string states free strings planar limit string interactions non-planar corrections (ongoing) matching interactions in light-cone string field theory on a pp-wave background and Yang-Mills theory computations

  5. Non planar corrections in Yang-Mills Theory String loop corrections to the energy of the 2-oscillator (2-impurity) states Symmetric traceless: Trace: In N=4 SYM there are 6 scalar fields Select a combination of them which is charged under a U(1) subgroup of the R Symmetry Dual BMN operator:

  6. Has not been computed using string theory! The only available formulation of string theory in which interactions can be computed is pp-wave light-cone string field theory

  7. Because of the presence of the RR-field, the only available formulation of string theory in which interactions can be computed is pp-wave light-cone string field theory M.Spradlin and A.Volovich, hep-th/0204146, hep-th/0206073, hep-th/0310033 (review) C.S.Chu, V.Khoze, M.Petrini, R.Russo and A.Tanzini, hep-th/0208146 Y.He, J.Schwarz, M.Spradlin and A.Volovich hep-th/0211198 A.Pankiewicz,hep-th/0208209 A.Pankiewicz and B.Stefanski,hep-th/0210246 R.Roiban,M.Spradlin and A.Volovich hep-th/0211220 S.Dobashi, H.Shimada and T.Toneya, hep-th/0209251 P.Di Vecchia, J.L.Petersen, M.Petrini, R.Russo and A.Tanzini, hep-th/0304025 J.Gomis, S.Moriyana and J.W.Park, hep-th/0301250 A.Pankiewicz, hep-th/0304232 P.Gutjahr and A.Pankiewicz, hep-th/0407098 S.Dobashi and T.Toneya, hep-th/0408058, hep-th/0406225 S. Lee and R. Russo, hep-th/0409261 G.Grignani, M.Orselli, B. Ramadanovich, G.W.Semenoff and D. Young, hep-th/0508126 G.Grignani, M.Orselli, B. Ramadanovich, G.W.Semenoff and D. Young, hep-th/0605080

  8. Plan of the talk 1. Light-cone String Field Theory (SFT) 2. The cubic vertex: Spradlin and Volovich (SVPS) vertex Di Vecchia et al. (DPPRT) vertex Dobashi and Yoneya (DY) vertex 3. Contact terms 4. Perturbation theory 5. A little history of the results 6. Main results: 1)one-loop correction to the energy of 2-oscillator states for each vertex. 2)cancellation of divergences for the trace state. 7. It’s just a factor of 2 ! 8. Conclusions

  9. Light-cone String Field Theory string field operator :fundamental object in light-cone string field theory creates or annihilates complete strings m-string Hilbert space • is a functional of x+, p+ and the worldsheet coordinates where XI()=XI(,=0) and likewise for other fields. In the momentum-space representation is a functional of where  is – i times the momentum conjugate to , i.e.: in the momentum-space representation

  10. The light-cone string field theory dynamics of  is governed by the non-relativistic Schroedinger equation light-cone string field theory Hamiltonian In terms of the string coupling the light-cone SFT Hamiltonian has the expansion In the free string theory limit it should be equal to the Hamiltonian coming from the string theory -model We will use perturbation theory to compute the energy of the states. free strings Consider the bosonic case and the Fourier modes of XI() and PI() for =0, xnand pn we write the free light-cone Hamiltonian in terms of these Fourier modes where

  11. The free pp-wave Hamiltonian written in terms of these Fourier modes becomes where the eigenfunctions of the Schroedinger equation are products of an infinite number of momentum eigenfunctions Nn(pn) where Nn is the excitation number of the nth oscillator with frequency n/. being a momentum eigenstates a coherent state the string field becomes

  12. to quantize the string field we promote the to operators acting on the string Fock space where it creates or destroys a complete string with excitation number {Nn} at =0 and The superalgebra generators are promoted to operators acting on the SFT Hilbert space. For example, the free Hamiltonian becomes where

  13. Turning on interactions: the cubic vertex The corrections to superalgebra generators, once interactions are turned on, are obtained following some guiding principles. 1. The interaction should couple the string worldsheet in a continuos way x+ =2(1+2) 2  =21 3 1 x+=0 =0 The interaction vertex for the scattering of 3 strings is constructed with a -functional enforcing worldsheet continuity. In pp-wave superstring however the situation is slightly more complicated but the basic principle governing the interaction is very simple:

  14. 2. The superalgebra has to be realized in the full interacting theory This is the essential difference with the bosonic string case and modifies the form of the vertex. Supercharges that square to the Hamiltonian receive corrections when adding interactions The picture remains geometric but in addition to a delta-functional enforcing continuity in superspace one has to insert local operators at the interaction point. These operators represent functional generalizations of derivative couplings prefactors There are two different set of superalgebra generators. 1. Kinematical generators : They are not corrected by interactions. The symmetries they generate are not affected by adding higher order terms to the action These generators remain quadratic in the string field  in the interacting field theory and act diagonally on Hm

  15. quartic in the string field cubic in the string field the corrections are such that , and still satisfy the superalgebra 2. Dynamical generators: They receive corrections in the presence of interactions and couple different number of strings

  16. leads to continuity condition in superspace requires insertion of interaction point operators: the prefactors To solve the constraints we use perturbation theory. Example: different from flat space In the plane wave geometry transverse momentum is not a good quantum number due to the confining harmonic potential. The expansion in gs, however, implies the same kinematical constraint as in flat space must be translationally invariant the relation is the same as in flat space, many of the techniques developed there can be used in the plane wave case as well The requirement that the superalgebra is satisfied in the interacting theory now gives rise to two kind of constraints: Kinematical constraints Dynamical constraints (anti)-commutation relations of kinematical and dynamical generators anti-commutation relations of dynamical generators alone

  17. In momentum space the conservation of transverse momentum by the interaction will be implemented by a -functional Analogously, from and and since P+ is a good quantum number and one concludes that the cubic interaction contains also

  18. Three string vertex in plane wave light-cone SFT and must be proportional to Dynamical constraint: the supersymmetry algebra Expanding the Hamiltonian and the supercharges at the order gs: the dynamical constraint on and is the same as in flat space

  19. do not depend on the string field  the constraints are solved by inserting prefactors into the ansatz for and where to determine the explicit form of the vertex it is essential to use a number basis eigenstates of rather then momentum eigenstates write explicitly y and perform the momentum integral

  20. prefactor kinematical part of the vertex, common to all dynamical generators and similar expressions for the fermionic part |Ebi To identify and it is sufficient to find their matrix elements between two incoming strings and one outgoing string. It is convenient to express not as an operator but as astate in the 3-string Hilbert space and similarly for |3i and h 3’| are related by worldsheet time-reversal We can write

  21. Neumann matrices we get performing the Gaussian integrals we find One can proceed analogously for the fermionic case by constructing the fermionic analog of the bosonic wave function one arrives at the solution

  22. General formulas for the kinematical part of the (SV) vertex and similarly for

  23. Two possible solutions for the prefactors. a) PSSV , the prefactor is odd under Z2 symmetry. The purely bosonic part of the prefactor is given by The fermionic part of the prefactor was derived by Pankiewicz and Stephanski and contains up to 8 fermionic and bosonic creation operators b) DPPRT , the prefactor is even under Z2 symmetry. A very simple way to realize the susy algebra is by acting with the free parts of the Hamiltonian and dynamical charges The purely bosonic part of the prefactor is given by The fermionic part contains at most 2 bosonic or fermionic creation operators

  24. c) Holographic SFT (Dobashi-Yoneya, Lee-Russo) The purely bosonic part of the prefactor is given by breaks maximally the Z2 symmetry • when restricted to the zero-modes provides the resultant supergravity effective theory in the pp-wave background. • gives the correct 3-point correlation functions on the SYM side derived by perturbation theory The only linear combination of the two vertices with these properties is the one which equally weights them. For all the vertices:the dynamical constraints do not fix the overall normalization of the vertex, this can depend on  and . In flat space the J- I generator is also dynamical imposes further constraints on the vertex and uniquely fixes the normalization. As J- Iis not part of the plane wave superalgebra an overall function f(,) remains unfixed in each vertex

  25. Contact Terms We want to compute the energy of string states using a quantum mechanical perturbation theory. The dynamical generators are expanded in terms of gs. Up to the O(gs2) we have The susy algebra up to the O(gs2) Necessarily there is an the so-called contact term Light-cone string field perturbation theory on a single string in a 2 oscillator state |ni uses QM perturbation theory to compute the correction to its light-cone energy

  26. The string join/split points of each vertex coincide In type IIA/B superstring on Minkowski space-time, it is known that the contact terms are necessary to cancel certain singularities in the integrations over parameters of light-cone SFT diagrams. In the conformal field theory they are also seen to arise as additional contributions needed to cancel certain singular surface terms which arise in the integration of correlators of vertex operators over the modular parameters of Riemann surfaces. J.Greensite and F.R.Klinkhamer NPB 304 (1988)

  27. Summary of results String loop corrections to the energy of the 2-impurity (2-oscillator) bosonic states Symmetric traceless: Trace: Dual string state: Yang-Mills Prediction

  28. A little history The gauge theory result is Roiban, Spradlin and Volovich, hep-th/0211220 It matches! But obtained from Reflection symmetry of the one-loop light-cone string diagram ?? P.Gutjhar and A. Pankiewicz, hep-th/0407098,no ½

  29. The gauge theory result is P.Gutjhar and A. Pankiewicz, hep-th/0407098 It does not match! • Expansion in half integer powers of ’ • Uses truncation to 2-impurities • Uses Spradlin-Volovich vertex • Fixes the pre-factor f=1 • Sets Q(4)=0

  30. Results: for any vertex • We confirmed Gutjhar and Pankiewicz result, the natural expansion parameter • is 2. In the trace state |[1, 1]ithe leading order is and it diverges Each term in the equation above diverges individually. With the ½ of the “reflection symmetry” no cancelation of divergences no 1/2

  31. 3. We provided a general proof for this result just using the supersymmetry algebra Then using with Q(4)=0 • The energy correction vanishes for a supersymmetric state Q(2)|ni=0 • Divergences are canceled in this formula and the relative factor between the two terms in the starting equation is fixed. • Q(2)|ni is proportional to the square root of the energy n

  32. Results: for each vertex • Spradlin and Volovich vertex We confirmed Gutjahr-Pankiewicz result 2. Di Vecchia et al. vertex

  33. Best matching of this quantity so far DY vertex is an improvement over its predecessors • No half integer powers up to ’7/2 • Uses truncation to 2-impurities • Fixes the pre-factor f=4/3 • Sets Q(4)=0 3. Dobashi-Yoneya vertex Divergences cancel and it works at 1-loop!

  34. 4. Some experimental results, it’s just a factor of 2 ! Introducing a factor of 2 in front of the contact term Both vertices match the gauge theory result up to two loops ! • Use truncation to 2-impurities • Fix the pre-factor f to suitable values • Set Q(4)=0

  35. Conclusions 1. Dobashi Yoneya vertex seems to be the most promising 2. Possible sources of discrepancies a)number of impurities in the intermediate states. b)Q(4) ≠ 0 a contact term which does not diverge? • The problem of matching non-planar YM and string interactions in the context of AdS/CFT is still open

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