1. 1 DLCQ STRINGS FROM QUIVER GAUGE THEORIES terzo incontro del P.R.I.N. “TEORIA DEI CAMPI SUPERSTRINGHE E GRAVITA`” Capri, 24-26 September 2004 Gianluca Grignani Universita` di Perugia Based on: hep-th/0409xxx, in collaboration with: G. De Risi (Bari University), M. Orselli (Perugia University), G. De Risi (Bari University), M. Orselli (Perugia University), and G. W. Semenoff (UBC),

2. 2 MOTIVATIONS 1.AdS/CFT duality tests 2.Effective Hamiltonians: big progress 4.Discrepancies (Gutjahr-Pankiewicz hep-th/0407098) 5.A new simplified setting for computing non-planar corrections to the anomalous dimension, the contribution of all orders in non-planar diagrams can be found analytically 3.In the pp-wave background accurate computations in string theory can be done and compared

3. 3 PLAN OF THE TALK Outline: DLCQ strings and SYM gauge theoriesOutline: DLCQ strings and SYM gauge theories Setting: PP Wave limit and Large Quiver limitSetting: PP Wave limit and Large Quiver limit Gauge Theory description of DLCQ stringsGauge Theory description of DLCQ strings Computation of the string spectrum from the anomalous dimension of the corresponding gauge theory statesComputation of the string spectrum from the anomalous dimension of the corresponding gauge theory states Examples of the exact computations of the interacting string spectrumExamples of the exact computations of the interacting string spectrum Conclusions and outlookConclusions and outlook

4. 4 Outline: DLCQ strings and SYM gauge theories It is often useful to compactify a null direction in light-cone quantization of strings this leads to Discrete Light Cone Quantization Discrete Light Cone Quantization (DLCQ) of the string theory sectors In this description the theory splits into sectors labelled by a discrete value of the quantized light-cone momentum string bit Interacting strings carry the quantized momenta, with the minimal momentum being carried by a string bit Matrix Theory description of M-theory Application: Matrix Theory description of M-theory Thanks to MRV (Mukhi,Rangamani,Verlinde hep-th/0204147) there is a nice pp- wave description of DLCQ strings The gauge theory/pp-wave correspondence could lead to a better understanding of string interactions

5. 5 The setting of the present talk is type IIB string theory which admits supersymmetric solutions of the type AdS 5 xM 5 In the usual correspondence the null direction one gets in AdS 5 XS 5 by just taking the BMN limit is non-compact and one cannot associate N =4 SYM with DLCQ strings Zooming into the equator of M 5, the space-time becomes a pp-wave, in the dual gauge theory this corresponds to select a sector with a large R -charge

6. 6 compact null direction In this talk, I will consider an N =2 gauge theory in a particular limit whose AdS dual in the same limit will become a pp-wave background with a compact null direction The radius of the null direction is a finite, controllable parameter of this background gauge theory / pp-wave pair This particular AdS background has a dual 4d conformal gauge theory. The above scaling limit will act on this gauge theory leading to a dual gauge theory / pp-wave pair double scaling This particular scaling limit will play a role similar to the now familiar double scaling limit in the usual BMN picture moosequiver The theory we will discuss is an N =2 superconformal “moose” or “quiver” theory in the large moose or quiver limit

7. 7 Several interesting aspects of the gauge theory/pp-wave correspondence will emerge as we explore this background In particular, we’ll find gauge theory operators of fixed momentum in the light-like direction Since we are in DLCQ the momentum is quantized and there is an integer k which labels that fixed momentum DLCQ momentum k The gauge theory operators can be identified with a string state in every sector of fixed DLCQ momentum k winding m We’ll also have states which wind around the null direction and so we’ll find operators that describe modes of the string winding m times on the DLCQ direction These operators satisfy the relation Level matching condition

8. 8 Setting: PP Wave limit and Large Quiver limit The gauge theory that we’ll study is obtained by placing N D3-branes transverse to the 6-dim space R 2 X(C 2 /Z M ) There are 2 integers, N is for the D-branes and M is for the order of the orbifold group The action of the orbifold group on R 2 XC 2 is This breaks N =4 susy to N =2. The theory on the brane world-volume is a N =2 4-dim superconformal field theory The R-symmetry group is U(1) R XSU(2) R and the gauge group becomes the direct product of M U(N) factors.

9. 9 The N =2 action can be derived from the parent N =4 SYM theory. Let’s start with N =4 with a U(NM) gauge group All fields are MNxMN matrices The orbifold group is the cyclic group Z M whose generator acts on the 6 scalar field of N =4 as With the notation Some components of these MNxMN matrix fields are then set to zero

10. 10 The elements of the bosonic fields which survive the projection are Each non-vanishing entry of the above matrices is an NxN matrix and corresponds to a bosonic field of the N =2 theory Analogous expressions hold for the fermionic superpartners An element of the residual gauge group that acts on all the above matrices by conjugation is

11. 11 The action for scalar fields is Where the interaction D- and F- term can be gotten from Only the F-term contribute to the one-loop corrections [hep-th/0205033 (Kristijansen,Plefka, Semenoff and Staudacher) and hep-th/0205089 (Constable, Freedman, Headrick, Minwalla, Motl, Postnikov and Skiba)]

12. 12 The resulting field content is as follows M vector multiplets in the adjoint of each U(N) I  I is a complex scalar field and the Weyl fermion   I is its superpartner. A  I is the gauge field and  I is the gaugino M bi-fundamental hypermultiplets, which, in N =1 notation, are A I and its superpartner  A I transform in the (N I,N I +1 ) representation of U(N) ( I ) xU(N) ( I +1). The pair B I and  B I transform in the complex conjugate representation(N I,N I +1 ) moosequiver This can be represented in a “moose” or “quiver” diagram

13. 13 Type II B String on AdS 5 xS 5 /Z M The holographic dual of this theory for finite N and M is type II B string theory on AdS 5 xS 5 /Z M with MN units of RR 5 form flux through the 5 sphere The radii of AdS 5 and S 5 are equal and given by g S is the type II B string coupling The YM theory coupling constant of the parent N =4 theory is identified with the coupling constant of the parent superstring theory on AdS 5 xS 5 double scaling limit We are going to consider the double scaling limit in which both N and M become large together with the ratio M/N fixed. In this way the radii of AdS 5 and S 5 are put to infinity while g S is kept small but finite I will show that this double scaling limit is the Penrose limit which obtains the pp-wave background

14. 14 The metric of AdS 5 xS 5 /Z M can be written as We parametrize the complex coordinates of the transverse plane C 3 /Z M in terms of the angle of S 5 The orbifold described by is obtained by the identifications To take the pp-wave limit it is useful to define the coordinates and introduce the light-cone coordinates This is the point where one chooses which equator goes into the definition of the pp-wave. By choosing the particle moving very fast along the direction involved in the orbifolding the pp-wave limit corresponds to a maximally supersymmetric background

15. 15 In the R   limit, the metric becomes the standard pp-wave metric the light-like direction x - turns out to be compact The important difference is that the light-like direction x - turns out to be compact To see this, recall that the action of the orbifold group on the angles  and  requires that the light-cone coordinates x + and x - have periodicity to infinity together, we find that R - is finite since in the double scaling limit when N and M are taken What we find is We end up with a pp-wave metric with a periodic x - We end up with a pp-wave metric with a periodic x -. As a consequence the corresponding light-cone momentum is quantized in units of 1/R - In other words: we are doing the Discrete Light Cone Quantization (DLCQ) of the string on a pp-wave background

16. 16 light-cone quanta k As is well-known the theory splits into sectors labelled by the discrete positive number of the light-cone quanta k winding modes m There can also be winding modes of the string on the null direction which are labelled by an integer m Note that the orbifold of the 5 sphere preserves half of the supersymmetries of the original AdS 5 xS 5 solution of string theory. Nonetheless in the Penrose limit we recover the maximally supersymmetric background

17. 17 Gauge Theory description of DLCQ strings DLCQ of the string on the pp-wave background is a slight generalization of BMN. One component of the light-come momentum is quantized as The other component is the light-cone Hamiltonian bosonic and fermionic transverse oscillations of the string respectively are the annihilation and creation operators for the discrete Since there are also wrapped states there is also a level matching condition which for states of the form reads level matching condition

18. 18 Matching Charges There are 3 important quantum numbers that can be mached between the string theory and its gauge theory dual.   Is the energy in string theory, which is the quantum operator generating a flow along the killing vector field i  t of the background. It corresponds to the conformal dimension  of operators in the gauge theory. JJ’ J and J’ are U(1) charges. J’ generates a U(1) which is in the SU(2) subgroup of the R-symmetry and has half-integers eigenvalues J generates the transformation that rotates A and B of opposite angles. The domain of the angle  is reduced from 2  to 2  /M by the orbifold identification The eigenvalues of J are integer multiple of M

19. 19 We can then recall the combination of ,  and t which were used to form the light- cone coordinates to deduce the light-cone momenta  (MJ+J’) We will focus on those states of the gauge theory where these quantum numbers remain finite in the double scaling limit. In particular  and (MJ+J’) are going to be large with their difference fixed We should look for gauge invariant states which have H=0 for the ground state of the string, H=1 for the oscillator excited states and so on.

20. 20 The important thing is that the bi-fundamental field A has H=0 and the simplest gauge invariant operator that can be made out of them is This operator has H=0 and  =M. This implies that it’s one unit of DLCQ momentum We can identify this state with the string ground state in the sector with one unit of DLCQ momentum (and no winding) string winding Pictorially, this operator is a string of fields winding around the quiver momentum state But in the string theory this is a momentum state

21. 21 Now we can construct all the other DLCQ momentum states by multiplying A 1 up to A M around the quiver k times and then taking the trace at the end k-times In this way one gets the ground state of the string in DLCQ momentum k. It’s like joining k string bits with k=1 to get a string state with 2p + =k/R - For a state with k=2

22. 22 zero-mode string oscillator states. The next step is to construct the zero-mode string oscillator states. These should have H=1. We should follow the BMN prescription and insert all the fields that are transverse to the R-symmetry that we picked up Let’s focus on . It’s an adjoint and can be inserted by sticking it into a quiver diagram at one particular point and then sum over the location of this  all over the quiver diagram In the k=1 sector we have the operator This operator is created by acting with the zero mode on the k=1 DLCQ ground state and similarly for The other four states,as usual, come from by replacing A I by a derivative of A I Thus,for k=1, we have identified the 8 zer- mode bosonic oscillator of the string. For general k the construction is analogous.

23. 23 BMN found that if you insert a single impurity and a phase, then cyclicity of the trace sets that operator to zero. In our case that does not happen because, for example, if we sum over the location of  I, we have a different field for every different index I, it’s an adjoint of a different group. This state is generically non-zero winding state of winding number m and DLCQ momentum 1. We should think of this state as a winding state of winding number m and DLCQ momentum 1. The level matching condition comes from realizing that the actual periodicity of the operator is I  I +M rather thatn I  I +kM. This requires that n = km A generalization of this state to DLCQ momentum k is given by this is the level matching condition level matching condition and m is the winding number

24. 24 The single oscillator state is no longer a protected operator. Its dimension  should get radiative corrections beyond the tree level in YM theory, even for planar diagrams. In fact, it must get such corrections if it is to match the string spectrum for planar diagrams In the double scaling limit all non-planar corrections vanish. Our YM computation predicts that the spectrum of this state in string theory does not receive string loop corrections We shall see that it produces this spectrum to one order in g 2 YM The winding state with two which corresponds to two oscillator states and have energies 2p - =2+corrections, reads The quantization condition that one finds is the level matching condition Analogously one can construct a general winding state with more excitation and fermionic states

25. 25 Spectrum of Strings from YM Theory According to the AdS/CFT duality, one can compute the energy spectrum of the string states by the anomalous dimensions of the corresponding gauge theory operators The scaling dimensions   of a set of conformal fields Ô  can be computed from the 2-point function which, at tree level, has the form operator mixing However, when perturbative corrections are considered, one has to keep into account operator mixing : operators with the same tree level dimension get mixed up when perturbative corrections are introduced The general form of the 2-point function at one-loop order is D 0 is the tree level anomalous dimension  is a renormalization constant S  is the tree level mixing matrix T  keeps into account the interactions The matrix elements S  can be computed as expectation value of a zero-dim Gaussian matrix model where all the fields are space-time independent

26. 26 dilatation operator For the BMN operators the matrix elements of the dilatation operator can be written as Just as in the N =4 SYM theory, the computation of anomalous dimensions is elegantly summarized by the action of an effective Hamiltonian. It can be shown that only the F-term of the potential account for the entire radiative correction to one-loop order for the conformal dimensions of products of scalar fields and the relevant Hamiltonian for operators with impurities of the type  is The fields that arise from the contractions are again linear combinations of the operators thus The dilatation matrix becomes The anomalous dimension of the operators O  can be obtained diagonalizing H   without explicitly knowing S 

27. 27 There is a basis of the operators O  determined by the quantum number k and the number and type of impurities  of traces = 1,...k One immediate result is that the DLCQ ground state of momentum k do not get corrections because it does not contain impurities This set of operators is related to chiral primary operators of N =4 SYM and shares that property For the states we presented before the tree level eigenvalue of the dilatation matrix is D 0 = kM +  of impurities

28. 28 one impurity = one oscillator state Consider the simplest states, those with k=1 The level matching condition n=km=m is trivially obeyed since the TR(...) is periodic of period M. This is kept into account by the Fourier transform The action of the dilatation operator is found by performing the Wick contractions In the large M limit, recalling that for this state 2p - =D-M This coincides with the string theory result providing the exact one loop correction There cannot be any string loop corrections in this case

29. 29 case k=2 and one impurity The eigenstates of the dilatation operator are The eigenvalues are The eigenstates have an interesting form. They are a mixture of one-trace and two-traces operators which one would normally associate with one-string state and two-string states. The mixing does not depend on the coupling constant so does not go away when the coupling constant is made small In the double scaling limit the eigenvalues are degenerate and the free string spectrum is exact, string loop corrections vanish In the double scaling limit the mixing between single and multi traces vanishes. This happens for any value of k The one-impurity states get YM loop corrections to all orders in perturbation theory from planar diagrams, but all non-planar diagrams vanish

30. 30 K=1, Two Impurities Any number of impurities, only one single trace state, no mixing is possible. It cannot be split into multi-trace operators by the action of the effective interaction Hamiltonian All the k=1 states correspond to free string states that cannot have non-planar corrections. This result reproduces the requisite string energy spectrum for the case of two oscillators up to one loop order. No string loop corrections. For the case of two impurities the spectrum is given by

31. 31 K=2, Two Impurities Consider the gauge theory operators with two impurity  fields that describe the string theory sector with light-cone momentum k=2. The basis of operators is There are linear combinations of these operators which are special. For example It is a periodic state in that The spectrum is Similarly the combination has the same property, obeys the same boundary condition and therefore has the same spectrum

32. 32 These states account for half of the allowed states. The two oscillator state of the string expanded to the leading order has the spectrum with the level matching condition From the latter n 1 and n 2 are either both even or both odd. In the above we have found two towers of states where they are both even. The first could be associated with a one string state with k=2 and two oscillators excited, both with even world-sheet momenta. The second corresponds to two one-string states, each string carrying k=1 There are two states left. There are two states left. Operator mixing can be diagonalized in the double scaling limit by taking the linear combinations with the boundary condition

33. 33 Introducing the variables x= I /M and y= J /M, O + and O - in the double scaling limit obey the equations where the antisymmetric step function defined as Introducing the center of mass and relative coordinates The variables separate and we can make the following ansatz The eigenvalues of the dilatation operator are

34. 34 The eigenvalues can be obtained by solving with be boundary condition Let’s focus on the equation for u(r), the eq. for v(r) is identical with g 2 replaced by –g 2 This is a trivial one-dimensional eigenvalue problem that leads to which determines the eigenvalue

35. 35 The light-cone momenta for these two states are By taking where n 1  n 2 are necessarily even, we get Imposing the boundary condition one finds that there are two possibilities for the integers m and n: 1) m is even and n is odd 2) m is odd and n is even. In both cases this implies that both n 1 and n 2 are odd integers Renaming the constant which governs the world-sheet energy Eq.n (  ) can be written as This is an exact result for string states with k=2. Free string spectrum The interaction term truncates at second order in the closed string coupling ()()

36. 36 only string states with two oscillators with world-sheet momenta n 1 and n 2 multiples of 3 behave as free string states K=3, Two Impurities There are 9 operators: 3 with one trace, 4 with two traces, 2 with three traces Can be constructed 5 linear combinations of these operators which are periodic of period 1 and have the following spectrum n 1 and n 2 must be multiple of 3 In order to be states periodic of period 1, n 1 and n 2 must be multiple of 3 no string loop corrections For the remaining 4 states, one should study the spectrum in the double scaling limit As in the k=2, we get a one-dim Schroedinger system of two equations The system can be decoupled to give a trascendental equation for the eigenvalue of the dilatation operator Solving the trascendental equation iteratively in g 2 2 we find that n 1 and n 2 must not be mulptiple of 3 and that there are string loop corrections to all orders in the genus expansion

37. 37 The eigenvalue of the light-cone Hamiltonian for these states up to 2-loops in the genus expansion is given by Higher loop corrections can be easily obtained by solving the trascendental equation

38. 38 Conclusions… The AdS/CFT correspondence, thought it has many spectacular successes, it is still a conjecture and it is not yet clear whether it is an exact correspondence or is only valid in some limits of two theories, so it is important to check it wherever possibleThe AdS/CFT correspondence, thought it has many spectacular successes, it is still a conjecture and it is not yet clear whether it is an exact correspondence or is only valid in some limits of two theories, so it is important to check it wherever possible We have provided a simple setting where this idea can be checked explicitlyWe have provided a simple setting where this idea can be checked explicitly From the duality between an N=2 quiver gauge theory and DLCQ type II string on a pp-wave background one finds these simple results:From the duality between an N=2 quiver gauge theory and DLCQ type II string on a pp-wave background one finds these simple results:  States with one oscillator and any value of the DLCQ momentum have a free string energy spectrum with no string loop corrections  States with any number of oscillators and 1 unit of light-cone momentum have a free energy spectrum with no string loop corrections

39. 39 …and outlook The existence of a positive definite discrete light-cone momentum greatly simplifies the gauge theory predictions b.The spin model corresponding to this quiver N =2 YM theory is still to be studied and also higher gauge theory loops... a.All the results found from the gauge theory should be easily checked from the string side of the correspondence. Success or failure of this matching would be a highly non trivial test of the AdS/CFT correspondence at the level of interacting strings  States with two oscillators and 2 units of light- cone momentum can have either a free spectrum or a simple one string loop correction  States with two oscillators and light-cone momentum k  3 can have either a free spectrum or computable corrections (at least for small k) to all orders in the genus expansion