By considering AdS 5 × S 5 string states with large angular momenta in S 5 one is able to provide non-trivial quantitative checks of the AdS/CFT duality. A string rotating in S 5 with two angular momenta J 1 ,J 2 is dual to an operator in N = 4 SYM theory whose conformal dimension can be computed by diagonalizing a (generalization of) spin 1/2 Heisenberg chain Hamiltonian. It was recently argued and verified to lowest order in a large J = J 1 + J 2 expansion, that the Heisenberg chain can be described using a non-relativistic low energy effective 2-d action for a unit vector field n i which exactly matches the corresponding large J limit of the classical AdS 5 × S 5 string action. In this paper we show that this agreement extends to the next order and develop a systematic procedure to compute higher orders in such large angular momentum expansion. This involves several non-trivial steps. On the string side, we need to choose a special gauge with a non-diagonal world-sheet metric which insures that the angular momentum is uniformly distributed along the string, as indeed is the case on the spin chain side. We need also to implement an order by order redefinition of the field n i to get an action linear in the time derivative. On the spin chain side, it turns out to be crucial to include the effects of integrating out short wave-length modes. In this way we gain a better understanding of how (a subsector of) the string sigma model emerges from the dual gauge theory, allowing us to demonstrate the duality beyond comparing particular examples of states with Understanding AdS/CFT duality beyond the BPS or near BPS [1] limit remains an important challenge. It was suggested in [2] that concentrating on string states with large quantum numbers, like angular momentum in AdS 5 , one finds a qualitative (modulo interpolating function of 't Hooft coupling λ) agreement between the AdS 5 string energies and anomalous dimensions of the corresponding gauge theory operators (see also [3,4,5]). About a year ago, it was observed [6] that semiclassical string states with several non-zero angular momenta (with large total S 5 momentum J) have a remarkable property that their energy admits an analytic expansion inλ ≡ λ J 2 at large J. * It was proposed, therefore, that the coefficients of such an expansion can be matched precisely with the perturbative anomalous dimensions of the corresponding scalar SYM operators computed in the same J → ∞,λ < 1 limit [6]. That would provide the first quantitative check of AdS/CFT duality far from the BPS limit. The reason for this expectation was that for such special solutions all string α ′ ∼ 1 √ λ corrections might be suppressed in the large J limit (as was explicitly checked for a particular case in [9]; see also [10] for a review). Then, the classical string energy would represent an exact string theory prediction in this limit. This proposal received a spectacular confirmation in [11,12] where the one-loop anomalous dimensions of the relevant scalar SYM operators were computed utilizing a rema...

Chiral primary operators annihilated by a quarter of the supercharges are constructed in the four dimensional N = 4 Super-Yang-Mills theory with gauge group SU (N ). These 1 4 -BPS operators share many nonrenormalization properties with the previously studied 1 2 -BPS operators. However, they are much more involved, which renders their construction nontrivial in the fully interacting theory. In this paper we calculate O(g 2 ) two-point functions of local, polynomial, scalar composite operators within a given representation of the SU (4) R-symmetry group. By studying these two-point functions, we identify the eigenstates of the dilatation operator, which turn out to be complicated mixtures of single and multiple trace operators.Given the elaborate combinatorics of this problem, we concentrate on two special cases. First, we present explicit computations for 1 4 -BPS operators with scaling dimension ∆ ≤ 7. In this case, the discussion applies to arbitrary N of the gauge group. Second, we carry out a leading plus subleading large N analysis for the particular class of operators built out of double and single trace operators only. The large N construction addresses 1 4 -BPS operators of general dimension. * ryzhovav@physics.ucla.edu 1 The possibility that certain non-chiral operators may have vanishing anomalous dimension was raised in [9].2 Higher n-point functions also agree with supergravity predictions in the large N limit [5]. 3 1 4 -BPS operators have been studied indirectly through OPEs of 1 2 -BPS chiral primaries, see [8], [9], [13], [14]. 4 I would like to thank Sergio Ferrara for bringing this to my attention.Apart from the double trace scalar composite operators in the [p, q, p] of the R-symmetry (flavor) group SU (4) (the free theory chiral primaries from the classification of [2]), there are other single and multiple trace scalar composite operators with the same SU (4) quantum numbers and the same Born level scaling dimension. Unlike in the 1 2 -BPS case where this phenomenon occurs [15], scalar composites in the [p, q, p] generally do not have a well defined scaling dimension. Therefore, one should first find their linear combinations which are eigenstates of the dilatation operator, which we call pure operators. To this end, we calculate two point functions of local, gauge invariant, polynomial, scalar composite operators in a given [p, q, p] representation; diagonalize the dilatation operator within each representation of SU (4); and find that some of the pure operators receive no O(g 2 ) corrections to their scaling dimension or normalization. These operators have the right SU (4) quantum numbers and protected ∆ = 2p + q, and are the only candidates for being the 1 4 -BPS chiral primaries from the classification of [2].Calculating the symmetry factors for Feynman diagrams is a formidable combinatorial problem for general representation [p, q, p] of SU (4), and general N of the gauge group SU (N ). So to keep the formulae manageable, in this paper we concentrate on two special cases. For low dimens...

We study the AdS/CFT correspondence for string states which flow into plane wave states in the Penrose limit. Leading finite radius corrections to the string spectrum are compared with scaling dimensions of finite R-charge BMN-like operators. We find agreement between string and gauge theory results.In the present paper we use this approach to determine the leading order finite radius

We consider N = 1 supersymmetric U (N ), SO(N ), and Sp(N ) gauge theories, with two-index tensor matter and added tree-level superpotential, for general breaking patterns of the gauge group. By considering the string theory realization and geometric transitions, we clarify when glueball superfields should be included and extremized, or rather set to zero; this issue arises for unbroken group factors of low rank. The string theory results, which are equivalent to those of the matrix model, refer to a particular UV completion of the gauge theory, which could differ from conventional gauge theory results by residual instanton effects. Often, however, these effects exhibit miraculous cancellations, and the string theory or matrix model results end up agreeing with standard gauge theory. In particular, these string theory considerations explain and remove some apparent discrepancies between gauge theories and matrix models in the literature.

A systematic construction is presented of 1/4 BPS operators in N = 4 superconformal Yang-Mills theory, using either analytic superspace methods or components. In the construction, the operators of the classical theory annihilated by 4 out of 16 supercharges are arranged into two types. The first type consists of those operators that contain 1/4 BPS operators in the full quantum theory. The second type consists of descendants of operators in long unprotected multiplets which develop anomalous dimensions in the quantum theory. The 1/4 BPS operators of the quantum theory are defined to be orthogonal to all the descendant operators with the same classical quantum numbers. It is shown, to order g 2 , that these 1/4 BPS operators have protected dimensions.