Alberto Santambrogio

Exact anomalous dimensions of N=4 Yang–Mills operators with large R charge

Abstract In a N =1 superspace formulation of N =4 Yang–Mills theory we obtain the anomalous dimensions of chiral operators with large R charge J→∞ keeping g2N/J2 finite, to all orders of perturbation theory in the planar limit. Our result proves the conjecture that the anomalous dimensions are indeed finite in the above limit. This amounts to an exact check of the proposed duality between a sector of N =4 Yang–Mills theory with large R charge J and string theory in a pp-wave background.

Exact results in planar Script N = 1 superconformal Yang-Mills theory

In the β-deformed = 4 supersymmetric SU(N) Yang-Mills theory we study the class of operators J = Tr(ΦiJΦk), i≠k and compute their exact anomalous dimensions for N,J→∞. This leads to a prediction for the masses of the corresponding states in the dual string theory sector. We test the exact formula perturbatively up to two loops. The consistency of the perturbative calculation with the exact result indicates that in the planar limit the one-loop condition g2 = h for superconformal invariance is indeed sufficient to insure the exact superconformal invariance of the theory. We present a direct proof of this point in perturbation theory. The J sector of this theory shares many similarities with the BMN sector of the = 4 theory in the large R-charge limit.

Two-point correlators in the beta-deformed N=4 SYM at the next-to-leading order.

We compute two-point functions of lowest weight operators at the next-to-leading order in the couplings for the β-deformed = 4SU(N) SYM. In particular we focus on the CPO Tr(Φ12) and the operator Tr(Φ1Φ2) not presently listed as BPS. We find that for both operators no anomalous dimension is generated at this order, then confirming the results recently obtained at lowest order in [5]. However, in both cases a finite correction to the two-point function appears.

Non-protected operators in N=4 SYM and multi-particle states of AdS_5 SUGRA

We study a class of non-protected local composite operators which occur in the R symmetry singlet channel of the OPE of two stress-tensor multiplets in N = 4 SYM. At tree level these are quadrilinear scalar dimension four operators, two single-traces and two double-traces. In the presence of interaction, due to a non-trivial mixing under renormalization, they split into linear combinations of conformally covariant operators. We resolve the mixing by computing the one-loop two-point functions of all the operators in an N = 1 setup, then diagonalizing the anomalous dimension matrix and identifying the quasiprimary operators. We find one operator whose anomalous dimension is negative and suppressed by a factor of 1/N 2 with respect to the anomalous dimensions of the Konishi-like operators. We reveal the mechanism responsible for this suppression and argue that it works at every order in perturbation theory. In the context of the AdS/CFT correspondence such an operator should be dual to a multiparticle supergravity state whose energy is less than the sum of the corresponding individual single-particle states.

F5 contributions to the non-Abelian Born–Infeld action from a supersymmetric Yang–Mills five-point function

Abstract We consider the N =4 supersymmetric Yang–Mills theory in four dimensions. We compute the one-loop contributions to the effective action with five external vector fields and compare them with corresponding results in open superstring theory. Our calculation determines the structure of the F 5 terms that appear in the non-Abelian generalization of the Born–Infeld action. The trace operation on the gauge group indices receives contributions from the symmetric as well as the antisymmetric part. We find that in order to study corrections to the symmetrized trace prescription one has to consistently take into account derivative contributions not only with antisymmetrized products ∇ [ μ ∇ ν ] but also with symmetrized ones ∇ ( μ ∇ ν ) .

Superspace approach to anomalous dimensions in N=4 SYM

Abstract In a N =1 superspace setup and using dimensional regularization, we give a general and simple prescription to compute anomalous dimensions of composite operators in N =4 , SU ( N ) supersymmetric Yang–Mills theory, perturbatively in the coupling constant g. We show in general that anomalous dimensions are responsible for the appearance of higher order poles in the perturbative expansion of the two-point function and that their lowest contribution can be read directly from the coefficient of the 1/ ϵ 2 pole. As a check of our procedure we rederive the anomalous dimension of the Konishi superfield at order g 2 . We then apply this procedure to the case of the double trace, dimension 4, superfield in the 20 of SU (4) recently considered in the literature. We find that its anomalous dimension vanishes for all N in agreement with previous results.

Four-point correlators of BPS operators in N=4 SYM at order g4

We study the large N degeneracy in the structure of the four-point amplitudes of 1 BPS operators of arbitrary weight k in perturbative N = 4 SYM theory. At one loop (order g 2 ) this degeneracy manifests itself in a smaller number of independent conformal invariant functions describing the amplitude, compared to AdS5 supergravity results. To study this phenomenon at the two-loop level (order g 4 ) we consider a particular N = 2 hypermultiplet projection of the general N = 4 amplitude. Using the formalism of N = 2 harmonic superspace we then explicitly compute this four-point correlator at two loops and identify the corresponding conformal invariant functions. In the cases of 1 -BPS operators of weight k = 3 and k = 4 the one-loop large N degeneracy is lifted by the two-loop corrections. However, for weight k > 4 the degeneracy is still there at the two-loop level. This behavior suggests that for a given weight k the degeneracy will be removed if perturbative corrections of sufficiently high order are taken into account. These results are in accord with the AdS/CFT duality conjecture.

The one-loop effective action of noncommutative super Yang–Mills is gauge invariant

Abstract We study the gauge transformation of the recently computed one-loop four-point function of N =4 supersymmetric Yang–Mills theory with gauge group U ( N ). The contributions from nonplanar diagrams are not gauge invariant. We compute their gauge variation and show that it is cancelled by the variation from corresponding terms of the one-loop five-point function. This mechanism is general: it insures the gauge invariance of the noncommutative one-loop effective action.

On the perturbative chiral ring for marginally deformed N = 4 SYM theories

For =1 SU(N) SYM theories obtained as marginal deformations of the =4 parent theory we study perturbatively some sectors of the chiral ring in the weak coupling regime and for finite N. By exploiting the relation between the definition of chiral ring and the effective superpotential we develop a procedure which allows us to easily determine protected chiral operators up to n loops once the superpotential has been computed up to (n-1) order. In particular, for the Lunin–Maldacena β–deformed theory we determine the quantum structure of a large class of operators up to three loops. We extend our procedure to more general Leigh–Strassler deformations whose chiral ring is not fully understood yet and determine the weight–two and weight–three sectors up to two loops. We use our results to infer general properties of the chiral ring.

Conformal invariance of the planar beta-deformed N=4 SYM theory requires beta real.

We study the = 1 SU(N) SYM theory which is a marginal deformation of the = 4 theory, with a complex deformation parameter β. We consider the large N limit and study perturbatively the conformal invariance condition. We find that finiteness requires reality of the deformation parameter β.