Non-renormalisation of extremal correlators in N=4 SYM theory

We show that extremal correlators of chiral primary operators in N=4 supersymmetric Yang-Mills theory with SU(N) gauge group are neither renormalised at first order in perturbation theory nor receive contribution from any instanton sector at leading order in the semiclassical expansion. This lends support to the strongest version of a new prediction recently put forward on the basis of the AdS/SCFT correspondence.Comment: 13 pages, LaTeX2e; definition of extremal correlators moved to the introduction, references added; final version to appear in Physics Letters

Holography and the Higgs branch of N=2 SYM theories

We present a proposal for the description of the Higgs branch of four-dimensional N=2 supersymmetric Yang-Mills theories in the context of the AdS/CFT correspondence. We focus on a finite Sp(N) N=2 theory arising as dual of a configuration of N D3-branes in the vicinity of four D7-branes and an orientifold 7-plane in type I' string theory. The field theory contains hypermultiplets in the second rank anti-symmetric and in the fundamental representations. The Higgs branch has a dual description in terms of gauge field configurations with non-zero instanton number on the world-volume of the D7-branes. In this setting the non-renormalisation of the metric on the Higgs branch implies constraints on the alpha' corrections to the D7-brane effective action, including couplings to the curvature and five-form field strength. In the second part of the paper we discuss non-renormalisation properties of BPS Wilson lines, which are closely related to the physics of the Higgs branch. Using a formulation of the four-dimensional N=2 theory in terms of a three-dimensional N=2 superspace we show that the expectation value of certain Wilson-line operators with hypermultiplets at the end points is independent of the length and thus coincides with the expectation value of the local operators parametrising the Higgs branch.Comment: LaTeX, 29 pages, 1 figure; v2: minor corrections, one reference adde

Membranes from monopole operators in ABJM theory: large angular momentum and M-theoretic AdS_4/CFT_3

We consider states with large angular momentum to facilitate the study of the M-theory regime of the AdS_4/CFT_3 correspondence. We study the duality between M-theory in AdS_4xS^7/Z_k and the ABJM N=6 Chern-Simons-matter theory with gauge group U(N)xU(N) and level k, taking N large and k of order 1. In this regime the lack of an explicit formulation of M-theory in AdS_4xS^7/Z_k makes the gravity side difficult, while the CFT is strongly coupled and the planar approximation is not applicable. To overcome these difficulties, we focus on states on the gravity side with large angular momentum J>>1 and identify the dual operators in the CFT, thereby establishing the AdS/CFT dictionary in this sector. Natural approximation schemes arise on both sides thanks to the presence of the small parameter 1/J. On the AdS side, we use the matrix model of M-theory on the maximally supersymmetric pp-wave background with matrices of size J/k. A perturbative treatment of this matrix model provides a good approximation to M-theory in AdS_4xS^7/Z_k when N^{1/3}<<J<<N^{1/2}. On the CFT side, we study the theory on S^2xR with magnetic flux J/k. A Born-Oppenheimer type expansion arises naturally for large J in spite of the theory being strongly coupled. The energy spectra on the two sides agree at leading order. This provides a non-trivial test of the AdS_4/CFT_3 correspondence including near-BPS observables associated with membrane degrees of freedom, thus verifying the duality beyond the previously studied sectors corresponding to either BPS observables or the type IIA string regime.Comment: 67 pages, 5 figures; V2: minor changes, references adde

On membrane interactions and a three-dimensional analog of Riemann surfaces

Membranes in M-theory are expected to interact via splitting and joining processes. We study these effects in the pp-wave matrix model, in which they are associated with transitions between states in sectors built on vacua with different numbers of membranes. Transition amplitudes between such states receive contributions from BPS instanton configurations interpolating between the different vacua. Various properties of the moduli space of BPS instantons are known, but there are very few known examples of explicit solutions. We present a new approach to the construction of instanton solutions interpolating between states containing arbitrary numbers of membranes, based on a continuum approximation valid for matrices of large size. The proposed scheme uses functions on a two-dimensional space to approximate matrices and it relies on the same ideas behind the matrix regularisation of membrane degrees of freedom in M-theory. We show that the BPS instanton equations have a continuum counterpart which can be mapped to the three-dimensional Laplace equation through a sequence of changes of variables. A description of configurations corresponding to membrane splitting/joining processes can be given in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann surface, consisting of multiple copies of R^3 connected via a generalisation of branch cuts. We discuss various general features of our proposal and we also present explicit analytic solutions.Comment: 64 pages, 17 figures. V2: An appendix, a figure and references added; various minor changes and improvement

Mixing of the RR and NSNS sectors in the BMN limit

This paper concerns instanton contributions to two-point correlation functions of BMN operators in N=4 supersymmetric Yang-Mills that vanish in planar perturbation theory. Two-point functions of operators with even numbers of fermionic impurities (dual to RR string states) and with purely scalar impurities (dual to NSNS string states) are considered. This includes mixed RR - NSNS two-point functions. The gauge theory correlation functions are shown to respect BMN scaling and their behaviour is found to be in good agreement with the corresponding D-instanton contributions to two-point amplitudes in the maximally supersymmetric IIB plane-wave string theory. The string theory calculation also shows a simple dependence of the mass matrix elements on the mode numbers of states with an arbitrary number of impurities, which is difficult to extract from the gauge theory. For completeness, a discussion is also given of the perturbative mixing of two-impurity states in the RR and NSNS sectors at the first non-planar level.Comment: latex, 29 pages, 4 figure

N=4 supersymmetric Yang-Mills theory and the AdS/SCFT correspondence

This dissertation reviews various aspects of the N=4 supersymmetric Yang--Mills theory in particular in relation with the AdS/CFT correspondence. The first two chapters are introductory. The first one contains a description of the general properties of rigid supersymmetric theories in four dimensions both at the classical and at the quantum level. The second chapter is a review of the main properties of the N=4 SYM theory under consideration. Original results are reported in chapters 3, 4 and 5. A systematic re-analysis of the perturbative properties of the theory is presented in the third chapter. Two-, three- and four-point Green functions of elementary fields are computed using the component formulation and/or the superfield approach and subtleties related to the gauge-fixing are pointed out. In the fourth chapter, after an introduction to instanton calculus in supersymmetric gauge theories, the computation of the one-instanton contributions to Green functions of gauge invariant composite operators in the semiclassical approximation is reported. The calculations of four-, eight- and sixteen-point Green functions of operators in the supercurrent multiplet are reviewed in detail. The final chapter is devoted to the AdS/SCFT correspondence. Some general aspects are discussed. Then the attention is focused on the relation between instantons in N=4 SYM and D-instanton effects in type IIB string theory. The comparison between instanton contributions to Green functions of composite operators in the boundary field theory and D-instanton generated terms in the amplitudes computed in type IIB string theory is performed and agreement between these two sources of non-perturbative effects is shown.Comment: 228 pages, Ph.D. thesis, LaTeX2e, uses the feynmp, fencyhdr, graphicx and AMS package

Properties of the Konishi multiplet in N=4 SYM theory

We study perturbative and non-perturbative properties of the Konishi multiplet in N=4 SYM theory in D=4 dimensions. We compute two-, three- and four-point Green functions with single and multiple insertions of the lowest component of the multiplet, and of the lowest component of the supercurrent multiplet. These computations require a proper definition of the renormalized operator and lead to an independent derivation of its anomalous dimension. The O(g^2) value found in this way is in agreement with previous results. We also find that instanton contributions to the above correlators vanish. From our results we are able to identify some of the lowest dimensional gauge-invariant composite operators contributing to the OPE of the correlation functions we have computed. We thus confirm the existence of an operator belonging to the representation 20', which has vanishing anomalous dimension at order g^2 and g^4 in perturbation theory as well as at the non-perturbative level, despite the fact that it does not obey any of the known shortening conditions.Comment: 23 pages, latex, no figure

On the logarithmic behaviour in N=4 SYM theory

We show that the logarithmic behaviour seen in perturbative and non perturbative contributions to Green functions of gauge-invariant composite operators in N=4 SYM with SU(N) gauge group can be consistently interpreted in terms of anomalous dimensions of unprotected operators in long multiplets of the superconformal group SU(2,2|4). In order to illustrate the point we analyse the short-distance behaviour of a particularly simple four-point Green function of the lowest scalar components of the N=4 supercurrent multiplet. Assuming the validity of the Operator Product Expansion, we are able to reproduce the known value of the one-loop anomalous dimension of the single-trace operators in the Konishi supermultiplet. We also show that it does not receive any non-perturbative contribution from the one-instanton sector. We briefly comment on double- and multi-trace operators and on the bearing of our results on the AdS/SCFT correspondence.Comment: 18 pages, Late

Instantons in supersymmetric Yang-Mills and D-instantons in IIB superstring theory

The one-instanton contributions to various correlation functions of supercurrents in four-dimensional N=4 supersymmetric SU(2) Yang-Mills theory are evaluated to the lowest order in perturbation theory.Expressions of the same form are obtained from the leading effects of a single D-instanton extracted from the IIB superstring effective action around the AdS5*S5 background. This is in line with the suggested AdS/Yang-Mills correspondence. The relation between Yang--Mills instantons and D-instantons is further confirmed by the explicit form of the classical D-instanton solution in the AdS5*S5 background and its associated supermultiplet of zero modes. Speculations are made concerning instanton effects in the large-N_c limit of the SU(N_c) Yang-Mills theory.Comment: 41 pages, LaTeX. Typos corrected and minor clarifications adde