Schlur polynomials, restricted Schur polynomials and the AdS/CFT correspondence

Stephanou, Michael Jared
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In this thesis we introduce novel technology pertaining to the calculation of certain classes of correlation functions in the half BPS, near BPS and multimatrix sectors (particularly the SU(2) sector) of N = 4 Super Yang-Mills (SYM) theory using the Schur polynomial and restricted Schur polynomial bases. This technology allows the detailed exploration of a host of physical questions within the AdS/CFT correspondence. We rst treat multi-point correlators of restricted Schur polynomials which constitute a particularly convenient basis for the SU(2) and general multi-matrix sectors of N = 4 SYM. We introduce a product rule for the restricted Schur polynomials, which, together with the two point correlator result of arXiv:0801.2061 allows us to compute exact multi-point correlators, in the free eld theory limit. This facilitates the exploration of physical questions such as elucidating the appropriate degrees of freedom for a perturbative description of quantum gravity in the sectors under consideration. We then treat the calculation of correlators of operators in the presence of large background operators with a R-charge of O(N2) that are dual to asymptotically AdS5 S5 backgrounds. Evaluating these correlators is a hard problem in general since the planar approximation fails. In this thesis we develop general techniques, known as the \cutting rules", that allow the computation of such correlators. Specializing to the LLM annulus geometry allows a number of concrete results to be derived. We then study the perturbation theory of these correlators and identify new perturbative expansion parameters replacing 1=N. Motivated by these results we explore new BMN-type sectors. Finally, we treat the problem of computing the anomalous dimensions of a class of (nearly) half BPS operators with a large R-charge. This problem is reduced to the problem of diagonalizing a Cuntz oscillator chain. Non-planar corrections must be summed to correctly construct the Cuntz oscillator dynamics. We explore whether these non-planar corrections that account for backreaction from the heavy operator rather than quantum corrections in the dual gravitational theory also spoil integrability as is generally the case with quantum corrections. We nd a limit in which our Cuntz chain continues to admit extra conserved charges suggesting that integrability might survive.