Operators in the d=4, N=4 SYM and the AdS/CFT correspondence

TL;DR

This paper explores the AdS/CFT correspondence focusing on operators in N=4 SYM, including BPS and non-BPS types, and compares their properties with supergravity and string theory predictions, providing new insights into their dualities.

Contribution

It identifies and analyzes quarter-BPS and BMN operators, computes their three-point functions, and matches string theory corrections with gauge theory results, advancing understanding of the duality beyond protected operators.

Findings

Quarter-BPS operators have non-renormalized scaling dimensions.

First 1/R^2 and 1/J corrections agree between string theory and gauge theory.

BMN operators' perturbative expansions match dual supergravity predictions.

Abstract

In this dissertation we explore various aspects of the AdS/CFT correspondence. As string quantization on general backgrounds with fluxes is very difficult, one often uses the duality at the level of canonical fields of supergravity and the corresponding half-BPS operators in SYM, since they both belong to the shortest multiplets of the superconformal group SU(2,2|4). In addition to half-BPS operators, there are others with non-renormalization properties. One such class of operators is the quarter-BPS operators, which are dual to threshold bound states of elementary supergravity excitations. Their scaling dimension is also determined by their internal quantum numbers. Extended superspace methods make it simple to identify and remove descendant pieces from quarter-BPS candidates. We also compute three-point functions involving quarter-BPS operators, and explain how their non-renormalization translates into statements about the dual supergravity quantities. But we can go beyond discussing protected operators. The GS superstring on AdS(5) x S(5) can be quantized exactly in the limit where the AdS(5) radius R goes to infinity and the R-charge J scales like R^2. BMN states are then dual to single trace operators with certain phases inserted (BMN operators). BMN operators are another natural generalization of half-BPS operators. The perturbative expansion of scaling dimensions of BMN operators is in powers of gN/J^2. Moreover, one can do perturbation theory around the infinite R, J ~ R^2 limit. Both expansions have the same regime of validity in string theory and in SYM. We calculate the first 1/R^2 and 1/J corrections, and find complete agreement between the dual quantities.