The Dilatation Operator of N=4 Super Yang-Mills Theory and Integrability

TL;DR

The work develops a comprehensive, algebraic construction of the dilatation operator in N=4 SYM, demonstrating that one-loop scaling dimensions are completely fixed by superconformal symmetry and that the planar dilatation generator maps to an integrable psu(2,2|4) spin chain. It then extends these methods to structured subsectors, deriving higher-loop deformations within closed sectors (notably su(2) and su(1,1) types) and arguing for integrability beyond one loop, including proposals for all-loop Bethe ansatz structures. The analysis connects to AdS/CFT through explicit calculations of plane-wave string Hamiltonians and comparisons with spinning-string energies, as well as via operator scaling dimensions for states with large quantum numbers. Wrapping effects, parity pairings, and the nuanced role of oscillator-based representations are treated to present a coherent, symmetry-driven framework for understanding the spectrum of planar N=4 SYM and its string duals. The results provide a robust platform for precision tests of the gauge/string duality and for exploring the rich integrable structure of this maximally supersymmetric theory.

Abstract

The dilatation generator measures the scaling dimensions of local operators in a conformal field theory. In this thesis we consider the example of maximally supersymmetric gauge theory in four dimensions and develop and extend techniques to derive, investigate and apply the dilatation operator. We construct the dilatation operator by purely algebraic means: Relying on the symmetry algebra and structural properties of Feynman diagrams we are able to bypass involved, higher-loop field theory computations. In this way we obtain the complete one-loop dilatation operator and the planar, three-loop deformation in an interesting subsector. These results allow us to address the issue of integrability within a planar four-dimensional gauge theory: We prove that the complete dilatation generator is integrable at one-loop and present the corresponding Bethe ansatz. We furthermore argue that integrability extends to three-loops and beyond. Assuming that it holds indeed, we finally construct a novel spin chain model at five-loops and propose a Bethe ansatz which might be valid at arbitrary loop-order! We illustrate the use of our technology in several examples and also present two key applications for the AdS/CFT correspondence.

Figures (33)

• Figure 1: Classical supertranslation variations of the fields in spinor notation. Left, vertical and right arrows correspond to generators $\dot{\mathfrak{Q}}$, $\mathfrak{P}$ and $\mathfrak{Q}$, respectively. We have dropped all commutators of fields which are suppressed for a vanishing coupling constant $g$.
• Figure 2: A contribution to the quantum expectation value of the operator $\mathcal{O}$ (Feynman graph).
• Figure 3: The contraction of a matrix-valued variation $\check\mathcal{W}$ and field $\mathcal{W}$.
• Figure 4: A fat graph. The circles are $\mathrm{U}(N)$ traces of the vertices and operators and the fat lines are propagators. For each face of the graph there is a closed empty trace $\mathop{\operator@font Tr}\nolimits 1=N$.
• Figure 5: The Dynkin diagram of $\mathfrak{psu}(2,2|4)$ convenient for $\mathcal{N}=4$ SYM.