On integrability in gauge/string correspondence

• Main Author: Chen, H.-Y.
• Published: University of Cambridge 2007
• Subjects: 530.1
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597543

In this Thesis, we present some direct quantitative tests for AdS/CFT correspondence using the newly discovered “integrability”. We shall begin by explaining the ideas of algebraic Bethe ansatz and scattering matrix in both gauge and string theories, and discussions how they enable us to extract the spectra of the scaling dimensions of the gauge invariant operators or the energies of the dual string states. In chapter 3, we explicitly apply the Bethe ansatz techniques in thermodynamically limit to the so-called <i>β</i>-deformation of ℵ = 4 SYM, and extract the one-loop anomalous dimensions for its long gauge invariant operators; we also construct the corresponding string solutions in the dual background, and show their energies precisely match with the gauge theory results. In the second part of Thesis, we consider a new asymptotic limit in both gauge and string theories, in such limit the elementary excitations are known as “magnons”; we shall also switch our focus to the building element of Bethe Ansatz, the scattering matrix between magnons. In chapter 4, we apply the magnon scattering matrix and explain how additional stable bound states can appear; both elementary magnons and their bound states have exact expressions for the dispersion relations. The classical string configuration dual to magnon bound state in gauge theory is constructed in chapter 5, where the connection between string sigma model on <i>R x S</i>³ and integrable complex sine-Gordon model is exploited. In chapter 6, the scattering matrix between the magnon bound states is considered via bootstrap method, and in the semiclassical limit, the result coincides with the scattering matrix between complex sine-Gordon solitons. This provides the direct verification for the proposed all-loop magnon scattering matrix and the lowest order “dressing factor”. In chapter 7, by applying the extended residual symmetry algebra psu(2|2)² x ℝ³, we classify all possible magnon bound states in terms of the constituent fields in ℵ = 4 SYM and justify the exactness of their dispersion relation.