Strings and Spins in Deformed AdS/CFT
Creators: name: Wyss, Leander
Publisher: name: University of Surrey
Year: 2023

description: In the first part of this thesis, we consider the scattering problem for the massless modes that appear in the context of AdS3/CFT2 from the perspective of integrable models. We start by considering the extension of the psu(1|1)^2 symmetry algebra by a Modified Poincaré algebra. Having found a consistent Hopf algebra structure, we construct suitable R-matrices, both for the undeformed and the q-deformed case of the centrally-extended su(1|1)^2 algebra, and study the found coproducts in detail. We find interesting connections between the boost operator J and the R-matrices. In addition, we encounter a non-coassociative structure for one of the cases we are interested in. This prompts us to make use of the notions of quasi-Hopf algebras and coassociators.<br>
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description: We then move from the framework of a 1+1-dimensional short representation to the analysis of the boost operator and its coproduct in a universal, representation-independent sense. This leads us to establish a classification of boost algebras and coproducts. We arrive at six different algebraic structures, each with its own universal coproduct for the boost operator. Finally, we put particular focus on the cases that can sensibly appear within the context of AdS3 physics.<br>
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description: Moving on from Hopf algebraic considerations, we stay within the context of AdS3 string theory, and consider a particular 3-parameter deformed background in the Landau-Lifshitz limit. We construct an effective field theory from the Polyakov action associated to this background. This way, we obtain a Lagrangian up to next-to-leading order in the energy parameter κ, allowing us to find kinematical quantities such as the dispersion relation of these theories. We then introduce one single complex field that parametrises the relevant coordinates of the theory (corresponding to one mode stemming from the S^3 part), allowing us to canonically quantise it in the standard way. Analysing the diagrammatics of the so-obtained quantum field theory, we find interesting peculiarities related to the propagator and the ground state, and proceed to compute two-body S-matrix elements, both at leading and subleading order in the string tension λ.<br>
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description: In the latter part of this thesis, we then move on to a study in linear algebra, as well as particular spin chains and their Hamiltonians. We first construct an algorithmic framework that allows us to infer the generalised eigensystem of a defective complex matrix by perturbing it in such a way that it becomes diagonalisable. In this case, the eigensystem is directly accessible, and from there we can then go back to the limiting case of non-diagonalisability by again turning off the perturbation. In this construction, we find curious particularities and differences for the case of singular and non-singular geometric multiplicity of the considered eigenvalue, the latter case needing more caution in its analysis. In order to apply this recipe to the eclectic spin chain, we make use of the Nested Coordinate Bethe Ansatz and find that the analysis of the spectrum of the twisted spin chain contains sufficient information about the generalised eigensystem of the eclectic spin chain for our developed methodology to be successful.<br>