The discovery of integrability in planar N=4 super Yang-Mills and ABJM has enabled a precise study of AdS/CFT. In the past decade integrability has been successfully applied to the spectrum of anomalous dimensions, which can now be obtained at any value of the coupling. However, in order to solve conformal field theories one also needs to understand their structure constants. Recently, there has been great progress in this direction with the all-loop proposal of Basso, Komatsu and Vieira. But there is still much to understand, as it is not yet possible to use that formalism to find structure constants of short operators at strong coupling. It is important to study wrapping corrections and resum them as the TBA did for the spectrum. It is also crucial to obtain perturbative data that can be used to check if the all-loop proposal is correct or if there are new structures that need to be unveiled. In this thesis we compute several structure constants of short operators at strong coupling, including the structure constant of Konishi with half-BPS operators. Still at strong coupling, we find a relation between the building blocks of superstring amplitudes and the tensor structures allowed by conformal symmetry. We also consider the case of extremal correlation functions and the relation of their poles to mixing with double-trace operators. We also study three-point functions at weak coupling. We take the OPE limit in a four-point function of half-BPS operators in order to shed some light on the structure of five-loop wrapping corrections of the Hexagon form factors. Finally, we take the first steps in the generalization of the Hexagon programme to other theories. We find the non-extremal setup in ABJM and the residual symmetry that it preserves, which we use to fix the two-particle form factor and constrain the four-particle hexagon. Finally, we find that the Watson equations hint at a dressing phase that needs to be further investigated.

In this thesis, we extend the numerical S-matrix bootstrap program to 1+1d theories with a boundary, where we bootstrap the 1-to-1 reflection matrix (R-matrix). We review the constraints that a physical R-matrix must obey, namely unitarity, analyticiy and crossing symmetry. We then carve out the allowed space of 2d R-matrices with the O(N) nonlinear sigma model and the periodic Yang Baxter solution in the bulk. We find a variety of integrable R-matrices along the boundary of the allowed space both with and without free parameters. The integrable models without a free parameter appear at vertices of the allowed space, while those with a free parameter occupy the whole boundary. We also introduce the extended analyticity constraint where we increase the domain of analyticity beyond the physical region. In some cases, the allowed space of R-matrices shrinks drastically and we observe new vertices which correspond to integrable theories. We also find a new integrable R-matrix through our numerics, which we later obtained by solving the boundary Yang--Baxter equation. Finally, we derive the dual to the extended analyticity problem and find that the formalism allows for R-matrices which do not saturate unitarity to lie on the boundary of the allowed region.

Wir untersuchen die perturbative Quantentheorie verschiedener integrabler Yang-Baxter-Deformationen des freien Superstrings auf AdS-Räumen. Dazu berechnen wir die Zwei-Körper-Streumatrix auf Baum-Niveau auf dem Weltenblatt mit Feynman-Diagramm-Methoden. Die verschiedenen Deformationen sind: (1) Alle abelschen Deformationen von AdS₅ ⨉ S⁵, die die Fixierung der Lichtkegel-Eichung erlauben. Diese sind dual zur nicht-kommutativen Super-Yang-Mills-Theorie und werden äquivalent durch TsT-Transformationen oder verwundene Randbedingungen beschrieben. Wir berechnen die bosonische Streumatrix auf Baum-Niveau für den BMN-String in uniformer Lichtkegel-Eichung. Die Streumatrix wird in den meisten Fällen durch einen Drinfeld-Verwindungen ausgedrückt; in einigen Fällen wird sie stattdessen durch eine verschobene Impulsabhängigkeit ausgedrückt. Abschließend vergleichen wir die aus diesen Ergebnissen abgeleiteten Bethe-Gleichungen mit denen des Modells mit verwundene Randbedingungen und stellen eine perfekte Übereinstimmung fest. Für Deformationen des GKP-Strings können wir aufgrund konzeptioneller Hindernisse keine deformierte Streumatrix um die Null-Cusp-Lösung bestimmen. (2) Die inhomogene oder eta-Deformation von AdS₅ ⨉ S⁵ entsprechend dem fermionischen Dynkin-Diagramm. Wir berechnen die Zwei-Körper-Streumatrix auf Baum-Niveau zu quadratischer fermionischer Ordnung in uniformer Lichtkegel-Eichung. Sie erfüllt die klassische Yang-Baxter-Gleichung, faktorisiert in zwei Blöcke und entspricht der exakten Streumatrix für ein Modell mit trigonometrisch quantendeformierter Symmetrie. (3) Inhomogene bi-Yang-Baxter-Deformationen von AdS₃ ⨉ S³ ⨉ T⁴ für mehrere Dynkin-Diagramme. Wir berechnen die Zwei-Körper-Streumatrix auf Baum-Niveau zu quadratischer fermionischer Ordnung in uniformer Lichtkegel-Eichung. Alle Deformationen ergeben die gleiche Streumatrix, die mit der erwarteten exakten Streumatrix bei trigonometrisch quantendeformierter Symmetrie übereinstimmt. / We study the perturbative quantum theory of various integrable Yang-Baxter deformations of the free superstring on AdS spaces. For this we compute the two-body tree-level scattering matrix on the worldsheet using Feynman diagram methods. The various deformations are: (1) All distinct Abelian deformations of AdS₅ ⨉ S⁵ allowing light-cone gauge fixing. These are dual to noncommutative super Yang-Mills theory and equivalently described through TsT transformations or twisted boundary conditions. We compute the bosonic tree-level scattering matrix for the BMN string in uniform light-cone gauge. The scattering matrix is expressed through a Drinfeld twist for most cases; for some cases it is expressed instead through a shifted momentum dependence. Lastly, we compare the Bethe equations derived from these results to the equations of the model with twisted boundary conditions and find perfect agreement. For deformations of the GKP string we are not able to determine a deformed scattering matrix around the null-cusp solution due to actions incompatible with perturbation theory in momentum space. (2) The inhomogeneous or eta deformation of AdS₅ ⨉ S⁵ corresponding to the fermionic Dynkin diagram. We compute the two-body tree-level scattering matrix up to second order in fermions in uniform light-cone gauge. It satisfies the classical Yang-Baxter equation, factorizes into two blocks and matches the exact scattering matrix for a model with trigonometrically quantum-deformed symmetry. (3) Inhomogeneous bi-Yang-Baxter deformations of AdS₃ ⨉ S³ ⨉ T⁴ for multiple Dynkin diagrams. We compute the two-body tree-level scattering matrix up to second order in fermions in uniform light-cone gauge. All deformations give the same scattering matrix, which matches the expected exact scattering matrix with trigonometrically quantum-deformed symmetry.

<p>Entanglement entropy (EE) is a quantum information theoretic measure that quantifies the correlations between a region and its surroundings. We study this quantity in the following two setups : </p> <ul> <li>We look at the dynamics of a free minimally coupled, massless scalar field in a deSitter expansion, where the expansion stops after some time (i.e. we quench the expansion) and transitions to flat spacetime. We study the evolution of entanglement entropy (EE) and the Rényi entropy of a spatial region during the expansion and, more interestingly, after the expansion stops, calculating its time evolution numerically. The EE increases during the expansion but the growth is much more rapid after the expansion ends, finally saturating at late times, with saturation values obeying a volume law. The final state of the subregion is a partially thermalized state, reminiscent of a Gibbs ensemble. We comment on application of our results to the question of when and how cosmological perturbations decohere.</li> <li>We study the EE in a theory that is holographically dual to a BTZ black hole geometry in the presence of a scalar field, using the Ryu-Takayangi (RT) formula. Gaberdiel and Gopakumar had conjectured that the theory of N free fermions in 1+1 dimensions, for large N, is dual to a higher spin gravity theory with two scalar fields in 2+1 dimensions. So, we choose our boundary theory to be the theory of N free Dirac fermions with a uniformly winding mass, m e^iqx, in two spacetime dimensions (which describes for instance a superconducting current in an N-channel wire). However, to O(m²), thermodynamic quantities can be computed using Einstein gravity. We aim to check if the same holds true for entanglement entropy (EE). Doing calculations on both sides of the duality, we find that general relativity does indeed correctly account for EE of single intervals to O(m²).</li> </ul>