The geometry of non-conformal supersymmetric non-linear sigma models in one and two dimensions are reviewed. Transformations of the Osp(1|2) subgroup of the superconformal group are derived and then used in finding geometrical constraints on the target space of an N=(1,1) sigma model reduced to an N=1 sigma model.

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This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model. In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

One of the main topics in the modern String Theory are the conjectured string/gauge (AdS/CFT) dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, i.e. the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality.The first part of this thesis is focused on the gravity side of the AdS5/CFT4 duality: we investigate the quantum integrability of the type IIB superstring on AdS5 x S5. In the pure spinor formulation we analyze the operator algebra by computing the operator product expansion of the Maurer-Cartan currents at the leading order in perturbation theory. With the same approach at one loop order, we show the path-independence of the monodromy matrix which implies the charge conservation law, strongly supporting the quantum integrability of the string sigma-model. We also verify that the Lax pair field strength remains well-defined at one-loop order being free from UV divergences. The same string sigma-model is analyzed in the Green-Schwarz formalism in the near-flat-space (NFS) limit. Such a limit remarkably simplifies the string world-sheet action but still leaving interesting physics. We use the NFS truncation to show the factorization of the world-sheet S-matrix at one-loop order. This property defines a two-dimensional field theory as integrable: it is the manifestation of the higher conserved charges. Hence, we have explicitly checked their presence at quantum level. The second part is dedicated to the AdS4/CFT3 duality: in particular the type IIA superstring on AdS4 x CP3. We compute the leading quantum corrections to the string energies for string configurations with a large but yet finite angular momentum on CP3 and show that they match the conjectured all-loop Bethe Ansatz equations.

Non-linear sigma models with extended supersymmetry have constrained target space geometries, and can serve as effective tools for investigating and constructing new geometries. Analyzing the geometrical and topological properties of sigma models is necessary to understand the underlying structures of string theory. The most general two-dimensional sigma model with manifest N=(2,2) supersymmetry can be parametrized by chiral, twisted chiral and semichiral superfields. In the research presented in this thesis, N=(4,4) (twisted) supersymmetry is constructed for a semichiral sigma model. It is found that the model can only have additional supersymmetry off-shell if the target space has a dimension larger than four. For four-dimensional target manifolds, supersymmetry can be introduced on-shell, leading to a hyperkähler manifold, or pseudo-supersymmetry can be imposed off-shell, implying a target space which is neutral hyperkähler. Different sigma models and corresponding geometries can be related to each other by T-duality, obtained by gauging isometries of the Lagrangian. The semichiral vector multiplet and the large vector multiplet are needed for gauging isometries mixing semichiral superfields, and chiral and twisted chiral superfields, respectively. We find transformations that close off-shell to a N=(4,4) supersymmetry on the field strengths and gauge potentials of the semichiral vector multiplet, and show that this is not possible for the large vector multiplet. A sigma model parametrized by chiral and twisted chiral superfields can be related to a semichiral sigma model by T-duality. The N=(4,4) supersymmetry transformations of the former model are linear and close off-shell, whereas those of the latter are non-linear and close only on-shell. We show that this discrepancy can be understood from T-duality, and find the origin of the non-linear terms in the transformations.

Cette thèse a pour sujet une classe de théories des champs intégrables appelées modèles avec fonction twist. Les principaux exemples de tels modèles sont les modèles sigma non-linéaires intégrables, tel le Modèle Principal Chiral, et leurs déformations. Un premier résultat obtenu est la preuve que le modèle dit de Bi-Yang-Baxter, qui est une déformation à deux paramètres du Modèle Principal Chiral, est lui aussi un modèle avec fonction twist. Il est ensuite montré que les déformations de type Yang-Baxter modifient certaines symétries globales du modèle non déformé en symétries de Poisson-Lie. Un autre chapitre concerne la construction d'une infinité de charges locales en involution pour tous les modèles sigma intégrables et leurs déformations : ce résultat repose sur le formalisme général partagé par tous ces modèles en tant que théories des champs avec fonction twist.La seconde partie de la thèse a pour sujet les modèles de Gaudin. Ceux-ci sont des modèles intégrables associés à des algèbres de Lie. En particulier, les théories des champs avec fonction twist sont liées aux modèles de Gaudin associés à des algèbres de Lie affines. Une approche standard pour l'étude du spectre des modèles de Gaudin quantiques sur des algèbres finies est celle de Feigin-Frenkel-Reshetikhin. Dans cette thèse, des généralisations de cette approche sont conjecturées, motivées et testées. L'une d'elles concerne les modèles de Gaudin finis dits cyclotomiques. La seconde porte sur les modèles de Gaudin associés à des algèbres affines. / This thesis deals with a class of integrable field theories called models with twist function. The main examples of such models are integrable non-linear sigma models, such as the Principal Chiral Model, and their deformations. A first obtained result is the proof that the so-called Bi-Yang-Baxter model, which is a two-parameter deformation of the Principal Chiral Model, is also a model with twist function. It is then shown that Yang-Baxter type deformations modify certain global symmetries of the undeformed model into Poisson-Lie symmetries. Another chapter concerns the construction of an infinite number of local charges in involution for all integrable sigma models and their deformations: this result is based on the general formalism shared by all these models as field theories with twist function.The second part of the thesis concerns Gaudin models. These are integrable models associated with Lie algebras. In particular, field theories with twist function are related to Gaudin models associated with affine Lie algebras. A standard approach for studying the spectrum of quantum Gaudin models over finite algebras is the one of Feigin-Frenkel-Reshetikhin. In this thesis, generalisations of this approach are conjectured, motivated and tested. One of them deals with the so-called cyclotomic finite Gaudin models. The second one concerns the Gaudin models associated with affine Lie algebras.

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.

No description available.

<p>In this thesis, two aspects of string theory are discussed, tensionless strings and supersymmetric sigma models.</p><p>The equivalent to a massless particle in string theory is a tensionless string. Even almost 30 years after it was first mentioned, it is still quite poorly understood. We discuss how tensionless strings give rise to exact solutions to supergravity and solve closed tensionless string theory in the ten dimensional maximally supersymmetric plane wave background, a contraction of AdS(5)xS(5) where tensionless strings are of great interest due to their proposed relation to higher spin gauge theory via the AdS/CFT correspondence.</p><p>For a sigma model, the amount of supersymmetry on its worldsheet restricts the geometry of the target space. For N=(2,2) supersymmetry, for example, the target space has to be bi-hermitian. Recently, with generalized complex geometry, a new mathematical framework was developed that is especially suited to discuss the target space geometry of sigma models in a Hamiltonian formulation. Bi-hermitian geometry is so-called generalized Kähler geometry but the relation is involved. We discuss various amounts of supersymmetry in phase space and show that this relation can be established by considering the equivalence between the Hamilton and Lagrange formulation of the sigma model. In the study of generalized supersymmetric sigma models, we find objects that favor a geometrical interpretation beyond generalized complex geometry.</p>

<p>This thesis considers two different aspects of string theory, the tensionless limit of the string and supersymmetric sigma models with extended supersymmetry. First, the tensionless limit is used to find a IIB supergravity background generated by a tensionless string. The background has the characteristics of a gravitational shock-wave. Then, the quantization of the tensionless string in a pp-wave background is performed and the result is found to agree with what is obtained by taking a tensionless limit directly in the quantized theory of the tensile string. Hence, in the pp-wave background the tensionless limit commutes with quantization. Next, supersymmetric sigma models and the relation between extended world-sheet supersymmetry and target space geometry is studied. The sigma model with N=(2,2) extended supersymmetry is considered and the requirement on the target space to have a bi-Hermitean geometry is reviewed. The Hamiltonian formulation of the model is constructed and the target space is shown to have generalized Kähler geometry. The equivalence between bi-Hermitean geometry and generalized Kähler follows, in this context, from the equivalence between the Lagrangian- and Hamiltonian formulation of the sigma model. Then, T-duality in the Hamiltonian formulation of the sigma model is studied and the explicit T-duality transformation is constructed. It is shown that the transformation is a symplectomorphism, i.e. a generalization of a canonical transformation. Under certain assumptions, the amount of extended supersymmetry present in the sigma model is shown to be preserved under the T-duality transformation. Next, extended supersymmetry in a first order formulation of the sigma model is studied. By requiring N=(2,2) extended world-sheet supersymmetry an intriguing geometrical structure arises and in a special case generalized complex geometry is found to be contained in the new framework.</p>