Duality and Integrability in Superstring and Gauge Field Theory

In the present thesis, we explore certain aspects of superstring and supersymmetric gauge field theory, independently as well as in the context of the holographic duality.The first part of the thesis is devoted to classical integrability and, in particular, to certain methods of analytic non-integrability, which are employed on various supergravity vacua. In Chapter 1, we introduce those tools of non-integrability, which consist of choosing an appro-priate string embedding and using differential Galois theory on the associated Hamiltonian system. The arena of all this, for the first chapter, is two classes of vacua in massive Type IIA supergravity, all of which are proven to be non-integrable, up to the trivial cases where the vacuum reduces to the Abelian and non-Abelian T-dual of known integrable backgrounds. Differential Galois theory, in this context, reduces to an algebraic form through Kovacic’s theorem, the proper use of which, on parametrized differential equations, is clarified in this application.In Chapter 2, we study integrability on the supergravity vacuum dual to the field-theoretical Ω-deformation of super Yang-Mills theory. The deformation manifests itself as turning on a Kalb-Ramond field on the dual supergravity vacuum and, by constructing appropriate string embeddings, we show that this space exhibits non-integrable dynamics. This, in turn, suggests that the Ω-deformation does not preserve classical integrability.In Chapter 3, we explore integrability on vacua in massive Type IIA supergravity, dual to six-dimensional superconformal quiver field theories. Analytic non-integrability illustrates that all vacua with a warped geometry, between Anti-de-Sitter space and the internal man-ifold, exhibit complete non-integrability, while in the special case of the unwarped space we prove the opposite to be true. In particular, we show that, besides the integrable dynamics on the symmetric Anti-de-Sitter subspace of the unwarped geometry, the σ-model on the internal manifold is an integrable deformation of the same model on the symmetric three-sphere, ultimately implying classical integrability of bosonic string theory on this special vacuum.The second part of the thesis is devoted to holography and, in particular, the AdS/CFT duality, which we exploit to study features of certain supersymmetric quantum field theories in two spacetime dimensions. More precisely, in Chapter 4, the final chapter, we study the duality between massive Type IIA supergravity vacua and two-dimensional quiver structures. After categorizing all kinds of gravity solutions, we demystify the ones that seem to reflect anomalous gauge theories. In particular, we prove that there are bound states of D-branes on the boundary of the space which provide the dual quiver theory with exactly the correct amount of matter in order to cancel its gauge anomalies. We also propose that the structure of the field theory should be complemented with additional bifundamental matter and, finally, we construct a BPS string configuration and use the old and new supersymmetric matter to build its dual ultraviolet operator.