Double field theory and exceptional field theory are formulations of supergravity that make certain dualities manifest symmetries of the action. To achieve this, the geometry is extended by including dual coordinates corresponding to winding modes of the fundamental objects. This geometrically unifies the spacetime metric and the gauge fields (and their local symmetries) in a generalized geometry. Solutions to these extended field theories take the simple form of waves and monopoles in the extended space. From a supergravity point of view they appear as 1/2 BPS objects such as the string, the membrane and the fivebrane in ordinary spacetime. In this thesis double field theory and exceptional field theory are introduced, solutions to their equations of motion are constructed and their properties are analyzed. Further it is established how isometries in the extended space give rise to duality relations between the supergravity solutions. Extensions to these core ideas include studying Goldstone modes, probing singularities at the core of solutions and localizing them in winding space. The relation of exceptional field theory to F-theory is also covered providing an action for the latter and incorporating the duality between M-theory and F-theory.

Quiver gauge theories are widely studied in the context of AdS/CFT, which establishes a correspondence between CFTs and string theories. CFTs in turn offer a map between quantum states and Gauge Invariant Operators (GIOs). This thesis presents results on the counting and correlators of holomorphic GIOs in quiver gauge theories with flavour symmetries, in the zero coupling limit. We first give a prescription to build a basis of holomorphic matrix invariants, labelled by representation theory data. A fi nite N counting function of these GIOs is then given in terms of Littlewood-Richardson coefficients. In the large N limit, the generating function simpli fies to an in finite product of determinants, which depend only on the weighted adjacency matrix associated with the quiver. The building block of this product has a counting interpretation by itself, expressed in terms of words formed by partially commuting letters associated with closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. We compute the free fi eld two and three point functions of the matrix invariants. These have a non-trivial dependence on the structure of the operators and on the ranks of the gauge and flavour symmetries: our results are exact in the ranks, and their expansions contain information beyond the planar limit. We introduce a class of permutation centraliser algebras, which give a precise characterisation of the minimal set of charges needed to distinguish arbitrary matrix invariants. For the two-matrix model, the relevant non-commutative algebra is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams.

In this thesis we exploit the setup of AdS3/CFT2 holography, and in particular the D1D5 two-dimensional CFT, to describe states dual to geometries relevant for the \fuzzball" proposal for the description of six-dimensional black hole microstates. Precise holographic dualities between CFT and bulk geometric objects are established and checked, both for 2 and 3-charge states. In particular, VEVs of CFT operators of small conformal dimension are checked to encode deviations from AdS3 geometry near the spacetime boundary. 4-point functions of the \heavy-heavy-light-light" type are also considered and matching is found between CFT and bulk computations via the usual AdS/CFT prescription, with the heavy states being dual to (simple) microstate geometries. In this context, the issue of the presence of spurious singularities at leading order in the large N limit is assessed and cancellations are found even without considering sub-leading corrections, at the cost of considering the full detail of the D1D5 CFT (i.e. including the Virasoro blocks of operators of small dimension charged under the internal SU(2)L SU(2)R R-symmetry group). Finally, more complicated 4-point functions, involving operators in the twisted sector of the CFT, are computed and the results are checked against known results in the literature with the aim of verifying the robustness of the (new) techniques used. Supersymmetric Ward identities are also derived, and checked for some cases, between correlators written in terms of bosons and in terms of fermions.

Information about the very early universe can be accessed from observations of the cosmic microwave background (CMB) radiation and the later formation of large-scale structure (LSS) that are produced from cosmological perturbations of the early universe. The most developed theoretical explanation for the origin of these perturbations is the theory of inflation, in which the early universe undergoes a period of accelerated expansion, amplifying quantum fluctuations to macroscopic size, which act as the seeds for the CMB anisotropies and the cosmic web of the LSS. The work in this thesis aims to connect the theory of inflation to properties of these observables in a highly detailed way, suitable for future high-precision astronomical surveys. After some introductory review chapters, we begin with new research on a study of inflation from string theory, deriving an observably-large value of the tensor-to-scalar ratio, which had been previously difficult to achieve theoretically. The next study investigates the link between the observed CMB power asymmetry and non-Gaussianity, including a novel non-zero value for the trispectrum. Next we study soft limits of non-Gaussian inflationary correlation functions, focussing first on the squeezed limit of the bispectrum and then generalizing to soft limits of higher-point correlation functions, giving results valid for multi-fi eld models of inflation.