We determine the conditions under which half-maximal matter coupled gauged supergravity in seven dimensions admits a chiral circle reduction to yield a matter coupled gauged supergravity in six dimensions with 8 real supersymmetry. Solving these conditions we nd that the SO(2; 2) and SO(3; 1) gauged 7D supergravities give a U(1)R, and the SO(2; 1) gauged 7D supergravity gives an Sp(1)R gauged chiral 6D supergravity coupled to certain matter multiplets. In the 6D models obtained, with or without gauging, we show that the scalar fields of the matter sector parametrize the coset SO(p + 1; 4)=SO(p + 1) SO(4), with the (p + 3) axions corresponding to its abelian isometries. We then derive the necessary and sufficient conditions for the existence of a Killing spinor in N = (1; 0) gauge 6D supergravity coupled to a single tensor multiplet, vector multiplets and hypermultiplets. We show that these conditions imply most of the eld equations. We also determine the remaining equations that need to be satised by an exact solution. In this framework, we nd a novel 1=8 supersymmetric dyonic string solution with nonvanishing hypermultiplet scalars. The activated scalars parametrize a 4 dimensional submanifold of a quaternionic hyperbolic ball. The key point is that we employ an identity map between this submanifold and the internal space transverse to the string worldsheet, thereby nding a higher dimensional generalization of Gell- Mann-Zweibach tear-drop solution.

We determine the conditions under which half-maximal matter coupled gauged supergravity in seven dimensions admits a chiral circle reduction to yield a matter coupled gauged supergravity in six dimensions with 8 real supersymmetry. Solving these conditions we nd that the SO(2; 2) and SO(3; 1) gauged 7D supergravities give a U(1)R, and the SO(2; 1) gauged 7D supergravity gives an Sp(1)R gauged chiral 6D supergravity coupled to certain matter multiplets. In the 6D models obtained, with or without gauging, we show that the scalar fields of the matter sector parametrize the coset SO(p + 1; 4) / SO(p + 1) X SO(4), with the (p + 3) axions corresponding to its abelian isometries. We then derive the necessary and sufficient conditions for the existence of a Killing spinor in N = (1; 0) gauge 6D supergravity coupled to a single tensor multiplet, vector multiplets and hypermultiplets. We show that these conditions imply most of the eld equations. We also determine the remaining equations that need to be satised by an exact solution. In this framework, we nd a novel 1=8 supersymmetric dyonic string solution with nonvanishing hypermultiplet scalars. The activated scalars parametrize a 4 dimensional submanifold of a quaternionic hyperbolic ball. The key point is that we employ an identity map between this submanifold and the internal space transverse to the string worldsheet, thereby nding a higher dimensional generalization of Gell- Mann-Zweibach tear-drop solution.

The LHC has begun collecting data and the first results have now been published. This is truly an exciting time in the field as we wait for the experimental data to exclude or verify new physics beyond the Standard Model. In order to make a precise prediction for the LHC one must go beyond the leading order of our perturbation series. In this thesis I present the extension of tools for the automation of one loop calculations for supersymmetric models. The second part of the thesis contains the application of these tools to neutralino pair production in the Minimal Supersymmetric Model.

<p>This Master  thesis considers certain aspects of Supersymmetric Quantum Mechanics in the context of Path integral approach. First we state all the basic mathematical structure involved, and carry out some basic Gaussian integrals for both commutative and non-commutative variables. Later in the thesis these simple results obtained are generalized to study the Supersymmetric sigma models on flat and curved space. And we will recover the beautiful relationship between the supersymmetric sigma  model and the geometry of the  target manifold in the form of topological invariants of the manifold, for the models on curved space.</p>

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This Master  thesis considers certain aspects of Supersymmetric Quantum Mechanics in the context of Path integral approach. First we state all the basic mathematical structure involved, and carry out some basic Gaussian integrals for both commutative and non-commutative variables. Later in the thesis these simple results obtained are generalized to study the Supersymmetric sigma models on flat and curved space. And we will recover the beautiful relationship between the supersymmetric sigma  model and the geometry of the  target manifold in the form of topological invariants of the manifold, for the models on curved space.

New four dimensional N = 2 field theories can be engineered from compactifying six dimensional (2, 0) superconformal field theory on a punctured Riemann surface. Hitchin’s equation is defined on this Riemann surface and the fields in Hitchin’s equation are singular at the punctures. Four dimensional theory is entirely determined by the data at the punctures. Theory without lagrangian description can also be constructed in this way. We first construct new four dimensional generalized superconformal quiver gauge theory by putting regular singularity at the puncture. The algorithm of calculating weakly coupled gauge group in any duality frame is developed. The asymptotical free theory and Argyres-Douglas field theory can also be constructed using six dimensional method. This requires introducing irregular singularity of Hithcin’s equation. Compactify four dimensional theory down to three dimensions, the corresponding N = 4 theory has the interesting mirror symmetry. The mirror theory for the generalized superconformal quiver gauge theory can be derived using the data at the puncture too. Motivated by this construction, we study other three dimensional theories deformed from the above theory and find their mirrors. The surprising relation of above four dimensional gauge theory and two dimensional conformal field theory may have some deep implications. The S-duality of four dimensional theory and the crossing symmetry and modular invariance of two dimensional theory are naturally related.

Gauge theories are one of the corner stones of modern theoretical physics. They describe the nature of all fundamental interactions and have been applied in multiple branches of physics. The most challenging problem of gauge theories, which has not been solved yet, is their strong coupling dynamics. A class of gauge theories that admits simplifications allowing to deal with the strong coupling regime are supersymmetric ones. For example, recently proposed method of supersymmetric localization allows to reduce expectation values of supersymmetric observables, expressed through the path integral, to finite-dimensional matrix integral. The last one is usually easier to deal with compared to the original infinite-dimensional integral. This thesis deals with the matrix models obtained from the localization of different 5D gauge theories. The focus of our study is N=1 super Yang-Mills theory with different matter content as well as N=1 Chern-Simons-Matter theory with adjoint hypermultiplets. Both theories are considered on the five-spheres. We make use of the saddle-point approximation of the matrix integrals, obtained from localization, to evaluate expectation values of different observables in these theories. This approximation corresponds to the large-N limit of the localized gauge theory. We derive <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B3%7D" /> behavior for the free energy of 5D N=1* super Yang-Mills theory at strong coupling. This result is important in light of the relation between 5D theory and the world-volume theories of M5-branes, playing a significant role in string theory. We have also explored rich phase structure of 5D SU(N) N=1 super Yang-Mills theory coupled to massive matter in different representations of the gauge group. We have shown that in the case of the massive adjoint hypermultiplet theory undergoes infinite chain of the third order phase transitions while interpolating between weak and strong coupling in the decompactification limit. Finally, we obtain several interesting results for 5D Chern-Simons theory, suggesting existence of the holographic duals to this theory. In particular, we derive <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B5/2%7D" /> behavior of the free energy of this theory, which reproduces the behavior of the free energy for 5D theories with known  holographic duals.

Recent results of string theory have shown that while the traditional cycles studied in Calabi-Yau 4-manifolds preserve half the spacetime supersymmetry, the more general class of Cayley cycles are novel in that they preserve only one quarter of it. Moreover, Cayley cycles play a crucial role in understanding mirror symmetry on Calabi-Yau 4-manifolds and Spin manifolds. Nonetheless, only very few nontrivial examples of Cayley cycles are known. In particular, it would be very useful to know interesting examples of Cayley cycles on the complex 4-torus. This thesis will develop key techniques for finding and constructing lattice periodic Cayley manifolds in Euclidean 8-space. These manifolds will project down to the complex 4-torus, yielding nontrivial Cayley cycles.

This work is focused on the different supersymmetric extensions of the Landau model. We aim to fully solve each model and describe its energy levels, wavefunctions, Hilbert space and define a norm on it, as well as find symmetry generators and transformations with respect to them. Several possible generalizations were considered before. It was found for Landau model on the so called Superflag manifold as well as planar Superflag and Superplane Landau models that standard norm on the Hilbert space is not positive definite. Later for planar cases it was found that it is possible to fix this by introducing a new norm which will be invariant and positive definite. Surprisingly this procedure brings up "hidden" symmetries for the known super Landau models. In the dissertation we apply the same procedure for Landau model on superpshere and Superflag manifolds. It turns out that superpsherical Landau model is equivalent to the Superflag model with one of the parameters fixed. Because the model on superpshere can be recovered from the Superflag we will do calculations of corrected norm only for the Superflag. After this we develop a different generalization of the Superplane Landau model. Starting with Lagrangian in a superfield form we introduce two arbitrary functions of superfields K(Φ) and V(Φ) into the Lagrangian. We follow with the component form of Lagrangian. The quantization of the model is possible, and we will show that there is a reparametrization which turn equation of motion of the first scheme into the second set. Standard metric is again non-positive definite and we apply already known procedure to correct it. It will not be possible to solve Schrodinger equations in general with undefined K and V, so we consider one specific case which give us Landau model on a sphere with N = 2 supersymmetry, which put it apart from the superspherical Landau model, which have a superpshere for a target space but do not possess supersymmetry.