Yang-Mills Theory in Gauge-Invariant Variables and Geometric Formulation of Quantum Field Theories
Abstract: This thesis explores two different aspects of geometry in quantum field theories. I the first part we are dealing with effective description of Yang-Mills theories based on gauge-invarint variables.For pure Yang-Mills we study the spin-charge separation varibles. The dynamics in these variables resembles the Skyrme-Faddev model. Thus the spin-charge separation is an important intermediate step between the fundamental Yang-Mills theory and the low-energy effective models, usedto model the low-energy dynamics of gluons. Similar methods may be useful for describing the Electroweak sector of the Standard Model in terms of gauge-invariantfield variables called supercurrents.We study the geometric structure of spin-charge separation in 4D Euclidean space (paper III) and elaborate onconnection with gravity toy model . Such reinterpretation gives a way to see how effective flat background metric is created in toy gravity model by studying the appearance of dimension-2 condensate in the Yang-Mills (paper IV). For Electroweak theory we derive the effective gauge-invariant Lagrangian by doing the Kaluza-Klein reduction of higher-dimensional gravitywith 3-brane, thus making explicit the geometric interpretation for gauge-invariant supercurrents. The analogy is then made more precise in the framework of exact supergravity solutions. Thus, we interpret the Higgs effect as spontaneous breaking of Kaluza-Klein gauge symmetry and this leads to interpretation of Higgs fieldas a dilaton (papers I and II).In the second part of the thesis we study rather simple field theories, called ``geometric'' or ``instantonic''.Their defining property is exact localization on finite-dimensional spaces -- the moduli spaces of instantons.These theories allow to account exactly for non-linearity of space of fields, in this respect they go beyond the standard Gaussian perturbation theory.In paper V we show how to construct a geometric theory of chiral boson by embedding it into the geometric field theory. In Paper VI we elaborate on the simplest geometric field theory -- the supersymmetric Quantum Mechanics and construct new non-perturbative topological observables that have a transparent meaning both in geometric and inthe Hamiltonian formalisms. In Paper VII we are motivated by making perturbations away from the simple instantonic limit. For that we need to carefully define the observables that are quadratic in momenta and develop the way to compute them in geometric framework. These correspond geometrically to bivector fields (or, in general, the polyvector fields).We investigate the local limit of polyvector fields and compare the geometric calculation with free-field approach.
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