Generalized abelian gauge field theory under rotor model

Gauge field theory with rank-one field [Formula: see text] is a quantum field theory that describes the interaction of elementary spin-1 particles, of which being massless to preserve gauge symmetry. In this paper, we give a generalized, extended study of abelian gauge field theory under successive rotor model in general [Formula: see text]-dimensional flat spacetime for spin-1 particles in the context of higher-order derivatives. We establish a theorem that [Formula: see text] rotor contributes to the [Formula: see text] fields in the integration-by-parts formalism of the action. This corresponds to the transformation of gauge field [Formula: see text] and gauge field strength [Formula: see text] in the action. The [Formula: see text] case restores back to the standard abelian gauge field theory. The equation of motion and Noether’s conserved current of the theory are also studied.

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.

Classical Gauge Field Theory

After a sketch of Lagrangian field theory on jet bundles, the notion of a gauge field is introduced as a section of an affine bundle which is naturally constructed without any involvement with structure groups. An original approach to gauge field theory in terms of covariant differentials (alternative to the jet bundle approach) is then developed, and the adaptations needed in order to deal with general theories are laid out. A careful exposition of the replacement principle allows comparisons with approaches commonly found in the literature.

Gauge Field Theory and Gravitation

The notion of 2-spinor soldering form allows a neat formulation, called the ‘tetrad-affine setting’, of a theory of matter and gauge fields interacting with the gravitational field. The latter is represented by a couple constituted by the soldering form and a 2-spinor connection. This approach is suited to describe matter fields with arbitrary spin and generic further internal structure. In particular one gets an approach to interacting Einstein-Cartan-Maxwell-Dirac fields, in which the only assumption is a complex bundle with 2-dimensional fibers: the needed bundles are obtained from it by natural geometric contructions, and any object which is not determined from these ‘minimal geometric data’ is assumed to be a dynamical field.

On Gauge Freedom and Interactions

The notion of gauge freedom is discussed in the context of the 2-spinor approach to Maxwell-Dirac fields. The various types of boson interaction terms arising in a generic gauge field theory are laid down, and detailed in the particular case of gauge boson interactions in the electroweak theory. Finally, several observations of ‘philosophical nature’ are offered, together with some ideas for future work.

Quantum Fields

The basics of a Lagrangian field theory of quantum fields are laid down by exploiting the differential geometric notions introduced through F-smoothness. Infinitesimal vertical symmetries and currents in this setting lead, in particular, to the notion of BRST symmetry. The above results are applied to a fairly detailed study of a sample gauge field theory which includes spinor fields and ghosts.