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Explorations in yang-mills matrix gauge theories with massive deformations

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2019
Oktay, Onur
We focus on two research projects on Yang-Mills (YM) matrix models with massive deformation terms, where fuzzy four-spheres, as well as fuzzy two-spheres appear as matrix configurations which are of interest. We first concentrate on an SU(N) YM gauge theory in 0+1-dimensions with five Hermitian matrices, a YM 5-matrix model, with a massive deformation term and search for matrix configurations of fuzzy four- spheres, which are formed by taking tensor products of certain irreducible and reducible representations of the isometry group SO(5) of the fuzzy four-spheres, which may be understood as new static configurations satisfying the classical equations of motion of this matrix model. The reducible representation of SO(5) that we employ is formed by following a Schwinger type construction which utilizes four pairs of fermionic annihilation-creation operators and their SO(5) irreducible representation (IRR) content is determined. It is shown that in addition to standard fuzzy four-spheres, the generalized fuzzy four-spheres, SΛ4, that recently appeared in the literature, also emerge as solutions to the YM 5–matrix model. We examine the quantization of the coset space O2 ≡ SU(4)/(SU(2) × U(1) × U(1)) via the coadjoint orbit method to provide a perspective on the structure of SΛ4 and employ the generalized coherent states associated to SO(6) ≈ SU(4) to discuss some aspects of both the basic and generalized fuzzy four-spheres. In the second part of the thesis, we examine a YM matrix model that can be contemplated as a massive deformation of the bosonic part of the Banks-Fischler-Shenker-Susskind (BFSS) model. An ansatz configuration involving fuzzy two- and four-spheres with collective time dependence is proposed to arrive at a set of reduced actions whose chaotic dynamics are revealed by calculating their Lyapunov spectrum, Poincaré sections and in particular largest Lyapunov exponents by using numerical solutions to their Hamiltonian equations of motion. We also analyze how the largest Lyapunov exponents change as a function of the energy.