Explorations in fuzzy physics and non-commutative geometry
Date of Award
Doctor of Philosophy (PhD)
A. P. Balachandran
Fuzzy physics, Noncommutative geometry, Supersymmetry, Quantum field theories
Physical Sciences and Mathematics | Physics
Fuzzy spaces arise as discrete approximations to continuum manifolds. They are usually obtained through quantizing coadjoint orbits of compact Lie groups and they can be described in terms of finite-dimensional matrix algebras, which for large matrix sizes approximate the algebra of functions of the limiting continuum manifold. Their ability to exactly preserve the symmetries of their parent manifolds is especially appealing for physical applications. Quantum Field Theories are built over them as finite-dimensional matrix models preserving almost all the symmetries of their respective continuum models.
In this dissertation, we first focus our attention to the study of fuzzy supersymmetric spaces. In this regard, we obtain the fuzzy supersphere [Special characters omitted.] through quantizing the supersphere, and demonstrate that it has exact supersymmetry. We derive a finite series formula for the [low *]-product of functions over [Special characters omitted.] and analyze the differential geometric information encoded in this formula. Subsequently, we show that quantum field theories on [Special characters omitted.] are realized as finite-dimensional supermatrix models, and in particular we obtain the non-linear sigma model over the fuzzy supersphere by constructing the fuzzy supersymmetric extensions of a certain class of projectors. We show that this model too, is realized as a finite-dimensional supermatrix model with exact supersymmetry.
Next, we show that fuzzy spaces have a generalized Hopf algebra structure. By focusing on the fuzzy sphere, we establish that there is a [low *]-homomorphism from the group algebra SU (2)* of SU (2) to the fuzzy sphere. Using this and the canonical Hopf algebra structure of SU (2)* we show that both the fuzzy sphere and their direct sum are Hopf algebras. Using these results, we discuss processes in which a fuzzy sphere with angular momenta J splits into fuzzy spheres with angular momenta K and L .
Finally, we study the formulation of Chern-Simons (CS) theory on an infinite strip of the non-commutative plane. We develop a finite-dimensional matrix model, whose large size limit approximates the CS theory on the infinite strip, and show that there are edge observables in this model obeying a finite-dimensional Lie algebra, that resembles the Kac-Moody algebra.
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Kurkcuoglu, Seckin, "Explorations in fuzzy physics and non-commutative geometry" (2004). Physics - Dissertations. Paper 36.