3. Electronic Theses and Dissertations (ETDs) - All submissions
Permanent URI for this communityhttps://wiredspace.wits.ac.za/handle/10539/45
Browse
2 results
Search Results
Item Bilocal approach to the infra-red fixed point of O (N) invariant theories in 3d and its relation to higher spins(2018) Mulokwe, MbavhaleloThe Klebanov-Polyakov Higher-Spin Anti-de Sitter/Conformal Field Theory conjecture posits that the free O(N) vector model is dual to the type A Vasiliev Higher Spin Gravity with the bulk scalar eld having conformal scaling dimension = 1. Similarly, the critical O(N) vector model in 3d is dual to type A Vasilev Higher Spin Gravity with bulk scalar having = 2 . This is a weak-weak duality and accordingly allows a setting where a reconstruction of bulk physics from the boundary CFT is possible. The Jevicki-Sakita collective eld theory provides an explicit realization of such a bulk reconstruction. In this thesis, we use the collective eld theory description of the large-N limit of vector models to study the O(N) infra-red interacting xed point. In particular, we compute the two-point functions for the non-linear sigma model (which is equivalent, in the infra-red, to the critical O(N) vector model) and the two-time bilocal propagator. The spectrum for the O(N) vector model is then obtained by looking at the poles of the connected Green's function. We then show that this same pole condition can be obtained from the homogeneous equation for the bilocal uctuations. We then discuss the single-time Hamiltonian formalism for the critical O(N) vector model. We derive a coupled integral equation for the single-time uctuations. This coupled integral equations allows us to write down the single-time pole condition. We show that the two-time pole condition is equivalent to the single-time pole condition. In addition, we also show that the two-time free bilocal propagator is equivalent to the single-time free bilocal propagator. A Lagrangian formulation of the single-time descripiv tion is given and we write down the single-time propagator. We then explain a puzzle which is that from our study of the non-linear sigma model and the the pole structure of both the two-time and the single-time propagators it would seem that both the = 1 and = 2 scalars are present. By studying the quadratic Hamiltonian determining the spectrum, we demonstrate how in the infra-red limit the state = 1 disappears from the spectum.Item The large-N limit of matrix models and AdS/CFT(2014-06-12) Mulokwe, MbavhaleloRandom matrix models have found numerous applications in both Theoretical Physics and Mathematics. In the gauge-gravity duality, for example, the dynamics of the half- BPS sector can be fully described by the holomorphic sector of a single complex matrix model. In this thesis, we study the large-N limit of multi-matrix models at strong-coupling. In particular, we explore the significance of rescaling the matrix fields. In order to investigate this, we consider the matrix quantum mechanics of a single Hermitian system with a quartic interaction. We “compactify” this system on a circle and compute the first-order perturbation theory correction to the ground-state energy. The exact ground-state energy is obtained using the Das-Jevicki-Sakita Collective Field Theory approach. We then discuss the multi-matrix model that results from the compactification of the Higgs sector of N = 4 SYM on S4 (or T S3). For the radial subsector, the saddle-point equations are solved exactly and hence the radial density of eigenvalues for an arbitrary number of even Hermitian matrices is obtained. The single complex matrix model is parametrized in terms of the matrix valued polar coordinates and the first-order perturbation theory density of eigenstates is obtained. We make use of the Harish-Chandra- Itzykson-Zuber (HCIZ) formula to write down the exact saddle-point equations. We then give a complementary approach - based on the Dyson-Schwinger (loop) equations formalism - to the saddle-point method. We reproduce the results obtained for the radial (single matrix) subsector. The two-matrix integral does not close on the original set of variables and thus we map the system onto an auxiliary Penner-type two matrix model. In the absence of a logarithmic potential we derive a radial hemispherical density of eigenvalues. The system is regulated with a logarithm potential, and the Dobroliubov-Makeenko-Semenoff (DMS) loop equations yield an equation of third degree that is satisfied by the generating function. This equation is solved at strong coupling and, accordingly, we obtain the radial density of eigenvalues.