The large-N limit of matrix models and AdS/CFT

Mulokwe, Mbavhalelo
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Random 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.