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

##### Date

2014-06-12

##### Authors

Mulokwe, Mbavhalelo

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##### Abstract

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.