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This thesis is devoted to study one of the most recent and
significant evolutions, maybe third revolution, of a theory formerly
known as string theory. This theoretical mutation is the observation
of duality between gravitational theories and gauge theories.
Through Maldacena conjecture, known as AdS/CFT correspondence there
is an equivalent description of quantum type IIB string theory on
asymptotically AdSxS or plane-wave backgrounds in terms of a 3+1
dimensional $U(N)$ gauge theory. Inspired by the idea of discrete
light-cone quantization (DLCQ), in this thesis we present a
description for quantum dynamics of the very string theory. In fact
we propose a prescription to construct the DLCQ Hamiltonian for that
string theory in the sector with $J$ units of light-cone momentum.
It is conjectured that the DLCQ Hamiltonian is exactly the
Hamiltonian of a 0+1 dimensional supersymmetric $U(J)$ gauge theory,
or a Matrix Theory (MT) for short. This matrix theory is the theory
for the stabilized phase of a tachyonic system of $J$ number non-BPS
D0-branes in the light-cone gauge on the plane-wave background with
compact null direction. We give some pieces of evidence in support
of this proposal. Thus we propose a Matrix theory for quantum type
IIB string theory on AdS or plane-wave backgrounds and study its
correspondence with supersymmetric conformal gauge theory
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