Abstract:

This thesis is devoted to study of two aspects or use of AdS/CFT conjecture: One
is the use of AdS/CFT in understanding the IRcutoff appearing in the resolution
of Hagedorn limiting temperature in String Theory via BFSS matrix model. And
other is the use of AdS/CFT in understanding of low energy condensed matter
system. A brief abstract of these studies are written separately below.
Hagedorn Phase Transition and Matrix Model for String Theory:
Hagedorn Temperature is the temperature at which the partition function diverges
due to the exponential growth in the density of states, which overtakes the Boltzmann
suppression factor. BFSS matrix model which describes Mtheory in some
limit, has membrane degrees of freedom. String like con gurations can be constructed
from basic degrees of freedom of the matrix model. Hagedorn temperature
can be reinterpreted in terms of new degrees of freedom as a phase transition. In
the first part of this thesis, we will construct two phases of matrix model ("string" phase and "clustered phase"), and compare their free energies to fi nd signature
of phase transition. It will be shown that, there exist phase transitions between
these two con gurations but only in presence of an IR cutoff. The low temperature
phase corresponds to a string (wrapped membrane) phase and so we call this
the Hagedorn phase transition. While the presence of an IR cutoff seemingly is
only required for perturbative analysis to be valid, the physical necessity of such
a cutoff can be seen in the dual supergravity side using AdS/CFT conjecture.
Interestingly the perturbative analysis also shows a second phase transition back
to a string phase. This is reminiscent of the GregoryLa amme instability.
Duality between Charged BTZ black hole and Luttinger liquids:
In the second part of thesis, we study properties of strongly coupled CFT's with
nonzero background electric charge in 1+1 dimensions by studying the dual gravity
theory  which is a charged BTZ black hole. We will calculate correlators of
operators dual to scalars, gauge elds and fermions are studied at both T = 0 and
T (not)= 0. In the T = 0 case we will also able to compare with analytical results
based on AdS2 and find reasonable agreement. In particular the correlation between
log periodicity and the presence of nite spectral density of gapless modes
will be seen. The real part of the conductivity (given by the currentcurrent correlator) also vanishes as w → 0 as expected. The fermion Green's function shows quasiparticle peaks with approximately linear dispersion but the detailed structure
is neither Fermi liquid nor Luttinger liquid and bears some similarity to a
"FermiLuttinger" liquid. This is expected since there is a background charge and
the theory is not Lorentz or scale invariant. A boundary action that produces
the observed nonLuttingerliquid like behavior (kindependent nonanalyticity at
w = 0) in the Green's function is discussed. 