7 research outputs found
Gauge/gravity duality at finite N
A dissertation submitted to the Faculty of Science, University of the
Witwatersrand, Johannesburg, in fulfilment of the requirements for the
degree of Doctor of Philosophy. March 2013.In the past decade, the gauge/gravity duality has been extensively explored
in the large N limit. In particular, the spectrum of anomalous dimensions have
been compared with the energy spectrum of the dual string theory showing
remarkable agreement. In this limit, for operators with a bare dimension of
order 1, planar diagrams give the leading contribution to the anomalous dimension.
To obtain the anomalous dimensions, one needs to diagonalize the
dilatation operator. One of the methods used to accomplish this is integrability.
This allows an exact computation of the spectrum of the anomalous
dimensions. There is by now a great deal of evidence that N = 4 supersymmetric
Yang-Mills (SYM) theory and N = 6 superconformal Chern Simons
(ABJ(M)) theory are integrable in the planar limit.
In this thesis we probe the gauge gravity duality at finite N using novel tools
developed from the representation theory of symmetric and unitary groups.
We start by studying the action of the nonplanar dilatation operator of N = 4
SYM theory and ABJ(M) theory. The gauge invariant operators we consider
are the restricted Schur polynomials. In the case of N = 4 SYM theory, we
obtain the spectrum of the anomalous dimension beyond the SU(2) sector
at one loop, and in the SU(2) sector at two loops. In both cases, we obtain
the spectrum at arbitrary (finite) N. We then obtain the spectrum of
anomalous dimensions in the SU(2) sector of ABJ(M) theory at two loops.
The class of gauge invariant operators we consider have classical dimension
of order O(N). In both theories, the spectrum of the anomalous dimensions
reduces to a set of decoupled harmonic oscillators at large N. This indicates,
for the first time, that N = 4 SYM theory and ABJ(M) theory exhibit nonplanar
integrability. We expect to recover non-perturbative quantum gravity
effects, from the gauge/gravity duality, when N is finite. The non-planar integrability
we discover here may play an important role in finite N studies of the
gauge/gravity duality, and hence may play an important role in understanding
non-perturbative string stringy physics. In addition, we study various classes
of correlators in ABJ(M) theory. In this context, we derive extremal n-point
correlators in ABJ(M) theory and we probe the giant graviton dynamics in
these theories
Nonplanar Integrability and Parity in ABJ Theory
In this article we study the action of the non-planar two-loop dilatation
operator in an SU(2)*SU(2) sub-sector of the ABJ Chern-Simons-matter theory.
The gauge invariant operators we consider are the restricted Schur polynomials.
As in ABJM theory, there is a limit in which the spectrum reduces to a set of
decoupled harmonic oscillators, indicating integrability in the large M and N
double limit of the theory. We then consider parity transformations on the
gauge invariant operators. In this case the non-planar anomalous dimensions
break parity invariance. Our analysis shows that (M-N) is related to the
holonomy in the string theory, confirming one of the main features of the
theory and its string dual. Furthermore, in the limit where ABJ theory reduces
to ABJM theory, parity invariance is restored.Comment: 1+12 pages, Corrected typos, more clarifications, figure and appendix
adde
Beyond the Planar Limit in ABJM
In this article we consider gauge theories with a U(N)X U(N) gauge group. We
provide, for the first time, a complete set of operators built from scalar
fields that are in the bi fundamental of the two groups. Our operators
diagonalize the two point function of the free field theory at all orders in
1/N. We then use this basis to investigate non-planar anomalous dimensions in
the ABJM theory. We show that the dilatation operator reduces to a set of
decoupled harmonic oscillators, signaling integrability in a nonplanar large N
limit.Comment: v2: minor revisison
Nonplanar integrability at two loops
In this article we compute the action of the two loop dilatation operator on
restricted Schur polynomials that belong to the su(2) sector, in the displaced
corners approximation. In this non-planar large N limit, operators that
diagonalize the one loop dilatation operator are not corrected at two loops.
The resulting spectrum of anomalous dimensions is related to a set of decoupled
harmonic oscillators, indicating integrability in this sector of the theory at
two loops. The anomalous dimensions are a non-trivial function of the 't Hooft
coupling, with a spectrum that is continuous and starting at zero at large N,
but discrete at finite N.Comment: version to appear in JHE