Non-perturbative effects in field theory and gravity

Nonperturbative effects are crucial to fully understand the
dynamics of quantum field theories including important topics such
as confinement or black hole evaporation. In this thesis we
investigate two systems where nonperturbative effects are of
paramount importance. In the first part we study the dynamics of
non-abelian gauge theories, while in the second part we try to shed
light on mysterious properties of black holes using a model
proposed earlier by Dvali and Gomez.\\ Non-abelian gauge theories
are the central element in the standard model of particle physics
and many dynamical aspects remain elusive. $\mathcal{N}=1$
supersymmetric Yang-Mills theories with $SU(N_C)$ allows for domain
walls with several curious properties. They are expected to have
gauge fields with a Chern-Simons (CS) term living on their
worldvolume, while in the 't Hooft limit of a large number of
colors many of their properties seem reminiscent of string
theoretic D-Branes. Similar domain walls were also conjectured to
be present in non supersymmetric Yang Mills theories. In our work,
we investigate this problem from several points of view. We
construct a toy model of how to localize a gauge field with a CS
term on a domain wall extending earlier work by Dvali and Shifman.
We then derive the peculiar properties of CS terms in terms of
effects of the underlying microscopic dynamics. Then we look at the
actual theory of interest. Here the main novelty is the focus on
the topological part of the Yang-Mills theory allowing us to make
robust statements despite working in a strongly coupled theory. We
construct the low energy effective action of both the
non-supersymmetric as well as the supersymmetric Yang Mills theory,
which due to the presence of a mass gap is a topological field
theory. This topological field theory encodes the Aharanov-Bohm
phases in the theory as well as phases due to intersection of flux
tubes. In this topological field theory we see that the worldvolume
theory of domain walls contains a level $N_C$ CS term. The presence
of this term was already conjectured in ealier works based on
string theoretic constructions. Here we give its first purely field
theoretical construction. Within this construction we also
illuminate differences between domain walls in the supersymmetric
and non-supersymmetric case.\\ Lastly we try to relate the effects
observed to similar effects in critical string theories and we also
speculate on whether the behaviour of these domain walls is due to
an analog of the fractional quantum hall effect.\\ In the second
part of this thesis we investigate non-perturbative aspects of
black hole physics. Here we consider a model for a low energy
description of black holes due to Dvali and Gomez, where black
holes are described in terms of a Bose-Einstein condensate (BEC) of
weakly interacting gravitons near a quantum critical point. We
focus on nonperturbative properties of a system of attractively
self-interacting non-relativistic bosons, which was proposed as a
toy model for graviton BECs by Dvali and Gomez. In this thesis we
investigate this system mostly relying on a fully non-perturbative
approach called exact diagonalization. We first investigate
entanglement properties of the ground state of the system, showing
that the ground state becomes strongly entangled as one approaches
the quantum critical point. In order to make this notion precise we
introduce the notion of fluctuation entanglement. We then compute
it in a Bogoliubov analysis and extract it from the exact
diagonlization procedure as well. We also consider the real time
evolution of the system. Here we are interested in finding an
analog of the conjectured fast scrambling property of black holes
originally introduced by Hayden and Preskill. We only consider the
weaker notion of quantum breaking and show that the toy model has a
quantum break time consistent with the fast scrambling time scale
conjectured in the black hole context. We then conclude by pointing
out several possible extensions of these results.