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Non-Perturbative Aspects of Supersymmetric
Gauge Theories and String Theory

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by

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John Ma Pierre

# Abstract

In this dissertation we explore various exact non-perturbative
results in supersymmetric gauge theories. We also discuss some issues
in strongly-coupled string theory such as the interactions of branes in
a matrix model of M-theory and the use of D-branes as probes and as microscopic
descriptions of black holes.
We begin with an investigation of *N = 1* supersymmetric
theories based on exceptional groups. The flat directions are most
easily parameterized using their correspondence with gauge invariant polynomials.
Symmetries and holomorphy tightly constrain the superpotentials, but due
to multiple gauge invariants other techniques are needed for their full
determination. We give an explicit treatment of *G*_{2}
and find gaugino condensation for *N*_{f }__<__
2, and an instanton generated superpotential for
*N*_{f }= 3. The analogy with *SU(N*_{c})
gauge theories continues with modified and unmodified quantum moduli spaces
for *N*_{f }= 4 and *N*_{f} = 5 respectively,
and a non-Abelian Coulomb phase for *N*_{f} __>__ 6.
Electric variables suffice to describe this phase over the full range of
*N*_{f}.

Next we apply the method of confining phase superpotentials
to derive the exact quantum moduli space of *N = 2* supersymmetric
Yang-Mills theory with the exceptional gauge group *G*_{2}.
Our findings are consistent with the spectral curve of the periodic Toda
lattice, but do not agree with the hyperelliptic curve suggested previously
in the literature. We also apply the method to theories with fundamental
matter, treating both the example of *SO(5)* and *G*_{2}.

Turning to string theory, we consider eight-brane configurations
in M(atrix) theory and compute their interaction potentials with gravitons,
membranes, and four-branes. We compare these results with the interactions
of D8-branes with D0-branes, D2-branes, and D4-branes in IIA string theory.
We find agreement between the two approaches for eight-brane interactions
with two-branes and four-branes. A discrepancy is noted in the case
with zero-branes.

Finally, we consider configurations of D6-branes
with D0-brane charge and compute interaction potentials with various D-brane
probes using a 1-loop open string calculation. These results are compared
to a supergravity calculation using the solution of an extremal black hole
carrying 0-brane and 6-brane charge. We find precise agreement of
the long distance interaction potentials and explore the short distance
behavior of the D-brane configuration.