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 G2 and find gaugino condensation for Nf < 2, and an instanton generated superpotential for Nf = 3. The analogy with SU(Nc) gauge theories continues with modified and unmodified quantum moduli spaces for Nf = 4 and Nf = 5 respectively, and a non-Abelian Coulomb phase for Nf > 6. Electric variables suffice to describe this phase over the full range of Nf.
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 G2. 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 G2.
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.