  In this thesis we discuss recent new insights in the structure of the moduli
space of flat connections of Yang-Mills theory on a 3-torus. Chapter 2
discusses the computation of Witten's index for 4-dimensional gauge theories,
and the paradox that arises in comparing various computations. This was
resolved by the discovery that for orthogonal and exceptional gauge groups,
periodic flat connections exist that are contained in seperate, disconnected
components of the moduli space. Chapter 3 and 4 discuss some aspects of the
construction of holonomies parametrising vacua on such disconnected components.
Chapter 5 demonstrates a construction of vacua and holonomies for gauge
theories with classical groups, with non-periodic (twisted) boundary
conditions, using an orientifold description. The new solutions with
exceptional and orthogonal gauge groups also occur in string theory. Chapter 6,
containing previously unpublished material, shows that they can be realised
within heterotic string theories as asymmetric orbifolds. The presence of
string winding states modifies the analysis for the gauge theory in a crucial
way, eliminating many possibilities. The remaining ones are related by string
dualities to known and new theories.
