  Geometric transitions between Calabi-Yau manifolds have proven to be a
powerful tool in exploring the intricate and interconnected vacuum structure of
string compactifications. However, their role in N=1, 4-dimensional string
compactifications remains relatively unexplored. In this work we present a
novel proposal for transitioning the background geometry (including NS5-branes
and holomorphic, slope-stable vector bundles) of 4-dimensional, N=1 heterotic
string compactifications through a conifold transition connecting Calabi-Yau
threefolds. Our proposal is geometric in nature but informed by the heterotic
effective theory. Central to this study is a description of how the cotangent
bundles of the deformation and resolution manifolds in the conifold can be
connected by an apparent small instanton transition with a 5-brane wrapping the
small resolution curves. We show that by a "pair creation" process 5-branes can
be generated simultaneously in the gauge and gravitational sectors and used to
describe a coupled minimal change in the manifold and gauge sector. This
observation leads us to propose dualities for 5-branes and gauge bundles in
heterotic conifolds which we then confirm at the level of spectrum in large
classes of examples. While the 5-brane duality is novel, we observe that the
bundle correspondence has appeared before in the Target Space Duality exhibited
by (0,2) GLSMs. Thus our work provides a geometric explanation of (0,2) Target
Space Duality.
