  We study the behavior of Donaldson's invariants of 4-manifolds based on the
moduli space of anti self-dual connections (instantons) in the perturbative
field theory setting where the underlying source manifold has boundary. It is
well-known that these invariants take values in the instanton Floer homology
groups of the boundary 3-manifold. Gluing formulae for these constructions lead
to a functorial topological field theory description according to a system of
axioms developed by Atiyah, which can be also regarded in the setting of
perturbative quantum field theory, as it was shown by Witten, using a version
of supersymmetric Yang-Mills theory, known today as Donaldson-Witten theory.
One can actually formulate an AKSZ model which recovers this theory for a
certain gauge-fixing. We consider these constructions in a perturbative quantum
gauge formalism for manifolds with boundary that is compatible with cutting and
gluing, called the BV-BFV formalism, which was recently developed by Cattaneo,
Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum
Master Equation and extend the result to a global picture when perturbing
around constant background fields. Additionally, we relate these constructions
to Nekrasov's partition function by treating an equivariant version of
Donaldson-Witten theory in the BV formalism. Moreover, we discuss the
extension, as well as the relation, to higher gauge theory and enumerative
geometry methods, such as Gromov-Witten and Donaldson-Thomas theory and recall
their correspondence conjecture for general Calabi-Yau 3-folds. In particular,
we discuss the corresponding (relative) partition functions, defined as the
generating function for the given invariants, and gluing phenomena.
