Title: Complex(ified) gauge theories on gravitational instantons

Abstract: First, I will introduce complexified versions of the Yang-Mills
instanton equation on 4 dimensional, oriented, Riemannian manifolds. I will
focus on the two most important cases, which are called the Haydys and the
Kapustin-Witten equations, respectively.  When the underlying manifold is a
quotient of the 4-dimensional Euclidean space, there is a stark difference
between the two equations that can be summarized as follows: We show that
while there is a large moduli of solutions of the Haydys equation, there are
no ``nontrivial'' finite energy solution of the Kapustin-Witten equation on
these spaces.  Furthermore, the Haydys moduli space carries interesting
geometric structures, in particular, it is hyperkahler. Most of our results
hold for more general spaces, called gravitational instantons. (The full
generalization to these spaces is a work in progress.)  This is a joint work
with Gonçalo Oliveira (UFF).