Abstract:
In the late 90's, S. K. Donaldson and R. P.
Thomas
suggested a program on gauge theory in
higher dimensions.
In alignment with this program, Thomas
introduced
the holomorphic Casson invariant for a
smooth projective Calabi-Yau
threefold.
The holomorphic Casson invariant is a
deformation invariant of a smooth
projective Calabi-Yau threefold.
Thomas defined it by constructing
algebro-geometrically a virtual moduli
cycle
of the moduli space of semi-stable sheaves
under some technical assumptions.
This can be viewed as a complex analogy of
the Taubes-Casson invariant
defined by C. H. Taubes.
However, it is not achieved as yet to
construct the invariant
in a "gauge theoretic", or an
analytic way.
In this seminar, we exhibit an analytic
approach to this invariant
by using perturbed Hermitian-Einstein
equations over a compact Kahler
threefold,
which we call Donaldson-Thomas equations.
We begin this seminar with some backgrounds
on the holomorphic Casson
invariant
and the Hermitian-Einstein equations.
Subsequently, we introduce the
Donaldson-Thomas equations
and describe local structures of the moduli
space of solutions
to the Donaldson-Thomas equations.
Furthermore, we discuss analytic results on
solutions to the equations,
aiming at constructing the holomorphic
Casson invariant
for compact Kahler threefolds.