More specifically, my current research interests are around the Vafa-Witten equations and the Kapustin-Witten ones on four-manifolds. They originated in an attractive theory called N=4 super (!) Yang-Mills theory in (Theoretical) Particle Physics, or more broadly in Superstring Theory. I've reached these objects with huge surprise and excitement from mathematical studies in higher-dimensional gauge theories such as the Donaldson-Thomas invariants and Spin(7)-instantons. Now I'm working on these from both algebraic and analytic aspects.
As for the algebraic studies,
Richard Thomas and I defined
Vafa-Witten invariants for projective surfaces from the moduli space of solutions to the Vafa-Witten equations by using virtual
C^{*}-localisations, and computed them in examples.
Surprisingly, conjectures by Vafa and Witten match with our calculations despite theirs only dealing with bundles.
Our projects for this are then to get more computable examples, perhaps including more semi-stables by Mochizuki/Joyce-Song formalisms, to examine the modularity of partition functions; and also to categorify these in the style of team Joyce on the theory of Donaldson-Thomas invariants. We also try to formulate
Kapustin-Witten invariants, using the same trick to the Vafa-Witten stuffs of ours; and invoking probably the Borisov-Joyce type theory on Calabi-Yau four-folds.
Analytic studies for them look rather horrible if one tries to define a Vafa-Witten invariant or such sort of things for general smooth four-manifold, since there are no a priori bounds for the Higgs fields for one reason. However, a recent breakthrough by Cliff Taubes in the studies of SL(2,C)-connections on closed four-manifolds, which includes the Kapustin-Witten case, wonderfully opened up a new way of understudying the boundary behaviour of the Higgs fields in the moduli spaces. Cliff introduces some real
codimension two singular sets, outside which a sequence of "partially rescaled" SL(2,C)-connections converges after
gauge transformations except bubbling out at finite set of points.
For the Vafa-Witten case, I proved the singular set is empty under certain no bubbling condition.
This enable us to relate L^2 bound of the Higgs fields with the irreduciblity of the connection in the equations.
More recently, Cliff obtained a nice convergence result for rescaled Higgs fields also in the Vafa-Witten equations, which surely plays a crucial role for the future analytic studies on the Vafa-Witten invariant.
With regard to the Kapustin-Witten equations, I figured out the singular sets for the underlying manifold being a compact Kahler surface are analytic subvarieties.
I'm now analysing those singular sets both in the Vafa-Witten and the Kapustin-Witten cases further ahead to hopefully get a nice compactification of their moduli spaces in rather comprehensible settings.
(folding back)