Email y-tanaka(
Gauge theory
Keywords Uhlenbeck compactification, virtual fundamental classes, (symmetric) obstruction theory, wallcrossing formulae
Office Interdisciplinary Research Building, Room 503 (Toyonaka campus)

One of my long term project is to construct an "analytic Donaldson-Thomas invariant", and to give a geometric interpretation of the MNOP conjecture suggested by physicists, which implies a strong relation between the Donaldson-Thomas invariants and the Gromov-Witten invariants on Calabi-Yau threefolds, at the level of the partition functions. Motivated by this, I'm currently interested in Vafa-Witten invariants and the Kapustin-Witten theory formulated on four-manifolds, and approach them by using techniques in algebraic geometry and geometric analysis. The study using techniques in algebraic geometry is joint work with Richard Thomas; we succeeded in defining Vafa-Witten invariants, and computed the partition functions of them in examples. It was a great surprise that our calculations match with conjectures by Vafa and Witten on the modularity of the partition functions more than 20 years ago. On the other hand, the analytic study of these has been wonderfully developed by a series of work by Cliff Taubes, and I figured out the structures of singular sets introduced by Cliff, in some cases. These Vafa-Witten invariants and Kapustin-Witten theory are in fact related to the Donaldson-Thomas invariant and "Spin(7) instantons invariant", so, with this point of view, I also try to "categorify" these invariants; and develop gauge theories on manifolds with special holonomy such as G_2 instantons and Spin(7) ones further ahead.