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Yuuji Tanaka: Lectures

September 10, 2018
TITLE: On the Vafa-Witten theory on closed four-manifolds

ABSTRACT: Vafa and Witten introduced a set of gauge-theoretic equations on closed four-manifolds around 1994 in the study of S-duality conjecture in N=4 super Yang-Mills theory in four dimensions. They predicted from supersymmetric reasoning that the partition function of the invariants defined through the moduli spaces of solutions to these equations would have modular properties. But little progress has been made other than their original work using results by Goettsche, Nakajima and Yoshioka.

However, it now looks worth trying to figure out some of their foreknowledge with more advanced technologies in analysis and algebraic geometry fascinatingly developed in these two decades. This talk discusses issues to construct the invariants out of the moduli spaces, and presents possible ways to sort them out by analytic and algebro-geometric methods; the latter is joint work with Richard Thomas.