Abstract: In the 1990's, Vafa-Witten tested S-duality of N=4 SUSY Yang-Mills
theory on a complex algebraic surface by studying modularity of a
certain partition function. Recently, a mathematical definition of
Vafa-Witten's invariants was given by Tanaka-Thomas. I outline a
method for calculating the instanton contribution to these invariants
using Mochizuki's theory of algebraic Donaldson invariants. For SU(2),
this leads to verifications of Vafa-Witten's original formula. For
SU(3), we find a new formula which corrects an error in the physics
literature. I will also discuss refinements to virtual \chi_y genus, elliptic
theory on a complex algebraic surface by studying modularity of a
certain partition function. Recently, a mathematical definition of
Vafa-Witten's invariants was given by Tanaka-Thomas. I outline a
method for calculating the instanton contribution to these invariants
using Mochizuki's theory of algebraic Donaldson invariants. For SU(2),
this leads to verifications of Vafa-Witten's original formula. For
SU(3), we find a new formula which corrects an error in the physics
literature. I will also discuss refinements to virtual \chi_y genus, elliptic
genus, and cobordism class. In the first half of the talk I'll give a brief
introduction to virtual intersection numbers. Joint work with Göttsche.