More than 10 years ago, I constructed representations of
Heisenberg algebras and affine algebras on the homology groups of
instanton moduli spaces on certain 4-manifolds. The proof was
mathematically rigorous, but was not conceptually satisfactory.
About two years ago, physicists, Alday-Gaiotto-Tachikawa (AGT) proposed
a much larger framework, which is conceptually satisfactory, but lacks
mathematical footing. I will explain their theory, and a mathematical
approach towards a conjecture.