Quiver varieties, moduli spaces and representation theory

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Together with Kota Yoshioka and Lothar Gottsche I have studied relation between Donaldson invariants and Nekrasov's partition function. Further more, Takuro Mochizuki have joined our group to continue the research.Donaldson invariants are defined as integration of natural cohomology classes over moduli spaces of instantons on 4-manifolds.When the underlying 4-manifold has b_+ =1, the invariants depends on the choice of a Riemannian metric. The wall-crossing formula gives the difference of Donaldson invariants with respect to two Riemannian metrics. We express the wall-crossing formua in terms of Nekrasov's partition function, when the rank of vector bundles is 2. This was proved via a study of the torus action on the moduli space when the underlying 4-manifold is a toric surface. This research is an expansion of one started in 2004, we give the proof that the same formula holds for arbitrary projective surfaces, not necessarily toric surfaces.We further study the K-theoretic version of instanton counting. We study the theta function associated with the Seiberg-Witten curve which is a mirror of K-theoretic instanton couting and show that the Seiberg-Witten curve can be reconstructed from the coordinate ring of moduli spaces. This result is the K-theoretic generalization of the Nekrasov's conjecture. We also prove the wall-crossing formula similar to above, but we do not understand how to define the invariants using the instanton moduli spaces, due to singularieties. So we restrict our attention to the case of projective surfaces, and define invariants as holomorphic Euler characteristic of natural line bundles over the algebra-geometric compactification of the moduli spaces.With helps of Takuro Mochizuki, we study higher rank (>2) cases, and we prove a recursive expression of the wall-crossing formula, and proved that it is again given by the Nekrasov's partition function.