Abstract. The Nekrasov conjecture predicts a relation between the partition function for N = 2 supersymmetric Yang-Mills theory and the Seiberg-Witten prepotential. For instantons on R 4 , the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces.

We study moduli of vector bundles on a two-dimensional neighbourhood Z k of an irreducible curve ℓ ∼ = P 1 with ℓ 2 = −k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We describe the topology of the regular and singular fibres, in particular we calculate their middle Betti numbers.

We describe semiuniversal deformation spaces for the noncompact surfaces Z k := Tot(O P 1 (−k)) and prove that any nontrivial deformation Z k (τ ) of Z k is affine.It is known that the moduli spaces of instantons of charge j on Z k are quasi-projective varieties of dimension 2j − k − 2. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation Z k (τ ) are empty.

We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangian submanifolds of the orbits.