Hyperkähler Manifolds and Seiberg–Witten Equations

The mathematical properties of the so-called gauged nonlinear $\sigma$-model in dimension 4 are studied. An important element of the construction is a nonlinear generalization of the Dirac operator on a 4-manifold such that the fiber of the spinor vector bundle, a copy of quaternions $\mathbb H$, is replaced by a hyperkähler manifold endowed with a hyperkähler Lie group action and an additional symmetry. This Dirac operator is used to define Seiberg–Witten moduli spaces. An explicit Weitzenböck formula for such a Dirac operator is derived and applied to describe some properties of the Seiberg–Witten moduli spaces.