Seminars & Colloquia Calendar

Hyperkaehler (HK) manifolds have attracted a lot of attention both by mathematicians and physicists due to their rich and intricate structure. Most prominently, they come equipped with three complex structures satisfying the quaternionic relations. Even though an HK manifold M is a differential-geometric object, the HK structure can be encoded in one complex-analytic object, the twistor space Z of M. By construction, it fibers over the complex line and the sections of this fibration provide the link to the HK structure on M. In this talk, we pursue and extend this complex-analytic approach for HK manifolds that admit an appropriate circle action. For each such HK manifold M, Haydys constructed a hyperholomorphic line bundle, i.e. it is holomorphic with respect to each of the three complex structures on M. In turn, this induces a holomorphic line bundle L on Z. By employing L, we define an energy functional on the space of sections of Z. We show that it is a natural complex-analytic extension of the moment map of the circle action on M. This gives new insights on the space of sections of Z if M is the moduli space of solutions to Hitchin's self-duality equations.