Infinite dimensional GIT and moment maps in differential geometry

Abstract

The starting point of this thesis is the following observation of Atiyah and Bott: The curvature of a connection on a bundle over a surface can be understood as a moment map for the action of the gauge group. Moreover, the moduli space of flat connections, or more generally of Yang--Mills connections, is closely related to the moduli space of holomorphic bundles obtained from geometric invariant theory. We discuss the various implications of this observation to the Yang--Mills equations and the symplectic vortex equations over Riemann surfaces. As main results, we obtain the analogue of the Ness uniqueness theorem, the Kempf-Ness theorem, the Hilbert-Mumford criterion and the moment-weight inequality in both settings. The main technical ingredients are long-time existence and convergence of the Yang--Mills and the Yang--Mills--Higgs heat flow. These are the parabolic flows associated to the corresponding moment map squared functionals in both setups. Donaldson introduced extensions of the Atiyah--Bott picture to actions of the diffeomorphism group. We begin with a self-contained exposition of his moment map framework and its applications to Teichmüller theory. This is the starting point for the following three projects, discussed in the remainder of this thesis. The first one generalizes Donaldson's construction of Teichmüller space to the moduli spaces of tuples of holomorphic differentials of mixed degree. These moduli spaces are closely related to Hitchin's higher Teichmüller components. A distant hope is, that this might lead to a new construction of the Hitchin component using the diffeomorphism group instead of the gauge group. The second project is joint work with Dietmar Salamon and Oscar Garcia-Prada. We show that the Ricci form yields a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This yields an extended Weil--Petersson symplectic form on the Calabi--Yau Teichmüller space of isotopy classes of complex structures with first Chern class zero and nonempty Kähler cone. The third project is joint work with Oscar Garcia-Prada, Luis Alvarez-Consul and Mario Garcia-Fernandez. We investigate variants of the Hitchin equations where the complex structure is not fixed and the gauge group is extended by the Hamiltonian diffeomorphism group. This leads to moduli spaces which naturally fiber over Teichmüller space with fibre being the corresponding Hitchin moduli space.