Symplectic quotients of unstable Morse strata for normsquares of moment maps

Abstract

Let K be a compact Lie group and fix an invariant inner product on its Lie algebra. Given a Hamiltonian action of K on a compact symplectic manifold X, the normsquare of the moment map defines a Morse stratification of X by locally closed symplectic submanifolds such that the stratum to which any x in X belongs is determined by the limiting behaviour of its downwards trajectory under the gradient flow with respect to a suitably compatible Riemannian metric on X. The open stratum indexed by 0 retracts K-equivariantly via this gradient flow to the minimum which is the zero-locus of the moment map (if this is not empty). The usual 'symplectic quotient' for the action of K on any other stratum is empty. Nonetheless, motivated by recent results in non-reductive geometric invariant theory, we find that the symplectic quotient construction can be modified to provide natural 'symplectic quotients' for the unstable strata labelled by nonzero indices. There is an analogous infinite-dimensional picture for the Yang--Mills functional over a Riemann surface with strata determined by Harder-Narasimhan type.