Schwinger terms, gerbes, and operator residues
We give an introduction into and exposition of Seiberg-Witten theory.
For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related boundedness properties of these operators and show in particular that, although they are not bounded on in general, they are always bounded on suitable weighted spaces.
Following joint work with Dyatlov [DyGu], we describe the semi-classical measures associated with generalized plane waves for metric perturbation of , under the condition that the geodesic flow has trapped set of Liouville measure .
We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs microlocally supported away from the trapping: , a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, , to be supported at the trapped set, giving when the a priori bound is .
Let be a compact Kähler manifold with integral Kähler class and a holomorphic Hermitian line bundle whose curvature is the symplectic form of . Let be a Hamiltonian, and let be the Toeplitz operator with multiplier acting on the space . We obtain estimates on the eigenvalues and eigensections of as , in terms of the classical Hamilton flow of . We study in some detail the case when is an integral coadjoint orbit of a Lie group.