Eta invariants of Dirac operators on Circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces

We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces on any 4-manifold bounding unions of circle bundles. This belated paper should be regarded as the analytical backbone of dg-ga/9711006. There, we indicated only what changes are needed to extend the methods of the present paper to Seifert fibrations and we focused only to topological and number theoretic aspects related to Froyshov invariants