From the text: Using an adiabatic collapse trick we determine, by two different methods, the eta invariants of many Dirac type operators on circle bundles over Riemann surfaces. These results, coupled with a delicate spectral flow computation, are then used to determine the virtual dimensions of moduli spaces of finite energy Seiberg-Witten monopoles on 4-manifolds bounding such circle bundles.
This paper is divided into three sections and four appendices. The first section is essentially a brief survey of known facts concerning the eta invariant: definition, the Atiyah-Patodi-Singer theorem, variational formulae and the spectral flow. We included these facts as a service to the reader, to eliminate any ambiguity concerning the various sign conventions. There does not seem to be general agreement on these conventions and, additionally, we used some “folklore” results which we could not indicate satisfactory references.
The second section contains the main steps in the computation of the eta invariants discussed above. We begin by describing the geometric background and the various Dirac operators. Then using variational formulae for the eta invariant and the adiabatic results of Bismut-Cheeger-Dai we compute in the second part the eta invariant of the Dirac operator on a circle bundle with very short fibers (Theorem 2.4).
In the third part, we compute the eta invariant of the adiabatic Dirac operator – a perturbation of the Dirac operator which arose in [L. I. Nicolaescu, Commun. Anal. Geom. 6, No. 2, 331-392 (1998; Zbl 0937.58025)]. This is achieved in Theorem 2.6 via a variational formula and a spectral flow computation. The computations of certain transgression terms involved in the variational formulae are deferred to appendices. An alternative method of computation is described in Appendix C.
The last part of this section is devoted to extending the previous computations to the Dirac operators coupled with flat line bundles. We use essentially the same variational strategy. However, new phenomena arise during the computation of some spectral flow contributions.
The third section is devoted to applications to Seiberg-Witten theory. The first two subsections describe the 3- and 4-dimensional Seiberg-Witten equations and some basic facts about them established in [T. Mrowka, P. Ozsvath and B. Yu, Commun. Anal. Geom. 5, No. 4, 685-791 (1997; Zbl 0933.57030)] and by the author [loc. cit.]. The third subsection is entirely devoted to the computation of a spectral flow. This is a very delicate job since one has to worry about eigenvalues changing sign in a nontransversal manner. In the last subsection we compute virtual dimensions of finite energy Seiberg-Witten moduli spaces on 4-manifolds founding circle bundles over Riemann surfaces and we conclude by comparing our answers in the special case of tunnelings to those by T. Mrowka, P. Ozsvath, B. Yu [loc. cit.].