Seiberg-Witten invariants and end-periodic Dirac operators
Let X be a smooth spin 4-manifold with homology of S^1 x S^3. In
our joint project with Tom Mrowka and Daniel Ruberman, we study the
Seiberg-Witten equations on X. The count of their solutions, called the
Seiberg-Witten invariant of X, depends on choices of Riemannian metric and
perturbation. A similar dependency issue is resolved in dimension 3 by
relating the jumps in the Seiberg-Witten invariant to the spectral flow of
the Dirac operator; the resulting invariant is then the Casson invariant.
dimension 4, we use Taubes' theory of end-periodic operators to relate the
jumps in the Seiberg-Witten invariant to the index theory of the Dirac
operator on a manifold with periodic end modeled on the infinite cyclic
cover of X. The resulting invariant is then a smooth invariant of X. Some
calculations and applications of this invariant will be discussed.