Torus Fibrations and Localization of Index II

The joint spectral flow and localization of the indices of elliptic operators

“…It follows from the fact in symplectic geometry, that the tubular neighborhood of a Lagrangian submanifold is isomorphic to its tangent bundle as symplectic manifolds, and that T * T n is actually the product space (T * S 1 ) n . More detail is in Section 6.4 of [FFY10].…”

“…He deformed Dirac operators by adding potential terms coming from Morse functions or Killing vectors. Recently Fujita-Furuta-Yoshida [FFY10] used its infinite dimensional analogue to localize the Riemann-Roch numbers of certain completely integrable systems and their prequantum data on their Bohr-Sommerfeld fibers. In this case the indices of Dirac operators on fiber bundles localize on some special fibers instead of points.…”

“…As a consequence we reprove and generalize theorems of Witten and Fujita-Furuta-Yoshida. [FFY10]). Let (X, ω) be a symplectic manifold of dimension 2n, T n → X → B a Lagrangian fiber bundle, and (L, ∇ L , h) its prequantum data.…”

$S^1$-equivariant local index and transverse index for non-compact symplectic manifolds

“…In [4], Furuta, Yoshida and the author gave a formulation of index theory of Diractype operator on open manifolds using torus fibration and the perturbation by Diractype operator along fibers. In [5] a refinement of it for a family of torus bundles with some compatibility conditions was given.…”

“…In [6] the authors used equivariant version of them to give a geometric proof of quantization conjecture for Hamiltonian torus action on closed symplectic manifolds. In this paper we give a formulation of S 1 -equivariant index theory for non-compact symplectic manifold with Hamiltonian S 1 -action based on the framework of [4]. The resulting index is a homomorphism from R(S 1 ) to Z, and if the manifold is closed, then the S 1 -equivariant index coincides with the Riemann-Roch character as a functional on R(S 1 ).…”