Abstract. This paper is the third of the series concerning the localization of the index of Dirac-type operators. In our previous papers we gave a formulation of index of Diractype operators on open manifolds under some geometric setting, whose typical example was given by the structure of a torus fiber bundle on the ends of the open manifolds. We introduce two equivariant versions of the localization. As an application we give a proof of Guillemin-Sternberg's quantization conjecture in the case of torus action.

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which is a generalization of the topological classification of quasi-toric manifolds by Davis and Januszkiewicz and of effective two-dimensional torus actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We investigate locally toric Lagrangian fibrations from the viewpoint of local torus actions. We give a necessary and sufficient condition in order that a local torus action becomes a locally toric Lagrangian fibration. Locally toric Lagrangian fibrations are classified by Boucetta and Molino up to fiber-preserving symplectomorphisms. We shall reprove the classification theorem of locally toric Lagrangian fibrations by refining the proof of the classification theorem of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes.Comment: 40 pages, 10 figures. Definition 1.1 is modified. Typos correcte

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Let π : (M, ω) → B be a (non-singular) Lagrangian torus fibration on a compact, complete base B with prequantum line bundle (L, ∇ L ) → (M, ω). For a positive integer N and a compatible almost complex structure J on (M, ω) invariant along the fiber of π, let D be the associated Spin c Dirac operator with coefficients in L ⊗N . Then, we give an orthogonal family { ϑ b } b∈B BS of sections of L ⊗N indexed by the Bohr-Sommerfeld points B BS , and show that each ϑ b converges to a delta-function section supported on the corresponding Bohr-Sommerfeld fiber π −1 (b) and the L 2 -norm of D ϑ b converges to 0 by the adiabatic(-type) limit. Moreover, if J is integrable, we also give an orthogonal basis {ϑ b } b∈B BS of the space of holomorphic sections of L ⊗N indexed by B BS , and show that each ϑ b converges to a deltafunction section supported on the corresponding Bohr-Sommerfeld fiber π −1 (b) by the adiabatic(-type) limit. We also explain the relation of ϑ b with Jacobi's theta functions when (M, ω) is T 2n . Contents Motivation and MainTheorems 1 1.1. Notations 4 2. Developing Lagrangian fibrations 4 2.1. Integral affine structures 4 2.2. Lagrangian fibrations 7 2.3. Lagrangian fibrations with complete bases 9 2.4. The lifting problem of the Γ-action to the prequantum line bundle 12 3. Degree-zero harmonic spinors and integrability of almost complex structures 15 3.1. Bohr-Sommerfeld points 15 3.2. Almost complex structures 16 3.3. The existence condition of non-trivial harmonic spinors of degree-zero 18 3.4. The Γ-equivariant case 27 4. The integrable case 29 4.1. Definition and properties of ϑ m N 29 4.2. The case when Z is constant 32 4.3. Adiabatic-type limit 33 5. The non-integrable case 36 References 40

In [8] we introduced the notion of a local torus actions modeled on the standard representation (we call it a local torus action for simplicity), which is a generalization of a locally standard torus action. In this note we define a lifting of a local torus action to a principal torus bundle, and show that there is an obstruction class for the existence of liftings in the first cohomology of the fundamental group of the orbit space with coefficients in a certain module.