Contact degree and the index of Fourier integral operators.

*(English)*Zbl 0929.58012Let \(X\) be a compact contact manifold, and \(L\subset T^*X\) an oriented contact line bundle on \(X\). Let \({\mathcal M}(X)\) be the group of components of the space of contact diffeomorphisms. Using the notion of quantization of the compact structure, the authors construct a contact degree, that is, a homomorphism \(c- \deg: {\mathcal M}(X) \to\mathbb{Z}\).

The contact degree is reduced to the index of a Dirac operator. Let \(\phi\) be an oriented contact diffeomorphism, and \(Z_\phi\) the mapping torus of \(\phi\), i.e., \(X\times[-1,1]\) with the ends identified by \(\phi\). The contact structure on \(X\) gives \(Z_\phi\) a natural Spin-\(\mathbb{C}\) structure. If \(\partial_\phi\) is the associated Dirac operator then \[ c-\deg (\phi)= \text{ind} (\partial_\phi). \] The last formula follows from the identification of \(c-\deg (\phi)\) with the spectral flow of the curve of Dirac operators on the contact manifold associated with an isotropy from any one partial Hermitian structure to its \(\phi\)-conjugate.

The above result leads to a geometric formula for the index of elliptic Fourier integral operators, which gives the answer to the long-open question of Weinstein for operators acting on a fixed manifold. More precisely, the following theorem is proved.

Theorem 3. If \(Y\) is a compact manifold and \(X=S^*Y\) is its cosphere bundle, then for any (oriented) contact diffeomorphism, \(\phi\), of \(X\), i.e., a canonical diffeomorphism of \(T^*X\setminus 0\), \[ \text{ind}(F_\phi)=c-\deg(\phi), \] where \(F_\phi\) is a Fourier integral operator associated to \(\phi\) with elliptic symbol corresponding to the positive trivialization of the Maslov bundle.

The contact degree is reduced to the index of a Dirac operator. Let \(\phi\) be an oriented contact diffeomorphism, and \(Z_\phi\) the mapping torus of \(\phi\), i.e., \(X\times[-1,1]\) with the ends identified by \(\phi\). The contact structure on \(X\) gives \(Z_\phi\) a natural Spin-\(\mathbb{C}\) structure. If \(\partial_\phi\) is the associated Dirac operator then \[ c-\deg (\phi)= \text{ind} (\partial_\phi). \] The last formula follows from the identification of \(c-\deg (\phi)\) with the spectral flow of the curve of Dirac operators on the contact manifold associated with an isotropy from any one partial Hermitian structure to its \(\phi\)-conjugate.

The above result leads to a geometric formula for the index of elliptic Fourier integral operators, which gives the answer to the long-open question of Weinstein for operators acting on a fixed manifold. More precisely, the following theorem is proved.

Theorem 3. If \(Y\) is a compact manifold and \(X=S^*Y\) is its cosphere bundle, then for any (oriented) contact diffeomorphism, \(\phi\), of \(X\), i.e., a canonical diffeomorphism of \(T^*X\setminus 0\), \[ \text{ind}(F_\phi)=c-\deg(\phi), \] where \(F_\phi\) is a Fourier integral operator associated to \(\phi\) with elliptic symbol corresponding to the positive trivialization of the Maslov bundle.

Reviewer: L.Skrzypczak (Poznań)

##### MSC:

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J20 | Index theory and related fixed-point theorems on manifolds |