A local index formula for metaplectic operators

Abstract: Let A be the algebra of unitary operators acting in $H=L_2(R^n)$ and generated by translations, orthogonal transformations, products with exponentials $e^{ikx}$ and fractional Fourier transforms. Equivalently, A is the algebra generated by quantizations of isometric affine canonical transformations in $T^*R^n$. We show that the well-known index one operator in $R^n$ (which is obtained from the creation and annihilation operators, see Higson-Kasparov-Trout 1998) denoted by D defines a spectral triple (A,H,D) in the sense of Connes. Our main result is an explicit formula for the Connes--Moscovici residue cocycle for this spectral triple. For the subalgebra in A generated by translations and exponentials, this gives a local index formula for noncommutative tori. This is joint work with Elmar Schrohe (Hannover)

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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