The local index formula of Connes and Moscovici and equivariant zeta functions for the affine metaplectic group.

Abstract: We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. This algebra includes as subalgebras the noncommutative tori and toric orbifolds. We introduce the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, \Lambda(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $\Psi(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta function $\zeta_B(z) = Tr (B|D|^{-2z})$ has a meromorphic extension to $\mathbb C$ with at most simple poles and decays rapidly along vertical lines. Our main result then is an explicit algebraic expression for the Connes-Moscovici cyclic cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.

(Joint work with Anton Savin, RUDN, Moscow)

mathematical physicscategory theorydifferential geometryfunctional analysisK-theory and homologyoperator algebrasquantum algebra

Audience: researchers in the discipline

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