Local Index Formulae on Noncommutative Orbifolds and Equivariant Zeta Functions for the Affine Metaplectic Group

TL;DR

This paper develops local index formulae for noncommutative orbifolds and tori using spectral triples and zeta functions, providing explicit algebraic expressions for cyclic cocycles and meromorphic extensions of zeta functions.

Contribution

It introduces a spectral triple framework for noncommutative orbifolds and derives explicit local index formulas and cyclic cocycles, extending noncommutative geometry tools to these spaces.

Findings

Spectral triple has simple dimension spectrum with meromorphic zeta functions.

Explicit algebraic expressions for Connes-Moscovici cyclic cocycle.

Local index formulas obtained for noncommutative tori and toric orbifolds.

Abstract

We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define 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 ${\zeta}_B(z) = {\rm Tr} (B|D|^{-2z})$ has a meromorphic extension to $\mathbb C$ with at most simple poles. 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.