Spectral triples with multitwisted real structure

TL;DR

This paper introduces multitwisted real structures in spectral triples, unifying various examples including quantum and noncommutative geometries, and extends the first-order condition to accommodate non-automorphic twists.

Contribution

It generalizes spectral triples with reality structures to include multitwisted real structures, enabling a unified framework for diverse noncommutative geometries.

Findings

Introduces multitwisted order one condition for Dirac operators.

Unifies known examples like quantum disc, quantum cone, and asymmetric tori.

Allows twists that do not implement algebra automorphisms.

Abstract

We generalize the notion of spectral triple with reality structure to spectral triples with multitwisted real structure, the class of which is closed under the tensor product composition. In particular, we introduce a multitwisted order one condition (characterizing the Dirac operators as an analogue of first-order differential operator). This provides a unified description of the known examples, which include rescaled triples with the conformal factor from the commutant of the algebra and (on the algebraic level) triples on quantum disc and on quantum cone, that satisfy twisted first order condition of \cite{BCDS16,BDS19}, as well as asymmetric tori, non-scalar conformal rescaling and noncommutative circle bundles. In order to deal with them we allow twists that do not implement automorphisms of the algebra of spectral triple.