A modular spectral triple for $\kappa$-Minkowski space

TL;DR

This paper constructs a spectral triple for two-dimensional $kappa$-Minkowski space, incorporating modular theory to define a Dirac operator and measure spectral dimension, aligning with classical expectations.

Contribution

It introduces a modular spectral triple for $kappa$-Minkowski space, utilizing a weight satisfying a KMS condition and a twisted commutator to ensure boundedness.

Findings

Spectral dimension equals classical dimension

Unique Dirac operator satisfying bounded twisted commutator

Incorporation of modular theory in noncommutative geometry

Abstract

We present a spectral triple for $\kappa$-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the $\kappa$-Poincar\'e algebra. The weight satisfies a KMS condition and its associated modular operator plays an important role in the construction. This forces us to introduce two ingredients which have a modular flavor: the first is a twisted commutator, used to obtain a boundedness condition for the Dirac operator, the second is a weight replacing the usual operator trace, used to measure the growth of the resolvent of the Dirac operator. We show that, under some assumptions related to the symmetries and the classical limit, there is a unique Dirac operator and automorphism such that the twisted commutator is bounded. Then, using the weight mentioned above, we compute the spectral dimension associated to the spectral triple and find that is equal to the classical dimension. Finally we briefly discuss the introduction of a real structure.