Product of real spectral triples

TL;DR

This paper develops a comprehensive method for constructing the product of real spectral triples across various dimensions and parities, addressing complexities like multiple real structures and representations.

Contribution

It introduces a systematic approach to forming product spectral triples that accounts for different cases and choices in real structures and Dirac operators.

Findings

Provides explicit construction methods for product triples

Addresses multiple real structures and gamma matrix representations

Lays groundwork for applications in noncommutative geometry

Abstract

We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.