The graded product of real spectral triples

TL;DR

This paper introduces a new graded tensor product for real spectral triples in non-commutative geometry, resolving issues of non-commutativity, non-associativity, and improper transformations present in the ungraded approach.

Contribution

It proposes a novel graded tensor product for real spectral triples that ensures the product is commutative, associative, and properly transforms under unitaries.

Findings

The graded tensor product resolves non-commutativity issues.

It ensures associativity of spectral triple products.

The new product always defines a proper spectral triple.

Abstract

Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple.