On the supersymmetric $N=1$ and $N=(1,1)$ spectral data : A multiplicativity property

TL;DR

This paper investigates the tensor product structures of supersymmetric N=(1,1) spectral data, identifying a unique multiplicative extension from N=1 to N=(1,1) spectral data within noncommutative geometry.

Contribution

It demonstrates that among six possible tensor products, only one preserves the extension's multiplicativity, establishing a unique tensor product for N=(1,1) spectral data.

Findings

Six tensor product choices identified for N=(1,1) spectral data

Only one tensor product preserves the extension's multiplicativity

Establishment of a unique tensor product under multiplicativity constraint

Abstract

We show that there are six different choice of tensor product of supersymmetric N=(1,1) spectral data in the context of supersymmetric quantum theory and noncommutative geometry. We also show that the procedure of extending a supersymmetric N=1 spectral data to N=(1,1) spectral data respects only one tensor product among these. We refer this as the multiplicativity property of the extension procedure. Therefore, if we demand that the extension procedure is multiplicative then there is a unique choice of tensor product of N=(1,1) spectral data.