# Physicists' $d=3+1$, $N=1$ superspace-time and supersymmetric QFTs from a tower construction in complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry and a purge-evaluation/index-contracting map

## TL;DR

This paper develops a complexified ${
m Z}/2$-graded $C^$-Algebraic Geometry framework to model 4D, N=1 supersymmetric quantum field theories, including the Wess-Zumino model and supersymmetric gauge theories, linking geometric structures with particle physics supersymmetry.

## Contribution

It introduces a towered superspace construction and a purge-evaluation map to embed supersymmetric QFTs into complexified $C^$-Algebraic Geometry, extending the geometric understanding of supersymmetry.

## Key findings

- Reproduces 4D, N=1 supersymmetric models within algebraic geometry.
- Defines a purge-evaluation map linking differential geometry and supersymmetry.
- Lays groundwork for nonabelian and extended supersymmetry generalizations.

## Abstract

The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics Theorem in Quantum Field Theory, which requires fermionic components of a superfield be anticommuting, lead us to the notion of towered superspace(-time) $\widehat{X}^{\widehat{\boxplus}}$ and the built-in purely even physics sector $X^{\mbox{physics}}$ from $\widehat{X}^{\widehat{\boxplus}}$. We use this to reproduce the $d=3+1$, $N=1$ Wess-Zumino model and the $d=3+1$, $N=1$ supersymmetric $U(1)$ gauge theory with matter --- as in, e.g., Chap.\,V and Chap.\,VI \& part of Chap.\,VII of the classical Supersymmetry \& Supergravity textbook by Julius Wess and Jonathan Bagger --- and, hence, recast physicists' two most basic supersymmetric quantum field theories solidly into the realm of (complexified ${\Bbb Z}/2$-graded) $C^\infty$-Algebraic Geometry. Some traditional differential geometers' ways of understanding supersymmetric quantum field theories are incorporated into the notion of a purge-evaluation/index-contracting map ${\cal P}:C^\infty(X^{\mbox{physics}})\rightarrow C^\infty(\widehat{X})$ in the setting. This completes for the current case a $C^\infty$-Algebraic Geometry language we sought for in D(14.1), footnote 2, that can directly link to the study of supersymmetry in particle physics. Once generalized to the nonabelian case in all dimensions and extended $N\ge 2$, this prepares us for a fundamental (as opposed to solitonic) description of super D-branes parallel to Ramond-Neveu-Schwarz fundamental superstrings