# On the Four Dimensional Holoraumy of the 4D, $\cal N$ = 1 Complex Linear Supermultiplet

## TL;DR

This paper explores the concept of holoraumy in supersymmetric field theories, extending it to the complex linear supermultiplet by analyzing the commutator of supercovariant derivatives on its components.

## Contribution

It introduces the extension of holoraumy to the complex linear supermultiplet through detailed calculations of supercovariant derivative commutators.

## Key findings

- Supports the existence of weight spaces in supersymmetric theories
- Calculates supercovariant derivative commutators for the complex linear supermultiplet
- Provides a foundation for understanding supermultiplet structures

## Abstract

We present arguments to support the existence of weight spaces for supersymmetric field theories and identify the calculations of information about supermultiplets to define such spaces via the concept of "holoraumy." For the first time this is extended to the complex linear supermultiplet by a calculation of the commutator of supercovariant derivatives on all of its component fields.