# A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra

## TL;DR

This paper presents a comprehensive explicit construction method for lower-bounded generalized twisted modules of grading-restricted vertex (super)algebras, satisfying a universal property, applicable to automorphisms and the identity case.

## Contribution

It introduces a direct construction approach for lower-bounded generalized twisted modules with a universal property, expanding the toolkit for vertex algebra module theory.

## Key findings

- Established convergence and commutativity of generating twisted fields.
- Defined twisted vertex operator maps for constructed modules.
- Proved that all lower-bounded generalized twisted modules are quotients of universal modules.

## Abstract

We give a general, direct and explicit construction of lower-bounded generalized twisted modules satisfying a universal property for a grading-restricted vertex (super)algebra $V$ associated to an automorphism $g$ of $V$. In particular, when $g$ is the identity, we obtain lower-bounded generalized $V$-modules satisfying a universal property. Let $W$ be a lower-bounded graded vector space equipped with a set of "generating twisted fields" and a set of "generator twist fields" satisfying a weak commutativity for generating twisted fields, a generalized weak commutativity for one generating twisted field and one generator twist field and some other properties that are relatively easy to verify. We first prove the convergence and commutativity of products of an arbitrary number of generating twisted fields, one twist generator field and an arbitrary number of generating fields for $V$. Then using the convergence and commutativity, we define a twisted vertex operator map for $W$ and prove that $W$ equipped with this twisted vertex operator map is a lower-bounded generalized $g$-twisted $V$-module. Using this result, we give an explicit construction of lower-bounded generalized $g$-twisted $V$-modules satisfying a universal property starting from vector spaces graded by weights, $\mathbb{Z}_{2}$-fermion numbers and $g$-weights (eigenvalues of $g$) and real numbers corresponding to the lower bounds of the weights of the modules to be constructed. In particular, every lower-bounded generalized $g$-twisted $V$-module (every lower-bounded generalized $V$-module when $g$ is the identity) is a quotient of such a universal lower-bounded generalized $g$-twisted $V$-module (a universal lower-bounded generalized $V$-module).