Fermionic construction of the $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra and its $\mathbb{Z}_2$-twisted module, I

TL;DR

This paper introduces a noncommutative generalization of vertex operator superalgebras called the $rac{ ext{Z}}{2}$-graded meromorphic open-string vertex algebra, exemplified by a fermionic construction with novel interaction and normal ordering properties.

Contribution

It defines a new class of noncommutative vertex algebras and provides an explicit fermionic example with a universal Fock space structure, extending the theory of vertex operator superalgebras.

Findings

Wick's theorem applies to the new algebraic structure.

Weak associativity is established for the constructed algebra.

Correlation functions have a closed-form expression.

Abstract

We define the $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra that is an appropriate noncommutative generalization of the vertex operator superalgebra. We also illustrate an example that can be viewed as a noncommutative generalization of the free fermion vertex operator superalgebra. The example is bulit upon a universal half-integer-graded non-anti-commutative Fock space where a creation operator and an annihilation operator satisfy the fermionic anti-commutativity relation, while no relations exist among the creation operators. The former feature allows us to define the normal ordering, while the latter feature allows us to describe interactions among the fermions. With respect to the normal ordering, Wick's theorem holds and leads to a proof of weak associativity and a closed formula of correlation functions.