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

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 built 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.