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

This paper continues with Part I. We define the module for a \(\frac{{\mathbb Z}}{2}\)-graded meromorphic open-string vertex algebra that is twisted by an involution and show that the axioms are sufficient to guarantee the convergence of products and iterates of any number of vertex operators. A module twisted by the parity involution is called a canonically \({\mathbb Z}_2\)-twisted module. As an example, we give a fermionic construction of the canonically \({\mathbb Z}_2\)-twisted module for the \(\frac{{\mathbb Z}}{2}\)-graded meromorphic open-string vertex algebra constructed in Part I. Similar to the situation in Part I, the example is also built on a universal \({\mathbb Z}\)-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 or among the zero modes. The Wick’s theorem still holds, though the actual vertex operator needs to be corrected from the naïve definition by normal ordering using the \(\exp (\Delta (x))\)-operator in Part I.