We present an extension of partition function of open Gromov Witten theory of CY 3-folds defined by the trace of an extended vertex operator and give a Representation theoretic interpretation of the trace of the vertex operator involved. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators. We extend the definition of the Fock space of level l, to let the level to be infinity. We prove a duality between gl∞ and a∞ = gl∞ of Howe type, which provides a decomposition of F∞ into irreducible representations obtained from joint highest weight vector for gl∞ and a∞. The decomposition of the Fock space F∞ into highest weight representations provide a method to calculate and interpret the extended trace. Alternatively, the trace can be interpreted as the character of a representation of gl(∞|∞) on the self-tensor product F ⊗ F of the ordinary Fock space. Thinking of the character as a super-trace we give another calculation as a result.