Zeitschrift für verallgemeinerte Lügentheorie und -anwendungen

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.

The Lie algebra of isometries of dimension superior than or equal to three is semi-simple if and only if the horizontal nullity space of the Nijenhuis tensor of the canonical connection is reduced to zero and the derivative ideal coincides with algebra.

The content of the system of linear equations makes it an interesting matter that can also be further expanded and deepened. The three methods most commonly used to solve systems of equations are substitution, elimination, and extended matrices. Replacement and elimination are simple methods that can efficiently solve most systems of two equations in a few direct steps. The extended matrix method requires several steps, but its application extends to a number of different systems. The elimination method is useful when the coefficient of one of the variables is equal (or its negative equivalent) in all equations.