Symmetric invariant pairings in vertex operator super algebras and Gramians Original Research Article

We define a symmetric bilinear pairing on any connected vertex operator super algebra with various vertex operator invariance properties including the translation invariance. The notion of quasi-primary vectors leads to an orthogonal double grading decomposition of the algebra. Hermitian pairings associated to certain conjugate-linear involutions are shown to be positive definite and integral for vertex operator algebras generated by vectors of weight one half. No Hilbert space structure on the vertex operator algebra is assumed. We deduce it. Our pairings of canonical basis vectors can be expressed in terms of Gramian determinants of matrices of binomial coefficients. Our construction can be carried out over the ring of integers, hence it makes sense over any field.