Functorial Hodge identities and quantization

By a uniform abstract procedure, we obtain integrated forms of the classical Hodge identities for Riemannian, Kähler and hyper-Kähler manifolds, as well as of the analogous identities for metrics of arbitrary signature. These identities depend only on the type of geometry and, for each of the three types of geometry, define a multiplicative functor from the corresponding category of real, graded, flat vector bundles to the category of infinite-dimensional $\mathbf{Z}_{2}$-projective representations of an algebraic structure. We define new multiplicative numerical invariants of closed Kähler and hyper-Kähler manifolds which are invariant under deformations of the metric.