Basics Polyakov’s Quantum Surface Theory on the Formalism of Functional Integrals and Applications

We propose an alternative view to the covariant Polyakov’s string path integral. In our new approach we clarified the role of the Liouville model on string theory for Q.C.D. and Dual Models. In the appendices, we present additional material, difficult to find in the specialized path integral literature on the detailed evaluation of the (Q.C.D.) fermion determinant, the path integral proof of the Atiyah-Singer index theorem on Riemannian Manifolds, and the role of the expansion on Polyakov’s String path integral, all containing important new results, insights on theses topics.