Strings and Membranes from Einstein Gravity, Matrix Models and W∞ Gauge Theories as paths to Quantum Gravity

It is shown how w∞, w1+∞ Gauge Field Theory actions in 2D emerge directly from 4D Gravity. Strings and Membranes actions in 2D and 3D originate as well from 4D Einstein Gravity after recurring to the nonlinear connection formalism of Lagrange-Finsler and Hamilton-Cartan spaces. Quantum Gravity in 3D can be described by a W∞ Matrix Model in D = 1 that can be solved exactly via the Collective Field Theory method. We describe why a quantization of 4D Gravity could be attained via a 2D Quantum W∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that non-critical W∞ (super) strings are devoid of BRST anomalies in dimensions D = 27 (D = 11), respectively, follows and which coincide with the critical (super) membrane dimensions D = 27 (D = 11). We establish the correspondence between the states associated with the quasi finite highest weights irreducible representations of W∞, W̄∞ algebras and the quantum states of the continuous Toda molecule. Schroedinger-like QM wave functional equations are derived and solutions are found in the zeroth order approximation. Since higher-conformal spin W∞ symmetries are very relevant in the study of 2D W∞ Gravity, the Quantum Hall effect, large N QCD, strings, membranes, ...... it is warranted to explore further the interplay among all these theories.