## Strings and Membranes from Einstein Gravity, Matrix Models and W_{∞} Gauge Theories as paths to Quantum Gravity

**Authors:** Carlos Castro

It is shown how w_{∞},w_{1+∞} 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 aW1 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_{∞},W_{∞} 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.

**Comments:** 49 pages, This article appeared in the Int. J. Mod. Phys. A 23, no. 24 (2008) 3891

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### Submission history

[v1] 27 Aug 2009

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