BRANES AND QUANTIZATION FOR AN A-MODEL COMPLEXIFICATION OF EINSTEIN GRAVITY IN ALMOST KAHLER VARIABLES

The general relativity theory is redefined equivalently in almost Kahler variables: symplectic form, theta[g], and canonical symplectic connection, (D) over cap [g] (distorted from the Levi-Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metric g on a manifold V enabled with a necessary type nonholonomic 2 + 2 distribution. Such nonholonomic symplectic variables allow us to formulate the problem of quantizing Einstein gravity in terms of the A-model complexification of almost complex structures on V, generalizing the Gukov-Witten method [1]. Quantizing (V, theta[g], (D) over cap [g]), we derive a Hilbert space as a space of strings with two A-branes which for the Einstein gravity theory are nonholonomic because of induced nonlinear connection structures. Finally, we speculate on relation of such a method of quantization to curve flows and solitonic hierarchies defined by Einstein metrics on (pseudo) Riemannian spacetimes.