Journal
INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
Volume 6, Issue 6, Pages 873-909Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0219887809003849
Keywords
Quantum gravity; Einstein gravity; nonholonomic manifolds; symplectic variables; nonlinear connections; strings and A-branes
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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.
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