[Paper Review] Branes and Quantization for an A-Model Complexification of Einstein Gravity in Almost Kahler Variables

Sergiu I. Vacaru|ArXiv.org|Oct 26, 2008

Advanced Differential Geometry Research36 references3 citations

TL;DR

This paper proposes a new approach to quantizing Einstein gravity by reformulating general relativity in almost Kähler variables—using a symplectic form and a canonical symplectic connection derived from a (pseudo)Riemannian metric on a nonholonomic manifold with a 2+2 distribution. It applies the Gukov–Witten A-model quantization framework to construct a Hilbert space of nonholonomic A-brane strings, linking quantum gravity to solitonic hierarchies and bi-Hamiltonian structures via N-adapted geometry.

ABSTRACT

The general relativity theory is redefined equivalently in almost Kahler variables: symplectic form and canonical symplectic connection (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 on a manifold 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 spacetime manifold, generalizing the Gukov-Witten method, see arXiv:0809.0305. Quantizing the complexified model, 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.

Motivation & Objective

• To reformulate general relativity in terms of almost Kähler variables—symplectic form and canonical symplectic connection—on nonholonomic manifolds with a 2+2 nonintegrable distribution.
• To apply the Gukov–Witten A-model quantization method to Einstein gravity, enabling a geometric quantization framework using branes.
• To construct a Hilbert space of strings as a space of nonholonomic A-branes, preserving the nonlinear connection structure induced by the metric.
• To relate the resulting quantum gravity model to curve flows, solitonic hierarchies, and bi-Hamiltonian structures defined by Einstein metrics.
• To generalize Fedosov-type quantization to gravity by using N-adapted connections and symplectic geometry on nonholonomic spacetimes.

Proposed method

• The paper uses a nonholonomic 2+2 distribution on a manifold V to define an almost Kähler structure via a symplectic form θ[g] and a canonical symplectic connection D̂[g], which is a distortion of the Levi-Civita connection by a tensor built from metric coefficients and their derivatives.
• It applies the A-model topological string theory framework to the almost Kähler geometry of Einstein gravity, treating the spacetime as a complexified manifold with A-branes coupled to the symplectic structure.
• The Hilbert space of quantum states is constructed as the space of strings stretched between two nonholonomic A-branes, specifically (gBcc, gB′), preserving the nonlinear connection structure.
• The method employs N-adapted differential geometry, including N-connection coefficients, adapted connections, and curvature forms, to maintain compatibility with the nonholonomic distribution and symplectic structure.
• It derives a Hermitian inner product on the Hilbert space Hg of nonholonomic A-model strings, ensuring unitarity and topological consistency.
• The approach generalizes Fedosov quantization to gravity by using canonical d-connections and curvature forms adapted to the N-structure, with explicit formulas for torsion, curvature, and Ricci tensors in N-adapted components.

Experimental results

Research questions

• RQ1 How can general relativity be equivalently reformulated in terms of almost Kähler variables, specifically a symplectic form and a canonical symplectic connection, on a nonholonomic manifold with a 2+2 distribution?
• RQ2 Can the Gukov–Witten A-model quantization procedure be consistently applied to Einstein gravity by using nonholonomic A-branes and symplectic geometry?
• RQ3 What is the structure of the Hilbert space of quantum states in this framework, and how is it realized as a space of strings between two nonholonomic A-branes?
• RQ4 How are curve flows and solitonic hierarchies related to the quantum gravity model constructed via nonholonomic A-branes and bi-Hamiltonian structures?
• RQ5 What are the N-adapted geometric and topological constraints that ensure unitarity and consistency of the quantum theory in this almost Kähler formulation?

Key findings

• The paper constructs a canonical almost symplectic d-connection D̂ that preserves the h-v splitting of the nonholonomic manifold and satisfies vanishing h-v and v-v torsion components, ensuring compatibility with symplectic geometry.
• The symplectic form θ[g] and the connection D̂[g] are uniquely determined by the (pseudo)Riemannian metric g, with the connection being a distortion of the Levi-Civita connection by a tensor derived from metric and its derivatives.
• A Hilbert space Hg is realized as the space of strings stretched between two nonholonomic A-branes (gBcc, gB′), with a Hermitian inner product defined on it, ensuring unitarity and topological consistency.
• The curvature 2-form of the canonical d-connection D̂ is computed in N-adapted components, with nontrivial components R̂ijk, P̂jka, and Ŝbcd, which encode the gravitational dynamics in the almost Kähler framework.
• The scalar curvature R̂ of the canonical d-connection is expressed as R̂ = g^βγ R̂_βγ, showing consistency with the standard Ricci scalar in the nonholonomic setting.
• The model links quantum gravity to solitonic hierarchies and bi-Hamiltonian structures through N-adapted symmetries and curve flow equations, generalizing previous results in Fedosov quantization of gravity.