[Paper Review] Working with Nonassociative Geometry and Field Theory

Gwendolyn E. Barnes, Alexander Schenkel+1 more|arXiv (Cornell University)|Jan 27, 2016

Black Holes and Theoretical Physics24 references6 citations

TL;DR

This paper develops a systematic formalism for differential geometry on noncommutative and nonassociative spaces using cochain twist deformation quantization, enabling explicit constructions of Yang-Mills and Einstein-Cartan gravity actions. It establishes real-valued, physically viable field theories on such spaces by defining nonassociative star-products, connections, curvatures, and antisymmetrized wedge products with consistent reality conditions and 3-cyclicity.

ABSTRACT

We review aspects of our formalism for differential geometry on noncommutative and nonassociative spaces which arise from cochain twist deformation quantization of manifolds. We work in the simplest setting of trivial vector bundles and flush out the details of our approach providing explicit expressions for all bimodule operations, and for connections and curvature. As applications, we describe the constructions of physically viable action functionals for Yang-Mills theory and Einstein-Cartan gravity on noncommutative and nonassociative spaces, as first steps towards more elaborate models relevant to non-geometric flux deformations of geometry in closed string theory.

Motivation & Objective

• To provide a concrete, explicit framework for differential geometry on noncommutative and nonassociative spaces arising from cochain twist deformation quantization.
• To address the lack of explicit constructions for bimodule operations, connections, and curvatures in nonassociative geometry for physically relevant field theories.
• To construct real-valued, physically consistent action functionals for Yang-Mills theory and Einstein-Cartan gravity on nonassociative spacetimes.
• To demonstrate that reality conditions and 3-cyclicity ensure the reality of the nonassociative Einstein-Cartan action in both even and odd dimensions.
• To generalize prior abstract categorical constructions to a more accessible, local, and computable form for applications in string theory and quantum gravity.

Proposed method

• Utilizes cochain twist deformation quantization of classical manifolds to generate noncommutative and nonassociative algebras via a twist element $ F $, with the star-product $ abla_{ ext{twist}} $ defined via the twist operator.
• Constructs explicit bimodule operations on trivial vector bundles with diagonal Hopf algebra action, enabling local descriptions of nonassociative geometry.
• Defines connections and curvatures on nonassociative vector bundles using the twisted differential calculus, with curvature $ R^{ab} $ derived from the nonassociative star-product.
• Introduces two bracketing conventions for the nonassociative wedge product $ igwedge_{ ext{twist}} $: $ E_{\text{left}}^{a_1\cdots a_k} $ and $ E_{\text{right}}^{a_1\cdots a_k} $, ensuring proper antisymmetrization and reality.
• Constructs the Yang-Mills action via the nonassociative field strength $ F^{ab} $, and the Einstein-Cartan action via the curvature $ R^{ab} $, both using the twisted wedge product and Hodge duals.
• Imposes reality conditions on the spin connection $ \omega^{ab} $ and vielbein $ E^a $, and assumes 3-cyclicity of the star-product to ensure the action functionals are real-valued.

Experimental results

Research questions

• RQ1 How can differential geometry be systematically formulated on nonassociative spaces arising from cochain twist deformation quantization?
• RQ2 What explicit expressions can be derived for bimodule operations, connections, and curvatures in nonassociative geometry on trivial vector bundles?
• RQ3 How can physically viable action functionals for Yang-Mills theory and Einstein-Cartan gravity be constructed on nonassociative spacetimes?
• RQ4 Under what conditions is the nonassociative Einstein-Cartan action real-valued in even and odd dimensions?
• RQ5 What role does 3-cyclicity of the star-product play in ensuring the consistency and reality of nonassociative field theory actions?

Key findings

• The paper provides explicit expressions for bimodule operations on trivial vector bundles over nonassociative spaces, derived from cochain twist deformation quantization.
• Connections and their curvatures on nonassociative vector bundles are constructed using the twisted differential calculus, with curvature defined via the nonassociative star-product.
• The Yang-Mills action is constructed using the nonassociative field strength $ F^{ab} $, with the action functional expressed in terms of the twisted wedge product and Hodge dual.
• The Einstein-Cartan action is formulated in both even and odd dimensions using two distinct bracketing conventions for the vielbein $ E^{a_i} $, ensuring proper antisymmetrization and reality.
• The nonassociative Einstein-Cartan action is proven to be real-valued in both even and odd dimensions under the assumptions of Hermitean twist and 3-cyclicity of the star-product.
• The 3-cyclicity property, which holds for Abelian twists and is shown in [25] for non-Abelian cases, allows rebracketing of the action to confirm its reality, as demonstrated in equations (4.18)–(4.20) and (4.22).