[Paper Review] Constraint Reduction in Algebra, Geometry and Deformation Theory

Marvin Dippell|arXiv (Cornell University)|Oct 9, 2023

Advanced Topics in Algebra4 citations

TL;DR

This paper introduces a unified framework for constraint reduction in algebra, geometry, and deformation theory, generalizing coisotropic reduction via constraint modules and algebras. It establishes a correspondence between geometric coisotropic submanifolds and algebraic Poisson normalizers, and develops a deformation quantization theory for constraint algebras using Hochschild cohomology, yielding a formal star product compatible with constraints.

ABSTRACT

To study coisotropic reduction in the context of deformation quantization we introduce constraint manifolds and constraint algebras as the basic objects encoding the additional information needed to define a reduction. General properties of various categories of constraint objects and their compatiblity with reduction are examined. A constraint Serre-Swan theorem, identifying constraint vector bundles with certain finitely generated projective constraint modules, as well as a constraint symbol calculus are proved. After developing the general deformation theory of constraint algebras, including constraint Hochschild cohomology and constraint differential graded Lie algebras, the second constraint Hochschild cohomology for the constraint algebra of functions on a constraint flat space is computed.

Motivation & Objective

• To develop a systematic algebraic and geometric theory of constraint reduction generalizing coisotropic reduction in Poisson geometry.
• To introduce constraint k-modules and constraint algebras as algebraic structures encoding constraints in deformation theory.
• To establish a categorical and cohomological framework for deformation quantization of constrained systems.
• To generalize the coisotropic reduction theorem to algebraic and formal deformation settings using constraint modules.
• To provide an algebraic characterization of the characteristic distribution and Poisson normalizer via constraint structures.

Proposed method

• Introduces constraint sets and constraint k-modules as foundational algebraic structures, generalizing ideals and modules under constraints.
• Defines constraint algebras and strong constraint algebras, with a focus on compatibility with Poisson and differential geometric structures.
• Constructs constraint vector bundles and sections, extending classical differential geometry to constrained settings.
• Develops constraint Cartan calculus, including differential forms, multivector fields, and symbol calculus for differential operators.
• Applies constraint structures to deformation theory via the constraint Hochschild cohomology complex.
• Establishes a formal deformation quantization framework using star products compatible with constraint algebras, linking to cohomological obstructions.

Experimental results

Research questions

• RQ1 How can coisotropic reduction in Poisson geometry be generalized algebraically using constraint modules?
• RQ2 What is the role of the Poisson normalizer in characterizing functions constant along characteristic leaves in constrained systems?
• RQ3 How can deformation quantization be extended to algebras equipped with constraints, and what cohomological obstructions arise?
• RQ4 What is the relationship between constraint vector bundles and the geometry of characteristic distributions?
• RQ5 Can a formal star product be constructed on a constraint algebra that respects the underlying geometric constraints?

Key findings

• The paper constructs a category of constraint k-modules that generalizes projective modules and provides a framework for free and projective resolutions under constraints.
• It proves that the Poisson normalizer of a coisotropic submanifold corresponds precisely to functions whose Hamiltonian vector fields are tangent to the submanifold, thus characterizing the characteristic distribution algebraically.
• A constraint deformation functor is defined, and its representability is linked to Hochschild cohomology, generalizing formal deformation theory to constrained settings.
• The second Hochschild cohomology of the constraint algebra on R^n is computed, showing that obstructions to deformation are controlled by the constraint structure.
• A generalized coisotropic reduction theorem is established: the quotient algebra BC/IC is isomorphic to C∞(Mred) as a Poisson algebra, extending classical results to the constraint setting.
• The paper provides a formal star product on constraint algebras that is compatible with the constraint ideal, generalizing deformation quantization to constrained systems.