[Paper Review] Deformation Theory of Courant Algebroids via the Rothstein Algebra

Frank Keller, Stefan Waldmann|arXiv (Cornell University)|Jul 3, 2008

Advanced Topics in Algebra15 references4 citations

TL;DR

This paper develops an algebraic deformation theory for Courant algebroids using two equivalent graded Poisson algebras of degree $-2$: the complex $\mathcal{C}^\bullet(\mathcal{E})$ and the Rothstein algebra $\mathcal{R}^\bullet(\mathcal{E})$. It establishes a Fedosov-type deformation quantization on the Rothstein algebra, showing that Courant structures can be deformed into elements of square zero under the star product, with obstructions lying in the same cohomology as in the classical theory. The key contribution is a quantization framework compatible with algebraic and singular geometries.

ABSTRACT

In this paper we define Courant algebroids in a purely algebraic way and study their deformation theory by using two different but equivalent graded Poisson algebras of degree -2. First steps towards a quantization of Courant algebroids are proposed by employing a Fedosov like deformation quantization.

Motivation & Objective

• To formulate Courant algebroids in a purely algebraic framework, independent of smooth manifolds, to allow for deformation theory in singular or orbifold contexts.
• To develop a deformation theory for Courant algebroids analogous to Crainic and Moerdijk's Lie algebroid deformation theory, using graded Poisson algebras of degree $-2$.
• To propose a quantization procedure via a Fedosov-like construction on the Rothstein algebra, linking classical deformation theory to quantum structures.
• To demonstrate that the deformation obstructions and infinitesimals for quantization coincide with those of the classical deformation complex, validating the algebraic framework.

Proposed method

• Define Courant algebroids algebraically over a commutative unital algebra $\mathcal{A}$ and a finitely generated projective $\mathcal{A}$-module $\mathcal{E}$ with a non-degenerate inner product.
• Construct two equivalent graded Poisson algebras of degree $-2$: $\mathcal{C}^\bullet(\mathcal{E})$ and $\mathcal{R}^\bullet(\mathcal{E})$, where elements of degree 3 satisfying $[m,m]=0$ correspond to Courant structures.
• Introduce a symbol calculus via a Poisson monomorphism from $\mathcal{C}^\bullet(\mathcal{E})$ to $\mathcal{R}^\bullet(\mathcal{E})$, which becomes an isomorphism under smoothness assumptions.
• Implement a Fedosov-type star product $\star_\kappa$ on the Rothstein algebra $\mathcal{R}^\bullet(\mathcal{L} \oplus \mathcal{L}')$, preserving bigrading and constructed using a connection and curvature terms.
• Define quantization by requiring $\mu \star_\kappa \mu = 0$ for a structure element $\mu$, or more weakly, $\mu \star_\kappa \mu$ central, to allow for non-vanishing higher-order corrections.
• Apply the framework to Lie-Rinehart pairs, showing that their canonical Courant structures admit deformation quantization under the proposed scheme.

Experimental results

Research questions

• RQ1 Can Courant algebroids be defined and their deformation theory formulated in a purely algebraic setting without relying on smooth manifolds?
• RQ2 How do the two graded Poisson algebras $\mathcal{C}^\bullet(\mathcal{E})$ and $\mathcal{R}^\bullet(\mathcal{E})$ relate, and can one replace the more complex $\mathcal{C}^\bullet(\mathcal{E})$ with the simpler $\mathcal{R}^\bullet(\mathcal{E})$ via symbol calculus?
• RQ3 Can a Fedosov-type deformation quantization be constructed on the Rothstein algebra to quantize Courant algebroid structures?
• RQ4 What are the obstructions to deforming a classical Courant structure into a quantum one, and do they match the classical deformation cohomology?
• RQ5 Does the proposed quantization framework recover known structures, such as the standard Courant bracket on $TM \oplus T^*M$, in the smooth case?

Key findings

• The deformation theory of Courant algebroids is fully algebraic and formulated via two equivalent graded Poisson algebras of degree $-2$, with Courant structures corresponding to degree-3 elements $m$ satisfying $[m,m]=0$.
• The symbol calculus establishes a Poisson monomorphism from $\mathcal{C}^\bullet(\mathcal{E})$ to $\mathcal{R}^\bullet(\mathcal{E})$, which is an isomorphism in the smooth case, allowing the use of the simpler Rothstein algebra for deformation problems.
• A Fedosov-type star product $\star_\kappa$ is constructed on the Rothstein algebra $\mathcal{R}^\bullet(\mathcal{L} \oplus \mathcal{L}')$, preserving the bigrading and depending on a connection and curvature term.
• For a Lie-Rinehart structure $\mu \in \mathcal{R}^{1,2}(\mathcal{L} \oplus \mathcal{L}')$, the condition $\mu \star_{1/2} \mu = 0$ holds without higher-order corrections, showing that the star product can preserve the quantum condition at second order.
• The obstructions to deformation quantization lie in the same cohomology groups as in the classical deformation theory, confirming consistency between classical and quantum frameworks.
• The framework successfully quantizes Lie-Rinehart pairs as a large class of examples, with the Courant bracket on $TM \oplus T^*M$ recovered in the smooth case.