[Paper Review] Dunkl operator and quantization of $\mathbb{Z}_2$-singularity

Gilles Halbout, Xiang Tang|arXiv (Cornell University)|Aug 28, 2009

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper constructs a universal deformation quantization of $ Z_2$-symplectic orbifolds by generalizing the Moyal star-product using Dunkl operators, which incorporate $ Z_2$-symmetry. It confirms a key part of the Dolgushev-Etingof conjecture by showing that deformations exist and are parametrized by codimension-2 components of the inertia orbifold, extending symplectic reflection algebras globally via Fedosov's method.

ABSTRACT

Let $(X,ω)$ be a symplectic orbifold which is locally like the quotient of a $\mathbb{Z}_2$ action on $ eals^n$. Let $A^{((\hbar))}_X$ be a deformation quantization of $X$ constructed via the standard Fedosov method with characteristic class being $ω$. In this paper, we construct a universal deformation of the algebra $A^{((\hbar))}_X$ parametrized by codimension 2 components of the associated inertia orbifold $\widetilde{X}$. This partially confirms a conjecture of Dolgushev and Etingof in the case of $\mathbb{Z}_2$ orbifolds. To do so, we generalize the interpretation of Moyal star-product as a composition of symbol of pseudodifferential operators in the case where partial derivatives are replaced with Dunkl operators. The star-products we obtain can be seen as globalizations of symplectic reflection algebras.

Motivation & Objective

• To construct a universal deformation of the star-algebra $A^{(( H))}_{X}$ on a $ Z_2$-symplectic orbifold $X = M/ Z_2$.
• To generalize the Moyal star-product by replacing partial derivatives with Dunkl operators to account for $ Z_2$-symmetry.
• To confirm a conjecture by Dolgushev and Etingof that deformations of the star-algebra are unobstructed and parametrized by codimension-2 components of the inertia orbifold $ X$.
• To provide a global realization of symplectic reflection algebras via a globalization of Dunkl-type pseudodifferential operators.

Proposed method

• Generalize the Moyal product using composition of symbols of difference-pseudodifferential operators with Dunkl operators replacing standard partial derivatives.
• Define two families of $ Z_2$-local bilinear operators satisfying associativity and $ Z_2$-equivariance, leading to a $ Z_2$-local star product.
• Use Fedosov's standard method on the complement of a tubular neighborhood of the codimension-2 fixed-point submanifold, where the star product is locally Moyal-like.
• Apply a modified Fedosov construction near the fixed-point submanifold of codimension 2, using the generalized Moyal product.
• Ensure compatibility across the overlap region by exploiting the $ Z_2$-locality of both the standard and generalized star products.
• Use generating functions and Taylor expansions to compute coefficients in the star product, identifying combinatorial structures via $ Z_2$-symmetric generating functions $1/(1-x)^{s+1}$ and $1/(1+x)^{s+1}$.

Experimental results

Research questions

• RQ1 Can a universal deformation of the star-algebra $A^{(( H))}_{X}$ on a $ Z_2$-orbifold be constructed, parametrized by codimension-2 components of the inertia orbifold?
• RQ2 How can the Moyal star-product be generalized to incorporate $ Z_2$-symmetry via Dunkl operators?
• RQ3 Does the generalized star-product yield a consistent, associative deformation quantization compatible with the $ Z_2$-action?
• RQ4 Can this construction be globally extended to compact symplectic orbifolds using Fedosov’s method?
• RQ5 Is the resulting deformation quantization isomorphic to a formal version of symplectic reflection algebras in the case $M = R^{2n}$?

Key findings

• A universal deformation of the star-algebra $A^{(( H))}_{X}$ is constructed, parametrized by $H^0(M^ G_2/ Z_2)(( H))$, confirming a key part of the Dolgushev-Etingof conjecture for $ Z_2$-orbifolds.
• The generalized star-product is associative and $ Z_2$-local, with associativity governed by combinatorial coefficients derived from generating functions $1/(1-x)^{ L_0+n_0+1}(1+x)^{ L_1+n_1}$.
• The coefficients $c_ u$ in the star-product expansion are identified as the $t^0$-coefficient of a product of Laurent series involving $a(x,1/t)$ and $t^{m+n}/(1 - t p_1)^{ L_0 + n_0 + 1}(1 + t p_1)^{ L_1 + n_1}$.
• The operator-symbol product formula is shown to be equivalent to the action of iterated divided difference operators $ D^{m+n}$, with evaluation on symmetric tuples of points reflecting $ Z_2$-symmetry.
• In the case $M = R^{2n}$, the resulting star-product is a formal version of symplectic reflection algebras, providing a global realization of these algebras.
• The construction achieves compatibility between the standard Fedosov quantization and the generalized Moyal product on the overlap of open sets, ensuring a well-defined global deformation.