On deformation quantizations of hypertoric varieties
Let \(X\) be complex analytic manifold with \(\mathcal O_X\) the sheaf of regular, holomorphic functions. Denote by \(\mathcal D_X\) the sheaf of differential operators, by \(\mathbf{k}=\mathbb C((\hbar))\) the field of formal Laurent series in \(\hbar\), and by \(\mathbf{k}(0)\) the subring \(\mathbb C\hbar\) of formal functions on \(\mathbb C\). Given \(m\in\mathbb Z\), define \(\mathcal W_{T^\ast\mathbb C^n}(m)\) to be the sheaf of formal power series \(\sum_{i\geq -m}\hbar^ia_i\in\mathbb O_{T^\ast\mathbb C^n}\) on the cotangent bundle \(T^\ast\mathbb C^n\) on \(\mathbb C^n\). Fix coordinates \(x_1,\dots,x_n\) on \(\mathbb C^n\) and dual coordinates \(\xi_1,\dots,\xi_n\) on \((\mathbb C^n)^\ast\). Set \(\mathcal W_{T^\ast\mathbb C^n}=\cup_{m\in\mathbb Z}W_{T^\ast\mathbb C^n}(m)\). Then \(W_{T^\ast\mathbb C^n}\) is given a structure of a sheaf of (noncommutative) \(\mathbf{k}\)-algebras on \(T^\ast\mathbb C^n\), and \(\mathcal O_{T^\ast\mathbb C^n}\) is a sheaf of Poisson algebras with a particular given Poisson bracket. Assume that \(X\) is a symplectic manifold with holomorphic 2-form \(\omega_X\). A \(W\)-algebra on \(X\) is a sheaf of \(\mathbf{k}\)-algebras \(\mathcal W_X\) together with a \(\mathbf{k}(0)\)-subalgebra \(\mathcal W_X(0)\) such that for each point \(x\in X\) there exists an open neighborhood \(U\) of \(x\in X\), a symplectic map \(f:U\rightarrow V\subset T^\ast\mathbb C^n\) and a \(\mathbf{k}\)-algebra isomorphism \(r:f^{-1}(\mathcal W_{T^\ast\mathbb C^n}|_V)\overset\sim\rightarrow\mathcal W|_U\) such that these work well together. Kontsevich and Polesello and Schapira have shown that one can construct a stack of \(W\)-algebroids on any symplectic manifold. This provides a quantization of the sheaf of holomorphic functions on the manifold. In certain cases, these stacks of \(W\)-algebroids are the algebroids associated to a sheaf of \(W\)-algebras. When the manifold is the Hamiltonian reduction of a space equipped with a sheaf of \(W\)-algebras, Kashiwara and Rouguier have shown that one can define a family of sheaves of \(W\)-algebras. This provides a lot of examples of such sheaves on nontrivial symplectic manifolds. This article study \(W\)-algebras on the simplest class of Hamitonian reductions, those coming from the action of a torus \(T\) on a symplectic vector space \(V\). These spaces \(Y(A,\delta)\), \(A\) a matrix encoding the action of \(T\) on \(V\) and \(\delta\in\mathbb X(T)\) a character of \(T\), are called hypertoric varieties. The cotangent space of any smooth toric variety can be realized as a dense, open subvariety of the corresponding hypertoric variety. To the data of a reductive group \(G\) acting on a symplectic vector space \(V\), we can associate a family of noncommutative algebras \(U_\chi\), where \(\chi\in(\mathfrak g)\) is a character of \(\mathfrak g=\text{Lie}(G),\) called quantum Hamiltonian reductions. The main goal of this article is to prove a localization theorem, analogous to the Beilinson-Bernstein localization theorem, giving an equivalence between the category of finitely generated modules for the quantum Hamiltonian reduction and a certain category of modules for a \(W\)-algebra. The authors define the categories and the localization functors: \(\text{Loc}:U_\chi\text{-mod}\rightarrow\text{Mod}_F^{\text{good}}(\mathcal A_\chi)\), \(\text{Loc}(M)=\mathcal A_\chi\otimes_{U_\chi}M\), and \(\text{Sec}:\text{Mod}^{\text{good}}_F(\mathcal A_\chi)\rightarrow U_{\chi}\text{-mod}\), \(\text{Sec}(\mathcal M)=\text{Hom}_{\text{Mod}^{\text{good}}_F(\mathcal A_\chi)}(\mathcal A_\chi,\mathcal M)\). Then the first main theorem tells when localization gives an equivalence of categories, proving in particular that localization always give an equivalence provided one is sufficiently far away from the G.I.T. walls. Also, a particular class of examples of hypertoric varieties are the minimal resolutions \((\mathbb C^2/\mathbb Z_m)^\sim\) of the Kleinian singularities of type \(A\). Restricting the parameters, the corresponding quantum Hamiltonian reductions are Morita equivalent to the rational Cherednik algebras \(H_h\) associated to cyclic groups. Then a corollary of the main result is that for \(h\) not lying on a G.I.T. wall, the functor \(\text{Loc}(e\cdot(\cdot))\) defines an equivalence of categories \(H_h\text{-mod}\tilde\rightarrow\text{Mod}^{\text{good}}_F(\mathcal A_h)\). The article is well written, even if it is not self contained and thereby demands a deeper knowledge of the field. The authors make the necessary references and definitions, and the theory and results are very interesting.
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