Pietro Polesello
Deformation quantization algebroids over a complex symplectic manifold X are locally given by rings of WKB operators, that is, microdifferential operators with an extra central parameter τ. In this paper, we will show that such algebroids are classified by H^2(X;k^*), where k^* is a subgroup of the group of invertible formal Laurent series in τ^-1.
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Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry
本文在同伦意义下将广义复流形重新表述为全纯symplectic formal stack，发展shifted symplectic机制证明coisotropic intersection继承shifted Poisson structure，并研究了广义复brane。
Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric "higher structures", such as Courant algebroids twisted by $Ω^2$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well known features …
In this paper, we develop a differential-graded symplectic (Batalin-Vilkovisky) version of the framework of Crawley-Boevey, Etingof and Ginzburg on noncommutative differential geometry based on double derivations to construct non-commutative analogues of the Courant algebroids introduced by Liu, Weinstein and Xu. Adapting geometric constructions of Ševera and Roytenberg for (commutative) graded symplectic supermanifolds, we express the BRST charge, given in our framework by a `homological double derivation', in terms of Van den Bergh's double Poisson algebras for graded bi-symplectic non-commu…
We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.
A log symplectic manifold is a Poisson manifold which is generically nondegenerate. We develop two methods for constructing the symplectic groupoids of log symplectic manifolds. The first is a blow-up construction, corresponding to the notion of an elementary modification of a Lie algebroid along a subalgebroid. The second is a gluing construction, whereby groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to classify all Hausdorff symplectic groupoids of log symplectic manifolds in a combinatorial fashion, in terms of a cer…
To a complex symplectic manifold X we associate a canonical quantization algebroid. Our construction is similar to that of Polesello-Schapira's deformation-quantization algebroid, but the deformation parameter is no longer central. If X is compact, the triangulated category of regular holonomic quantization modules is a linear complex Calabi-Yau category of dimension 1 + dim X.
We review recent results and ongoing investigations of the symplectic and Poisson geometry of derived moduli spaces, and describe applications to deformation quantization of such spaces.
Mirror Symmetry for Quiver Algebroid Stacks
本文提出箭袋栈概念及其 $A_\infty$ 范畴表示理论，利用全局中间代理粘合不相交 Lagrangian，并在穿孔椭圆曲线中构造出镜像 nc 局部射影平面。
On a complex symplectic manifold, we construct the stack of quantization-deformation modules, that is, (twisted) modules of microdifferential operators with an extra central parameter, a substitute to the lack of homogeneity. We also quantize involutive submanifolds of contact manifolds.
The sheaf of rings of WKB operators provides a deformation-quantization of the cotangent bundle to a complex manifold. On a complex symplectic manifold $X$ there may not exist a sheaf of rings locally isomorphic to a ring of WKB operators. The idea is then to consider the whole family of locally defined sheaves of WKB operators as the deformation-quantization of $X$. To state it precisely, one needs the notion of algebroid stack, introduced by Kontsevich. In particular, the stack of WKB modules over $X$ defined in Polesello-Schapira (see also Kashiwara for the contact case) is better understoo…
On the Bezrukavnikov-Kaledin quantization of symplectic varieties in characteristic $p$
本文研究特征p下辛流形的Bezrukavnikov-Kaledin量子化，并证明其与微分运算代数的中心约简之间存在Morita等价关系。
We study sheaves on holomorphic spaces of loops and apply this to the study of the complex, defined in \cite{BdSHK}, governing deformations of the \emph{Poisson vertex algebra} structure on the space of holomorphic loops into a Poisson variety. We describe this complex in terms of the (continuous) de Rham-Lie cohomology of an associated Lie algebroid object in locally linearly compact topological (alias \emph{Tate}) sheaves of modules on $\mathcal{L}^{+}M$. In particular this allows us to easily compute the cohomology of the above in the case where $π$ is symplectic - we obtain de Rham cohomol…
We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing …
We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.
Consider a complex symplectic manifold $X$ and the algebroid $W_X$ of quantization-deformation. For two regular holonomic modules $L_i$ ($i=0,1$) supported by smooth Lagrangian manifolds, we prove that the complex $Rhom_{W_X}(L_1,L_0)$ is constructible and perverse and dual to the complex $Rhom_{W_X}(L_0,L_1)$.
Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X+1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X.
We introduce geometric quantization in the setting of shifted symplectic structures. We define Lagrangian fibrations and prequantizations of shifted symplectic stacks and their geometric quantization. In addition, we study many examples including symplectic groupoids, Hamiltonian spaces and moduli spaces of flat connections. In the case of Hamiltonian spaces we prove a derived analog of the "quantization commutes with reduction" principle.
We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams.
We study modules over the algebroid stack $\W[\stx]$ of deformation quantization on a complex symplectic manifold $\stx$ and recall some results: construction of an algebra for $\star$-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regular holonomic $\W[\stx]$-modules, finiteness and duality for the composition of ``good'' kernels. As a corollary, we get that the derived category of good $\W[\stx]$-modules with compact support is a Calabi-Yau category. We also give a conjectural Riemann-Roch type formula in this …
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