Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric $R$-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for $\mathcal A_3$ and $\mathcal A_4$ in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for $\mathcal A_n$ via sequences of cluster mutations in the special $\mathbb A_n$-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.
We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the $\mathcal A_n$-groupoid of upper-triangular matrices to express roots of the characteristic equation $\det(\mathbb A-λ\mathbb A^{\text{T}})=0$, with $\mathbb A\in \mathcal A_n$, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra $sl_n$. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement…
We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique formal symplectic groupoid `with separation of variables' over an arbitrary Kaehler-Poisson manifold.
We give an explicit form of the symplectic groupoid that integrates the semiclassical standard Podles sphere. We show that Sheu's groupoid, whose convolution C*-algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of the symplectic groupoid. By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.
We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Poisson manifold. We study various families of examples. In particular, a global symplectic groupoid for a general class of two-dimensional Poisson domains is constructed.
Using Fedosov's approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kaehler-Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.
We introduce a new kind of groupoid--a pseudo étale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the idea that symplectic and Poisson geometries are the semiclassical limits of the corresponding quantum geometries, we quantize these noncommutative Poisson manifolds in the framework of deformation quantization.
To each natural star product on a Poisson manifold $M$ we associate an antisymplectic involutive automorphism of the formal neighborhood of the zero section of the cotangent bundle of $M$. If $M$ is symplectic, this mapping is shown to be the inverse mapping of the formal symplectic groupoid of the star product. The construction of the inverse mapping involves modular automorphisms of the star product.
Given a formal symplectic groupoid $G$ over a Poisson manifold $(M, π_0)$, we define a new object, an infinitesimal deformation of $G$, which can be thought of as a formal symplectic groupoid over the manifold $M$ equipped with an infinitesimal deformation $π_0 + επ_1$ of the Poisson bivector field $π_0$. The source and target mappings of a deformation of $G$ are deformations of the source and target mappings of $G$. To any pair of natural star products $(\ast, \tilde\ast)$ having the same formal symplectic groupoid $G$ we relate an infinitesimal deformation of $G$. We call it the deformation …
A set of all linear transformations with a fixed Jordan structure $J$ is a symplectic manifold isomorphic to the coadjoint orbit $\mathcal O (J)$ of $GL(N,C)$. Any linear transformation may be projected along its eigenspace to (at least one) coordinate subspace of the complement dimension. The Jordan structure $\tilde J$ of the image is defined by the Jordan structure $J$ of the pre-image, consequently the projection $\mathcal O (J)\to \mathcal O (\tilde J)$ is the mapping of the symplectic manifolds. It is proved that the fiber $\mathcal E$ of the projection is a linear symplectic space and t…
In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.
As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results is also addressed. In particular, the paper includes an extension to space-times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a "differential" version of the classical and quantum master equations. The quantization…
Let $G$ be a connected complex semi-simple Lie group and ${\mathcal{B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${\mathcal{B}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of P…
This paper is about the role of Planck's constant, $\hbar$, in the geometric quantization of Poisson manifolds using symplectic groupoids. In order to construct a strict deformation quantization of a given Poisson manifold, one can use all possible rescalings of the Poisson structure, which can be combined into a single "Heisenberg-Poisson" manifold. The new coordinate on this manifold is identified with $\hbar$. I present an explicit construction for a symplectic groupoid integrating a Heisenberg-Poisson manifold and discuss its geometric quantization. I show that in cases where $\hbar$ canno…
Poisson manifolds may be regarded as the infinitesimal form of symplectic groupoids. Twisted Poisson manifolds considered by Severa and Weinstein [math.SG/0107133] are a natural generalization of the former which also arises in string theory. In this note it is proved that twisted Poisson manifolds are in bijection with a (possibly singular) twisted version of symplectic groupoids.
We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is "multiplicatively integrable", i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying t…
On general symplectic manifolds we describe a correspondence between symplectic transformations and their phase functions. On the quantum level, this is a correspondence between unitary operators and phase functions of the WKB-approximation. We represent generic functions via symplectic area of membranes and consider related geometric properties of the noncommutative phase product. An interpretation of the phase product in terms of symplectic groupoids and the groupoid extension of Lagrangian submanifolds are described. The membrane representations of corresponding Lagrangian phase functions a…
The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the `exchange r-matrix' that governs the corresponding Poisson brackets is established. Generalizing earlier results related to diagonal monodromy, the exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter equation, which is found to admit an interpretation in terms of (new) Poisson-Lie groupoids. Dynamical exchange r-matrices for which right multiplication yields a classical or a Poisson-Lie symmetry on the chiral WZ…
Dirac submanifolds are a natural generalization in the Poisson category for symplectic submanifolds of a symplectic manifold. In a certain sense they correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable locus of a Poisson involution. In this paper, we provide a general study for these submanifolds including both local and global aspects. In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable locuses. We discuss the Poisson involutions on a s…
A chain of one-dimensional Schrödinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $α$. If $α\not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlevé equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. O…
Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli space D(G,S) is the symplectic double of the Poisson moduli space of framed G-local systems on S. Its symplectic form is upgraded to a K2-symplectic structure for which the special coordinates are K2-Darboux coordinates.
