Noncommutative Symplectic Geometry Research Articles

A double Poisson algebra structure on Fukaya categories

Let $M$ be an exact symplectic manifold with $c_1(M)=0$. Denote by $\mathrm{Fuk}(M)$ the Fukaya category of $M$. We show that the dual space of the bar construction of $\mathrm{Fuk}(M)$ has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology $\mathrm{HC}^\bullet(\mathrm{Fuk}(M))$, which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology.

On a family of quivers related to the Gibbons–Hermsen system

We introduce a family of quivers $Z_{r}$ (labeled by a natural number $r\geq 1$) and study the non-commutative symplectic geometry of the corresponding doubles $\mathbf{Q}_{r}$. We show that the group of non-commutative symplectomorphisms of the path algebra $\mathbb{C}\mathbf{Q}_{r}$ contains two copies of the group $\mathrm{GL}_{r}$ over a ring of polynomials in one indeterminate, and that a particular subgroup $\mathcal{P}_{r}$ (which contains both of these copies) acts on the completion $\mathcal{C}_{n,r}$ of the phase space of the $n$-particles, rank $r$ Gibbons-Hermsen integrable system and connects each pair of points belonging to a certain dense open subset of $\mathcal{C}_{n,r}$. This generalizes some known results for the cases $r=1$ and $r=2$.

Lie bialgebra structure on cyclic cohomology of Fukaya categories

Let M be an exact symplectic manifold with contact type boundary such that c1(M) = 0. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure.

Modular Operads and Batalin-Vilkovisky Geometry

This is a copy of the article published in IMRN (2007). I describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic geometry of the corresponding tree-level cyclic operad. I show, in particular, that the algebras over the Feynman transform of a twisted modular operad P are in one-to-one correspondence with solutions to quantum master equation of Batalin-Vilkovisky geometry on the affine P-manifolds. As an application I give a construction of characteristic classes with values in the homology of the quotient of Deligne-Mumford moduli spaces. These classes are associated naturally with solutions to the quantum master equation on affine S[t]-manifolds, where S[t] is the twisted modular Det-operad constructed from symmetric groups, which generalizes the cyclic operad of associative algebras.

Noncommutative geometry and compactifications of the moduli space of curves

In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Witten’s conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.

Noncommutative geometry and quiver algebras

We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T ∗ B , which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, Π = Π ( B ) , that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space Π / [ Π , Π ] with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where Π is the preprojective algebra associated with a quiver of non-Dynkin type, we give a complete description of the Gerstenhaber algebra structure on the Hochschild cohomology of Π in terms of the Lie algebra Π / [ Π , Π ] .

Non‐commutative moduli spaces of topological D‐branes

AbstractWe give a general construction of extended moduli spaces of topological D‐branes as non‐commutative algebraic varieties. This shows that noncommutative symplectic geometry in the sense of Kontsevich arises naturally in String Theory.

Necklace Lie algebras and noncommutative symplectic geometry

Recently, V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry, [12]. In this note we generalize his argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg [13]. Using results of W. Crawley-Boevey and M. Holland [10], [8] and [9] we give a combinatorial description of all the relevant couples \((\alpha,\lambda)\) which are coadjoint orbits. We give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness.

Quantum mechanics as a matrix symplectic geometry

The main purpose of this work is to describe the quantum analog of the usual classical symplectic geometry and then to formulate quantum mechanics as a noncommutative symplectic geometry. First, we describe a discrete Weyl-Schwinger realization of the Heisenberg group and we develop a discrete version of the Weyl-Wigner-Moyal formalism. We also study the continuous limit and the case of higher degrees of freedom. In analogy with the classical case, we present the noncommutative (quantum) symplectic geometry associated with the matrix algebraMN(C) generated by the Schwinger matrices.

Quantum mechanics as noncommutative symplectic geometry

The authors construct consistent differential calculi on the algebra generated by operators satisfying the canonical commutation relation. This leads to a mathematical framework in which quantum mechanics can be understood as (noncommutative) symplectic geometry. They point out the possibility to describe interactions 'geometrically' as deformations of the differential calculus.

Non-commutative Symplectic Geometry, Quiver varieties,$\,$ and$\,$ Operads

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich's formalism of `non-commutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of P-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.