Dirac structures on the space of connections

We shall give a twisted Dirac structure on the space of irreducible connections on a SU(n)-bundle over a three-manifold, and give a family of twisted Dirac structures on the space of irreducible connections on the trivial SU(n)-bundle over a four-manifold. The twist is described by the Cartan 3-form on the space of connections. It vanishes over the subspace of flat connections. So the spaces of flat connections are endowed with ( non-twisted ) Dirac structures. The Dirac structure on the space of flat connections over the three-manifold is obtained as the boundary restriction of a corresponding Dirac structure over the four-manifold. We discuss also the action of the group of gauge transformations over these Dirac structures.

Homology Groups and Categorical Diagonalization

We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finitely generated free modules on a commutative ring. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. We found that the fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping cone is homotopy equivalent to zero. On the other hand, the fixed object is regarded as a categorified eigenvalue of the chain complex in the context of the categorical diagonalization introduced by B.Elias and M. Hogancamp arXiv:1801.00191v1. It is found that (co)homology groups are constructed as the eigenvalue of a chain complex.

Categorical Perspective on Quantization of Poisson Algebra

We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized quantization categories including matrix regularization, strict deformation quantization, prequantization, and Poisson enveloping algebra, respectively. It is shown that the categories of strict deformation quantization, prequantization, and matrix regularization with some conditions are categorical equivalence. On the other hand, the categories of Poisson enveloping algebra is not equivalent to the other categories.

Differential operators and contravariant derivatives in Poisson geometry

We inquire into the relation between the curl operators, the Poisson coboundary operators and contravariant derivatives on Poisson manifolds to study the theory of differential operators in Poisson geometry. Given an oriented Poisson manifold, we describe locally those two differential operators in terms of Poisson connection whose torsion is vanishing. Moreover, we introduce the notion of the modular operator for an oriented Poisson manifold. For a symplectic manifold, we describe explicitly the modular operator in terms of the curvature 2-section of Poisson connection, analogously to the Weitzenb$ddot{rm o}$ck formula in Riemannian geometry.

Formal Deformation Quantization for Super Poisson Structures on Super Calabi-Yau Twistor Spaces

It is known that Wolf constructed a lot of examples of Super Calabi-Yau twistor spaces. We would like to introduce super Poisson structures on them via super twistor double fibrations. Moreover we define the structure of deformation quantization for such super Poisson manifolds.

Geometric quantization of Dirac manifolds

We define prequantization for Dirac manifolds to generalize known procedures for Poisson and (pre) symplectic manifolds by using characteristic distributions obtained from 2-cocycles associated to Dirac structures. Given a Dirac manifold $(M,D)$, we construct Poisson structure on the space of admissible functions on $(M,D)$ and a representation of the Poisson algebra to establish the prequantization condition of $(M,D)$ in terms of a Lie algebroid cohomology. Additional to this, we introduce a polarization for a Dirac manifold $M$ and discuss procedures for quantization in two cases where $M$ is compact and where $M$ is not compact.

Equivalence of the categories of modules over Lie algebroids

We study geometric representation theory of Lie algebroids. A new equivalence relation for integrable Lie algebroids is introduced and investigated. It is shown that two equivalent Lie algebroids have equivalent categories of infinitesimal actions of Lie algebroids. As an application, it is also shown that the Hamiltonian categories for gauge equivalent Dirac structures are equivalent as categories.

On correlations and mutual entropy in quantum composed systems

We study the correlations of classical and quantum systems from the information theoretical points of view. We analyze a simple measure of correlations based on entropy (such measure was already investigated as the degree of entanglement by Belavkin, Matsuoka and Ohya). Contrary to naive expectation, it is shown that separable state might possesses stronger correlation than an entangled state.

Notes on Geometric Morita equivalence of twisted Poisson manifolds

This paper is devoted to the study of Morita equivalence for twisted Poisson manifolds. We review some Morita invariants and prove that integrable twisted Poisson manifolds which are gauge equivalent are Morita equivalent. Moreover, we introduce the notion of weak Morita equivalence and show that if two twisted Poisson manifolds are weak Morita equivalent, there exists a one-to-one correspondence between their twisted symplectic leaves.