The Riemann-Hilbert maps of certain meromorphic linear systems with Poncaré rank 1 are diffeomorphisms ν u , parametrized by the regular elements of a Cartan subalgebra u ∈ h reg (R) of u(n), from the space Herm(n) of n × n Hermitian matrices to the space Herm + (n) of n × n positive definite Hermitian matrices. In this paper, we propose an extension of the family of Riemann-Hilbert maps from h reg (R) to its de Concini-Procesi wonderful compactification M(R) via isomonodromy deformation. We then study the map ν rel corresponding to a caterpillar point on M(R), and prove that (up to a gauge transformation) it coincides with the Alekseev-Meinrenken diffeomorphism Γ AM from Herm(n) to Herm + (n), a map uniquely characterized by distinguished linear algebra properties. We also discuss the applications of our results in Poisson geometry and representation theory.

In this paper, we show that the quantum Stokes matrices, of certain meromorphic linear system of ordinary differential equations, give rise to a family of Drinfeld isomorphisms from quantum groups to the undeformed universal enveloping algebra of gl(n). In particular, we compute explicitly the Drinfeld isomorphisms corresponding to caterpillar points on the parameter space. Our computation unveils a relation between the asymptotics of confluent hypergeometric functions and the Gelfand-Zeitlin subalgebras.As by products, we show that the Drinfeld isomorphisms at caterpillar points coincide with the Appel-Gautam isomorphisms. Then by going to the semiclassical limit, we place the results in this paper into the context of Poisson geometry. As an application, we prove the conjecture of Appel and Gautam, i.e., their isomorphisms are canonical quantization of the Alekseev-Meinrenken diffeomorphisms arising from the linearization problem in Poisson geometry.

We study Maurer-Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson g-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term L∞-algebra is a homotopy Poisson manifold of degree n − 1, we obtain a Courant algebroid from a 2-term L∞-algebra g via the degree 2 symplectic NQ-manifold. By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie-quasiPoisson groupoid from a 2-term L∞-algebra, which is proposed to be the geometric structure on the dual of a Lie 2-algebra. These results lead to a construction of a new 2-term L∞-algebra from a given one, which could produce many interesting examples.

In this paper, we show that the Jacobiator J of a pre-Courant algebroid is closed naturally. The corresponding equivalence class [J ♭ ] is defined as the Pontryagin class, which is the obstruction of a pre-Courant algebroid to be deformed into a Courant algebroid. We construct a Leibniz 2-algebra and a Lie 2-algebra associated to a pre-Courant algebroid and prove that these algebraic structures are isomorphic under deformations. Finally, we introduce the twisted action of a Lie algebra on a manifold to give more examples of pre-Courant algebroids, which include the Cartan geometry.

Articles you may be interested inThe classical exchange algebra of a Green-Schwarz sigma model on supercoset target space with Z 4 m grading Consistent boundary conditions for Alexandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) sigma models and the corresponding boundary theories are analyzed. As their mathematical structures, we introduce a generalization of differential graded symplectic manifolds, called twisted QP manifolds, in terms of graded symplectic geometry, canonical functions, and QP pairs. We generalize the AKSZ construction of topological sigma models to sigma models with Wess-Zumino terms and show that all the twisted Poisson-like structures known in the literature can actually be naturally realized as boundary conditions for AKSZ sigma models. C 2014 AIP Publishing LLC. [http://dx.idea is inspired by analysis of the consistency of boundary conditions of AKSZ sigma models, 2,8,25,41 which are topological sigma models constructed by supergeometric methods and a topological open membrane. 22,38 A similar structure appears in the Batalin-Vilkovisky (BV) formalism of string field theory. 21 Another motivation comes from the canonical transformation in symplectic supergeometry, which is also called twisting. 40 It can be viewed as a higher analogue of a Poisson function, 33,45 and it defines a generalization of the Dirac structure 12 . Moreover, a canonical function describes the boundary condition structures of the AKSZ sigma models, which have played key roles in the derivation of the deformation quantization from the Poisson sigma model. 30 and 7 A canonical function leads to the concept of a QP pair, which is a certain tower of two (twisted) differential graded symplectic manifolds. This unifies the various concepts that were separately analyzed above, and it includes many geometric structures such as the Lie (2-)algebras, the (twisted or quasi) Poisson structures, the (homotopy) Lie algebroids, the (twisted) Courant algebroids, the Nambu-Poisson structures, and others. Some of these will be used below as examples.Analysis of a canonical transformation on a QP manifold naturally leads to what we call a twisted QP manifold. It is a mathematical framework for a unified understanding of the so-called Wess-Zumino terms, together with twisted Poisson-like structures. A general method to get a twisted QP manifold is given by the deformation theory. Some examples will be presented to illuminate this twisting process. Moreover a new geometric structure, the strong Courant algebroid, is proposed.The defining structure of a QP pair guarantees consistency of the bulk structure and the boundary conditions. In general, a quantum theory on a manifold X in n + 1 dimensions with given boundaries may have the same structure as the corresponding quantum theory on the boundary ∂X in n dimensions. 3 When applying this so-called bulk-boundary correspondence of quantum field theories to the AKSZ sigma models, we find that it is necessary to extend the AKSZ sigma models to the "twisted" AKSZ sigma models. This is a general...