Xiaomeng Xu

The Pontryagin class for pre-Courant algebroids

In this paper, we show that the Jacobiator J J of a pre-Courant algebroid is closed naturally. The corresponding equivalence class [J ♭ ] [ 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.

Twisted Courant algebroids and coisotropic Cartan geometries

In this paper, we show that associated to any coisotropic Cartan geometry there is a twisted Courant algebroid. This includes in particular parabolic geometries. Using this twisted Courant structure, we give some new results about the Cartan curvature and the Weyl structure of a parabolic geometry. As more direct applications, we have Lie 2-algebra and 3D AKSZ sigma model with background associated to any coisotropic Cartan geometry.

Canonical functions, differential graded symplectic pairs in supergeometry, and Alexandrov-Kontsevich-Schwartz-Zaboronsky sigma models with boundaries

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.

Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids

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

Generalized Classical Dynamical Yang-Baxter Equations and Moduli Spaces of Flat Connections on Surfaces

\mathfrak g

Current Algebras from DG Symplectic Pairs in Supergeometry

g-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term

Chern–Simons, Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

L_\infty