[Paper Review] Canonical Functions and Differential Graded Symplectic Pairs in Supergeometry and AKSZ Sigma Models with Boundary

Noriaki Ikeda, Xiaomeng Xu|arXiv (Cornell University)|Jan 21, 2013

Homotopy and Cohomology in Algebraic Topology29 references3 citations

TL;DR

This paper introduces twisted QP manifolds and QP pairs as a generalized framework for understanding boundary conditions in AKSZ sigma models with Wess-Zumino terms. It shows that all known twisted Poisson-like structures arise naturally as boundary conditions, unifying geometric structures via canonical functions and graded symplectic geometry, with applications to topological field theories and deformation quantization.

ABSTRACT

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.

Motivation & Objective

• To clarify the mathematical structure of consistent boundary conditions in AKSZ sigma models with Wess-Zumino terms.
• To generalize differential graded symplectic geometry by introducing canonical functions and QP pairs as a new framework.
• To unify various geometric structures—such as twisted Poisson, Courant, and Nambu-Poisson structures—under a single formalism.
• To establish a correspondence between bulk AKSZ models and boundary theories via twisted QP manifolds, extending the AKSZ construction.
• To propose a new geometric object: the strong Courant algebroid, as a generalization of Courant algebroids with higher homotopy structure.

Proposed method

• Introduce canonical functions on graded symplectic manifolds as a tool to describe boundary conditions in AKSZ sigma models.
• Define QP pairs as a tower of two twisted differential graded symplectic manifolds, generalizing QP manifolds to include boundary data.
• Construct twisted QP manifolds via deformation theory and canonical transformations, generalizing standard QP structures.
• Apply the AKSZ construction to twisted QP manifolds to produce topological sigma models with Wess-Zumino terms.
• Use derived Poisson brackets and homological vector fields to ensure consistency of bulk and boundary actions.
• Demonstrate that the boundary action of a twisted AKSZ model with a Wess-Zumino term reproduces known models like the $H_4$-twisted Courant sigma model.

Experimental results

Research questions

• RQ1 How can consistent boundary conditions for AKSZ sigma models with Wess-Zumino terms be systematically characterized?
• RQ2 What is the role of canonical functions in encoding boundary data within graded symplectic geometry?
• RQ3 How do QP pairs generalize QP manifolds to describe both bulk and boundary structures simultaneously?
• RQ4 Can all known twisted Poisson-like structures be derived as boundary conditions of a single class of topological sigma models?
• RQ5 What new geometric structures, such as the strong Courant algebroid, emerge from this generalized framework?

Key findings

• All known twisted Poisson-like structures, including WZ-Poisson and twisted Courant algebroids, are shown to naturally arise as boundary conditions of AKSZ sigma models.
• The canonical function formalism provides a unified mechanism to derive boundary actions from bulk QP structures, ensuring consistency of the classical master equation.
• Twisted QP manifolds are constructed via deformation theory and canonical transformations, generalizing standard QP manifolds to include higher-degree symplectic and homological data.
• The boundary action of the twisted AKSZ model with a Wess-Zumino term reproduces the $H_4$-twisted Courant sigma model, confirming consistency with known physical models.
• The derived Poisson bracket on the boundary space $\mathcal{L}$ is generally degenerate, but the bulk and boundary theories remain physically consistent due to satisfaction of the classical master equation.
• The framework extends the AKSZ construction to a 'twisted' version, generalizing the Chern-Simons/WZW correspondence in the context of topological field theories.