- The paper presents a novel use of BFV formalism to extend the AKSZ construction for reduced topological sigma models.
- It details the integration of BV quantization with graded symplectic and BFV manifolds to compute correlators in constrained systems.
- The methodology is demonstrated through examples on Poisson actions and Courant algebroids, offering fresh insights in quantum geometry.
AKSZ Construction from Reduction Data
Introduction
The paper focuses on extending the application of the AKSZ (Aleksandrov-Kontsevich-Schwartz-Zaboronsky) formalism to encode the process of geometric reduction within the framework of topological sigma models. The AKSZ construction is pivotal in the field of theoretical physics, offering a systematic approach to the formulation of topological field theories. The novelty of this work lies in its utilization of the BFV (Batalin-Fradkin-Vilkovisky) construction to facilitate this task, where the BFV framework assists in handling constrained graded symplectic manifolds.
Background and Methodology
BV Manifolds and AKSZ Construction
The research explores BV quantization and its extension through the AKSZ formalism. AKSZ construction deals with topological field theories by defining a sigma model based on a graded symplectic manifold with an associated homological hamiltonian. This formalism is pivotal due to its capacity to integrate various geometric structures like Poisson and Courant algebroids into topological field theory formulation.
BFV Manifolds
The BFV manifold approach is employed to handle reduction data in geometrical structures. A BFVn manifold is characterized as a graded symplectic space enriched with a specific degree symplectic form and a graded BFV charge. In this formalism, the reduction of target geometry is encapsulated using a higher-dimensional manifold, which then acts as the target for the AKSZ construction.
Reduction Process
The reduction process examined involves Lie group actions leading to simplified models. The method is applicable across various scenarios such as encoding Poisson structures and Courant algebroids. The reduction by group actions particularly for n = 1 and n = 2, related to Poisson and Courant sigma models, respectively, highlights the paper's methodological innovation. The AKSZ-BFV system facilitates the computation of correlators by considering the interplay between BV and BFV manifolds, which is substantiated through multiple examples.
Examples and Applications
Poisson Actions and Reduction
The study provides detailed explorations for Poisson actions, demonstrating the construction of BFV1 manifolds from reduction data. The exemplification spans Poisson Lie group actions and invariant momentum maps, elucidating their algebraic structure and implications within the symplectic geometry framework.
Courant Algebroids
In the context of n = 2, Courant algebroids are analytically reduced using the presented framework. The approach involves encoding the quotient operation via group actions into the structure of the Courant algebroid, effectively capturing the formal parallels in higher-dimensional algebraic topology.
Theoretical Implications
The implications of this research significantly enhance the understanding of symplectic reductions and their applications in the wider scope of graded symplectic geometry. The BFV-AKSZ model presents a coherent method to produce new models of reduced geometrical structures, expanding the scope of sigma models and offering a comprehensive perspective on the integration of symplectic and graded manifold theories into quantum field theory (QFT).
Conclusion
The work outlined in "AKSZ construction from reduction data" is integral for advancing the theoretical framework of AKSZ sigma models in conjunction with BFV construction for constrained systems. By systematically examining reduction processes, this approach refines the capacity to both analyze and construct topological field theories with significant theoretical and practical applications in quantum geometry and field theory. As a path forward, further exploration into integrating these models with algebraic topology and QFT dynamics remains both promising and imperative.