Overview
This study addresses the integration and quantization of specific geometric structures, namely 1-shifted Lagrangians and, more broadly, 1-shifted coisotropics, within the context of 1-shifted symplectic differentiable stacks. It builds upon and extends prior work concerning the quantization of reduced coisotropic submanifolds and their integration into Lagrangian subgroupoids.
Research Context
The problem of quantizing the reduction of a coisotropic submanifold within a Poisson manifold has been previously explored. Cattaneo and Felder developed a homotopy Poisson algebra and a deformation quantization for this purpose. The degree-zero cohomology of this construction corresponds to the reduced Poisson algebra, while the full homotopy structure retains information pertaining to the formal embedding, even when the reduced space exhibits singularities or is trivial.
On the integration side, Cattaneo demonstrated that a coisotropic submanifold integrates to a Lagrangian subgroupoid. This subgroupoid arises within any symplectic groupoid that integrates the ambient Poisson manifold. Current understanding recognizes that both these constructions fit within the framework of shifted symplectic geometry. Specifically, the Lagrangian subgroupoid represents a $1$-shifted Lagrangian morphism of differentiable stacks, and the associated $0$-shifted Poisson structure is described by the homotopy Poisson algebra.
However, the class of $1$-shifted Lagrangians is broader than just such morphisms.
Approach
This research extends the established picture to encompass general $1$-shifted Lagrangians and, more broadly, $1$-shifted coisotropics. The investigation is conducted within $1$-shifted symplectic differentiable stacks.
The approach involved several key steps:
- Establishing an integration theorem for the infinitesimal descriptions of these structures. These infinitesimal descriptions are given in terms of twisted Dirac structures.
- Constructing a natural Poisson bracket on the invariant functions associated with these structures.
- Under specific regularity hypotheses, equipping the associated Lie algebroid complex with a flat $P_\infty$-structure.
- Under the same regularity hypotheses, equipping the Lie algebroid complex with a curved $A_\infty$-quantization. This quantization is demonstrated to be independent of auxiliary choices, up to isomorphism.
Findings
The study establishes an integration theorem specifically for the infinitesimal description of $1$-shifted Lagrangians and $1$-shifted coisotropics, which are characterized by twisted Dirac structures. A natural Poisson bracket was successfully constructed on the invariant functions pertinent to these extended structures.
Under specified regularity hypotheses, the research found that the associated Lie algebroid complex can be equipped with a flat $P_\infty$-structure. Concurrently, under the same regularity hypotheses, a curved $A_\infty$-quantization was constructed for this complex. This quantization exhibits independence from auxiliary choices, up to isomorphism.
The induced bracket in degree zero, resulting from this framework, was found to recover the reduced Poisson bracket. Furthermore, the study indicates that suitable cohomological vanishing conditions yield a deformation quantization of invariant functions.
Why This Matters
This research extends the Cattaneo–Felder picture, providing a framework for more generalized reduction procedures. The same infinitesimal data, derived within this extended framework, governs both the integration and quantization aspects of these more general structures.