Poisson AKSZ theories and their quantizations

TL;DR

This paper extends the AKSZ construction of topological field theories to include target manifolds with degenerate Poisson structures, connecting quantization to operadic formality and proposing a properadic framework.

Contribution

It generalizes AKSZ theories to degenerate Poisson structures and links their quantization to the formality of the E_d operad, introducing a properadic perspective.

Findings

Classical AKSZ theories described via dioperads.

Quantization relates to extending dioperads to properads.

Conjecture on properadic description of E_d formality quasiisomorphisms.

Abstract

We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads. The quantization problem is posed in terms of extending from dioperads to properads. We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms.