A-type Sigma Models from Differential Poisson Geometry

Abstract

We study the differential Poisson sigma model (DPSM) in the symplectic case and show that its classical reduction defines a distinguished class of A-type models on symplectic targets, not necessarily Kähler. The DPSM is a covariant first-order sigma model whose graded target is the parity-shifted tangent bundle $T[1]M$ of a Poisson manifold $M$. Its graded Poisson tensor encodes a differential Poisson bracket on $C(T[1]M)\cong {\Omega }^{\bullet }(M)$, written covariantly in terms of a connection $\Gamma$ and its transpose $\tilde{\Gamma }$. In the nondegenerate case, the Jacobi identities force $\Gamma$ to be flat, while the quartic coupling of the reduced action is given by the curvature of $\tilde{\Gamma }$, induced by the torsion of $\Gamma$. Thus, the DPSM selects a symplectic class in which the A-model curvature coupling acquires a first-order Poisson origin. We describe this class through examples and obstructions; ${\mathbb{CP}}^n$ and K3 surfaces are excluded, while affine symplectic targets, symplectic tori, and the Kodaira--Thurston manifold furnish explicit examples. The graded parent geometry on $T[1]M$ equips ${\Omega }^{\bullet }(M)$ with a differential graded Poisson algebra structure; in particular, the underlying differential graded Lie algebra defines a strict $L_{\infty }$-algebra on the observable complex. This chain-level structure is not manifest in the usual Kähler formulation of the A-model.

v2: corrected the grading/sign conventions for the differential Poisson bracket on forms, streamlined the observable-algebra discussion, and fixed minor typos