Shifted Poisson unfoldings and quantum anomalies

Families of classical field theories often depend on geometric parameters.

A basic question is whether the corresponding classical observables can be compared by flat parallel transport, and whether this comparison survives quantization.

We study this problem for families of shifted Poisson structures.

To such a family we attach a Poisson transverse controller $\mathbb U_\pi$, a homotopy stabilizer encoding transverse symmetries which preserve the Poisson Maurer--Cartan element up to coherent homotopy.

Its flat splittings are precisely transversal shifted Poisson unfoldings, and they act on vertical symmetries, Poisson cohomology and local deformation theory.

We then formulate the $\hslash$-adic lifting problem for quantized objects; its degree-two obstruction classes are the transport anomalies.

The construction is realized for star-products, BV observables, factorization algebras and AKSZ theories.

For the Poisson sigma model, anomaly-free quantization of the flat transport makes the Cattaneo--Felder/Kontsevich boundary product horizontal over the parameter space.