Tensor hierarchy from deformation quantisation

RESUMO

We show that a formal deformation quantisation of degree-2 differential graded symplectic (QP) manifolds gives rise to the complete classical kinematics and dynamics of NS-NS supergravity, or, equivalently, its duality-covariant formulation as double field theory (DFT). Our construction extends the standard tensor hierarchy, encoding gauge parameters and their redundancies, to a complex encoding the physical fields, field strengths, and Bianchi identities. We present two realisations of such a complex: 1) a differential graded Lie algebra built from the algebra of functions on the QP-manifold; 2) a cochain complex that reproduces the tensor hierarchy representations conjectured in the literature and can be interpreted as a classical BV complex of the linearised theory. The deformation that gives rise to the dilaton comes from a normal-ordered graded Moyal--Weyl star product. This naturally extends the duality structure group from $\mathrm{O}(D,D)$ to $\mathrm{O}(D,D)\times\mathbb{R}^+$. Finally, our framework provides a direct route to constructing the DFT action via local double Lorentz invariance, and offers a transparent algebraic foundation for curvature tensors in generalised Cartan geometry.