Quantum microformal morphisms of supermanifolds: an explicit formula and further properties

We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially oscillatory integral operators or Fourier integral operators of a particular kind. They act on oscillatory wave functions, whose algebra extends the algebra of formal power series in Planck's constant. In the classical limit, quantum microformal morphisms reproduce, as the main term of the asymptotic, the nonlinear pullbacks of functions with respect to `classical' microformal morphisms. We found these nonlinear transformations of functions in search of $L_{\infty}$-morphisms for homotopy Poisson structures.