We define a new perverse t-exact pullback operation on derived categories of constructible sheaves which generalizes most perverse t-exact functors in sheaf theory, such as microlocalization, the Fourier-Sato transform and vanishing cycles. This operation is defined for morphisms of algebraic stacks equipped with a relative exact (-1)-shifted symplectic structure, and can be used to define cohomological Donaldson-Thomas invariants in a relative setting. We prove natural functoriality properties for perverse pullbacks, such as smooth and finite base change, compatibility with products and Verdier duality.

115 pages