We construct new six-functor formalisms capturing cohomological invariants of varieties with potentials. Starting from any six-functor formalism $C$, encoded as a coefficient system, we associate a new six-functor formalism $C_{ \text{ exp}}$. This requires in particular constructing the convolution product symmetric monoidal structure at the $\infty$-categorical level. We study $C_{ \text{ exp}}$ and how it relates to $C$. We also define motives in $C_{ \text{ exp}}$ attached to varieties with potential and study their properties.
Categories: math.AG