Formal moduli problems with cohomological constraints

We aim to generalize Lurie-Pridham’s famous $\mathrm{\infty }$-correspondence between formal moduli problems and differential graded Lie algebras to the context where some cohomological conditions are imposed. Specifically, a natural equivalence between formal moduli problems with cohomological constraints and derived cohomology jump functors is provided, thereby answering affirmatively a question posed by N. Budur and B. Wang in \cite{1}.