From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC moduli space") correspond to deformations. (More accurately the MC moduli space is formulated as a deformation functor but these details are not important for this question.)

Is anything known about the putative $L_\infty$-algebra whose MC moduli space corresponds to deformations of some fixed $L_\infty$-algebra? I am also interested in the corresponding question for $A_\infty$-algebras.

This seems to have been considered in the physics literature, albeit in a context with more structure: Hata-Zwiebach write down an "actional" whose critical points are solutions of the quantum master equation i.e. loop $L_\infty$-algebras. Insofar as their "actional" defines a (loop) $L_\infty$-algebra (which Hata and Zwiebach suggest but is not entirely obvious to me), that (loop) $L_\infty$-algebra would be the object we are looking for, albeit in the case where the original $L_\infty$-algebra in question --- whose deformations we are considering --- has this extra structure making it into a loop $L_\infty$-algebra.

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