On the Uniqueness of L$_\infty$ bootstrap: Quasi-isomorphisms are Seiberg-Witten Maps

In the context of the recently proposed L$_\infty$ bootstrap approach, the question arises whether the so constructed gauge theories are unique solutions of the L$_\infty$ relations. Physically it is expected that two gauge theories should be considered equivalent if they are related by a field redefinition described by a Seiberg-Witten map. To clarify the consequences in the L$_\infty$ framework, it is proven that Seiberg-Witten maps between physically equivalent gauge theories correspond to quasi-isomorphisms of the underlying L$_\infty$ algebras. The proof suggests an extension of the definition of a Seiberg-Witten map to the closure conditions of two gauge transformations and the dynamical equations of motion.