derived critical locus

Question

I am looking for discussion in the literature that properly formalizes the heuristic idea that a BV-BRST complex is a model for the "derived critical locus of a function on an $\infty$-Lie algebroid".

The kind of statement that I am after would be in the following style:

Pass to the ambient $\infty$-topos of $\infty$-sheaves on the $\infty$-site of formal duals to commutative cochain dg-algebras in non-positive degree over a field of characteristic 0 (for some topology, which I think won't matter much for the following question): the context of dg-geometry. There is then a derived functor $dgAlg^{op} \to Sh_\infty(dgAlg_-^{op})$ that interprets unbounded dg-algebras as objects in this $\infty$-topos, and this I shall make use of in the following.

In there we should have a canonical morphism

$$ \theta : \mathbb{A}^1 \to \mathbb{L}\Omega^1_K(-) $$

from the line to the $\infty$-sheaf of cotangent complexes, that sends over $A \in dgAlg_-$ an element $a \in Q A \simeq \mathbb{A}^1(A)$ to $d a$, for $Q A$ a cofibrant replacement.

Now consider an $\infty$-Lie algebroid, for instance as a simple standard example the homotopy quotient of a Lie algebra action on an ordinary affine, for which sugestive notation would be $X//\mathfrak{g}$. The dg-algebra corresponding to this dually is the corresponding Chevalley-Eilenberg algebra / BRST complex (in non-negative degree). Then a morphism

$$ S : X//\mathfrak{g} \to \mathbb{A}^1 $$

is a $\mathfrak{g}$-invariant "action functional". The composite

$$ d S : X//\mathfrak{g} \stackrel{S}{\to} \mathbb{A}^1 \stackrel{\theta}{\to} \mathbb{L} \Omega^1_K(-) $$

would be its differential. The derived critical locus of $S$ ought to be the homotopy fiber $hofib (d S)$ (over the global point given by the 0-forms).

Is the BV-BRST complex in $dgAlg$ of the data $(X, \mathfrak{g}, S)$ a model for $hofib (d S)$ ?

Or do you know writeups of details about statements of a similar flavor?

Answer 1

Update Sept. 27: Gabriele Vezzosi has just posted a preprint on the arXiv that could be of interest for that question (below is my original answer).

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I think there is a paper by Frédéric Paugam ("Histories and observables in covariant field theory") where this is discussed.

EDIT: There is also a book in progress by the same author where he discusses the notion of derived critical space for the Euler-Lagrange equation (this is in Chapter 10).

Answer 2

You might look at Costello-Gwilliam book (especially 'Derived Euler-Lagrange equations' and 'Derived critical locus' - in the Appendix)

http://math.northwestern.edu/~costello/factorization_public.html (Wayback Machine)