# derived critical locus

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?

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I'm looking through the tags: this is the only one tagged 'derived'. It seems (meanwhile) more expressive and well-used tags exists, e.g., 'derived-algebraic-geometry', also 'dg-algebras' seems like a good additional tag. To add some arXiv tag(s) could be nice, too. Also, in my experience adjective-like tags as 'derived' are not optimal and this one could get easily misused (say, some, UG tagging some HW 'derived functions') So, I'd like to ask you to retag avoiding 'derived'. If you should be against a retag; please also reply. Thanks in advence! –  quid Feb 11 '13 at 12:09
Since comment notification does not work very reliably in my experience, I write another one to draw attention to the preceeding one. In case you noted the preceeding one already, sorry for the noise. –  quid Feb 12 '13 at 18:27
Sure, thanks. I don't remember, but I can't imagine that I intentionally made "derived" a tag. Did I do that? Maybe I just forgot a hyphen? In any case, I entirely agree and have taken note of it. –  Urs Schreiber Feb 12 '13 at 20:20
Thank you for the reply. I cannot tell if you created it. I think there used to be a second one tagged with this tag, which I retaged somewhat recently as there it seemed clearer to me how to retag. Here I was not sure which top-level tag(s) would be best and also it was brought to my attention that me retagging various things is perceived as too much 'noise' (in general). So I did not simply go ahead and retag for this one. –  quid Feb 13 '13 at 21:38

## 2 Answers

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).

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).

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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

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Last time I looked at this it was using the term, but not giving a formalization or deriving the details. (Maybe it has meanwhile?) I think there is some technical fine-print to be taken care of here. But maybe I found the answer myself. I discuss it here: ncatlab.org/schreiber/show/derived%20critical%20locus –  Urs Schreiber Aug 6 '11 at 8:48