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on my personal web I am starting a page derived critical locus (schreiber) with some notes.
I think so far I can convince myself of the claim that the page currently ends with (without proof). My next goal is to show that the homotopy fibers discussed there are given by BV-BRST complexes. But I have to interrupt now.
maybe I made some progress with understanding the BV-complexes formally as derived critical loci: as homotopy fibers of sections $d S : \mathfrak{a} \to T^* \mathfrak{a}$ of the cotangent bundle on an $\infty$-Lie algebroid $\mathfrak{a}$ (a formal dual to a BRST complex).
New, rewritten notes are at derived critical locus (schreiber).
In fact I think I understand the full story if I assume that homotopy pushouts of my unbounded commutative dg-algebras are computed by mapping cones as usual. This is what one expects, but one needs to be a bit careful with what model structure exactly one uses to present the derived geometry, and what assumptions on projectivity are being made. This I need to think more about.
New entry derived critical locus at the main $n$Lab to record the Vezzosi’s paper. I am a bit surprised that the page lists that it is linked from derived critical locus while I have not put that self-referencing link.
Oh, wow.
Thanks, I had not seen that.
I have brought into derived critical locus the core of my old notes (from my personal web) aiming to show that the BV-BRST complex really is (the formal dual of) the derived critical locus in dg-geometry of a function on a Lie algebroid (BRST complex).
Looking at this material from 2011 again I notice two things:
In the example I don’t check the smoothness assumption made in this prop.
meanwhile there ought to be a reference that provides all the required model-category theoretic background in the entry in an easily citable way.
I don’t quite have the time right now to dig into this again. If anyone has a hint, I’d be grateful.
The Costello-Gwilliam reference seems to now be this book (pdf); does anyone have a more precise location for the claim referred to in the nLab page? Or is it just the general philosophy of the approach?
a more precise location for the claim
In the book it is now the beginning of section 4.8.1
(Back in 2011 I was pointing to their wiki, which however no longer exists. I have added the section pointer to the entry now.)
Vincent Schlegel kindly pointed out to me that, as stated, the computation gave the critical locus in $C \times \mathbb{A}^1$ instead of in $C$, while the latter is really the further pullback along $C \to C \times \mathbb{A}^1$. I have fixed this.
Dually the point is that in $Sym_{\mathcal{O}(C)}\left( \mathcal{O}(C) \oplus \cdots \right)$ there are “two copies” of $\mathcal{O}(C)$, and they eventually need to be identified. Indeed that’s what is necessary to yield the desired conclusion (which tacitly made that identification).
I have fixed this now.
Re #6, we have a dedicated page Factorization algebras in perturbative quantum field theory, which points to two volumes.
Thanks. I have made the reference line point to that page. I’d have to check which volume is relevant.
Thanks, Urs.
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