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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2011

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2011
    • (edited Mar 9th 2011)

    maybe I made some progress with understanding the BV-complexes formally as derived critical loci: as homotopy fibers of sections dS:𝔞T *𝔞d S : \mathfrak{a} \to T^* \mathfrak{a} of the cotangent bundle on an \infty-Lie algebroid 𝔞\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.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeSep 27th 2011
    • (edited Sep 27th 2011)

    New entry derived critical locus at the main nnLab 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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2011

    Oh, wow.

    Thanks, I had not seen that.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2017

    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:

    1. In the example I don’t check the smoothness assumption made in this prop.

    2. 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.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 1st 2017

    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?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2017

    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.)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2017
    • (edited Oct 2nd 2017)

    Vincent Schlegel kindly pointed out to me that, as stated, the computation gave the critical locus in C×𝔸 1C \times \mathbb{A}^1 instead of in CC, while the latter is really the further pullback along CC×𝔸 1C \to C \times \mathbb{A}^1. I have fixed this.

    Dually the point is that in Sym 𝒪(C)(𝒪(C))Sym_{\mathcal{O}(C)}\left( \mathcal{O}(C) \oplus \cdots \right) there are “two copies” of 𝒪(C)\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.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 2nd 2017

    Re #6, we have a dedicated page Factorization algebras in perturbative quantum field theory, which points to two volumes.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2017

    Thanks. I have made the reference line point to that page. I’d have to check which volume is relevant.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 2nd 2017

    Thanks, Urs.