derived critical locus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:33:49Z http://mathoverflow.net/feeds/question/57058 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57058/derived-critical-locus derived critical locus Urs Schreiber 2011-03-02T00:10:12Z 2011-09-27T07:01:53Z <p>I am looking for discussion in the literature that properly <em>formalizes</em> the heuristic idea that a BV-BRST complex is a model for the "derived critical locus of a function on an $\infty$-Lie algebroid".</p> <p>The kind of statement that I am after would be in the following style:</p> <p>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 <a href="http://ncatlab.org/nlab/show/dg-geometry" rel="nofollow">dg-geometry</a>. 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.</p> <p>In there we should have a canonical morphism</p> <p>$$\theta : \mathbb{A}^1 \to \mathbb{L}\Omega^1_K(-)$$</p> <p>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.</p> <p>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 <em>non-negative</em> degree). Then a morphism</p> <p>$$S : X//\mathfrak{g} \to \mathbb{A}^1$$</p> <p>is a $\mathfrak{g}$-invariant "action functional". The composite</p> <p>$$d S : X//\mathfrak{g} \stackrel{S}{\to} \mathbb{A}^1 \stackrel{\theta}{\to} \mathbb{L} \Omega^1_K(-)$$</p> <p>would be its differential. The <em>derived critical locus</em> of $S$ ought to be the homotopy fiber $hofib (d S)$ (over the global point given by the 0-forms).</p> <p>Is the BV-BRST complex in $dgAlg$ of the data $(X, \mathfrak{g}, S)$ a model for $hofib (d S)$ ?</p> <p>Or do you know writeups of details about statements of a similar flavor?</p> http://mathoverflow.net/questions/57058/derived-critical-locus/64841#64841 Answer by DamienC for derived critical locus DamienC 2011-05-12T21:34:55Z 2011-09-27T07:01:53Z <p>Update Sept. 27: Gabriele Vezzosi has just posted a <a href="http://arxiv.org/abs/1109.5213" rel="nofollow">preprint on the arXiv</a> that could be of interest for that question (below is my original answer). </p> <hr> <p>I think there is <a href="http://arxiv.org/abs/1010.3210" rel="nofollow">a paper by Frédéric Paugam</a> ("Histories and observables in covariant field theory") where this is discussed. </p> <p>EDIT: There is also a <a href="http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf" rel="nofollow">book in progress</a> by the same author where he discusses the notion of derived critical space for the Euler-Lagrange equation (this is in Chapter 10). </p> http://mathoverflow.net/questions/57058/derived-critical-locus/71825#71825 Answer by gabriele for derived critical locus gabriele 2011-08-01T20:11:07Z 2011-08-01T20:11:07Z <p>You might look at Costello-Gwilliam book (especially 'Derived Euler-Lagrange equations' and 'Derived critical locus' - in the Appendix)</p> <p><a href="http://math.northwestern.edu/~costello/factorization_public.html" rel="nofollow">http://math.northwestern.edu/~costello/factorization_public.html</a></p>