These are notes on derived critical loci, created for a Seminar on derived critical loci at Utrecht University in spring 2011. This follows up on a previous Seminar on derived differential geometry. See there for more background.
Given an ordinary space $C$ and a function $S : C \to \mathbb{A}^1$, the critical locus $C_{\{d S = 0\}}$ of $S$ is the subspace on which the de Rham differential of $S$ vanishes, the fiber product
We want to consider this situation in the context of derived geometry and compare with the toolset of BRST-BV complexes. The basic idea is indicated in (CostelloGwilliam).
In higher geometry such $C$ is generally a cohesive ∞-groupoid. A simple motivating example is the groupoid of connections over some spacetime (the configuration space of a gauge theory). A first-order approximation to $C$ is its ∞-Lie algebroid $\mathfrak{c} \hookrightarrow C$. In the context of gauge theory its function algebra $\mathcal{O}(\mathfrak{c})$ is called a BRST complex.
We demonstrate that the derived critical locus of $S$ restricted to $\mathfrak{c}$
is the object whose function algebra is essentially the BRST-BV complex for $S$. See below for more detailed discussion.
Recall the general setup of derived geometry over a given (∞,1)-algebraic theory $T$: we take formal duals of a small collection of $\infty$-algebras over $T$ to be our test spaces and then let general derived spaces be ∞-stacks over these test spaces.
For derived differential geometry let $T =$ CartSp be the Lawvere theory for smooth algebras, regarded as an (∞,1)-algebraic theory. Write $Smooth Alg_\infty$ for its (∞,1)-category of ∞-algebras over an (∞,1)-algebraic theory.
We may present this by the model structure on simplicial presheaves $[CartSp, sSet]$, left Bousfield localized at the morphism of the form
Let $CartSp \hookrightarrow C \hookrightarrow Smooth Alg_\infty$ be a small full sub-(∞,1)-category equipped with the structure of a subcanonical (∞,1)-site.
We write
for the (∞,1)-category of (∞,1)-sheaves over $C$.
The following definition recalls the setup of dg-geometry over a field $k$ of characteristic 0.
Write
and
for the category of commutative cochain dg-algebras over $k$, in non-positive degree and without restrition on degrees, respectively, the latter equipped with the model structure on dg-algebras whose weak equivalences are the quasi-isomorphisms and whose fibrations are the degreewise surjections.
Notice that the derived hom-spaces in both cases are given by
For $(cdgAlg_k^-)^{op}$ equipped with any subcanonmical sSet-site structure, write
for the (∞,1)-sheaf (∞,1)-topos over it.
The $(\infty,1)$-Yoneda extension of the canonical inclusion
yields an (∞,1)-adjunction
For $\mathbf{H}$ an (∞,1)-topos and $X \in \mathbf{H}$ any object, the over-(∞,1)-category
is itself an (∞,1)-topos – the over-(∞,1)-topos over $X$ – which is to be thought of as the little topos incarnation of $X$, sitting by an etale geometric morphism $\pi_X$ over $\mathbf{H}$.
We consider this now in the context of dg-geometry.
Let $\mathcal{O}(C) \in cdgAlg_k$, write $\mathcal{O}(C) Mod$ for the model structure on dg-modules over $\mathcal{O}(C)$.
Write
for the category of commutative monoids in $\mathcal{O}(C)$-modules.
There is a model category structure on $cdgAlg_{\mathcal{O}(C)}$ whose fibrations and weak equivalences are those of the underlying $\mathcal{O}(C)$-modules such that the free-forgetful adjunction
is a Quillen adjunction.
This is
This follows with the general discussion at dg-geometry. We indicate how to see it directly.
We observe that the adjunction exhibits the transferred model structure on the left. By the statement discussed there, it is sufficient to check that
$\mathcal{O}(C) Mod$ is a cofibrantly generated model category.
This follows because the model structure on dg-modules (as discussed there) is itself transferred along
from the cofibrantly generated model structure on cochain complexes.
$U$ preserves filtered colimits.
This follows from the general fact $U : CMon(\mathcal{C}) \to \mathcal{C}$ creates filtered colimits for $\mathcal{C}$ closed symmetric monoidal (see here) and that $A Mod$ is closed symmetric monoidal (see here).
To check this explicitly:
Let $A_\bullet : D \to cdgAlg_k$ be a filtered diagram. We claim that there is a unique way to lift the underlying colimit $\lim_\to U A_\bullet$ to a dg-algebra cocone: for $a \in A_i \to \lim_\to U A_\bullet$ and $b \in A_j \to \lim_\to U A_\bullet$ there is by the assumption that $D$ is filtered a $A_i \to A_l \leftarrow A_j$. Therefore in order for the cocone component $U A_l \to \lim_{\to} U A_\bullet$ to be an algebra homomorphism the product of $a$ with $b$ in $\lim_\to U A_\bullet$ has to be the image of this product in $A_l$. This defines the colimiting cocone $A_l \to \lim_\to A_\bullet$.
The left hand has functorial fibrant replacement (this is trivial, since every object is fibrant) and functorial path objects.
This follows by the same argument as for the path object in $cdgAlg_k$ here this can be taken to be $(-)\otimes_k \Omega^\bullet_{poly}(\Delta[1])$.
Let $\mathbf{H}$ be the $(\infty,1)$-topos for dg-geometry discussed above, and $C \in \mathbf{H}$. Write $\mathcal{O}(C) \in cdgAlg_k$ for a cofibrant representative of the image of $C$ under $\mathcal{O}$.
Then the function algebra adjunction from def 1 induces a relative function algebra adjunction
This follows with the above, with the general properties of (∞,1)-adjunctions on slices and the discussion at dg-geometry.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos with differential cohesion and with line object $\mathbb{A}^1$.
Write $\mathbf{\flat}_{dR} \mathbf{B}\mathbb{A}^1$ for the canonical de Rham coefficient object and and
for the intrinsic curvature characteristic form.
Given a morphism
we write
for its differential.
Write $T^* C \to C$ for the object that represents $\mathbf{H}(C,\mathbf{\flat}_{dR} \mathbf{B}\mathbb{A}^1)$ in $\mathbf{H}_{/C}^{et}$, see at cohesive (infinity,1)-topos – infinitesimal cohesion – structure sheaves.
In good cases the object $T^*_f C$ defined this way is the formal dual of the tangent complex of the function algebra $\mathcal{O}(C)$. This is the actual definition to be used in the following
In the context of dg-geometry, for $C \in \mathcal{H}$ an object we define
to be the free $\mathcal{O}(C)$-algebra on the tangent complex of $\mathcal{O}(C)$.
The derived critical locus of $S : C \to \mathbb{A}^1$ in $\mathbf{H}$ is the (∞,1)-pullback
computed in $\mathbf{H}/C$.
If $C$ is $\mathcal{O}$-perfect (…) in that $\mathcal{O}$ preserves this pullback, this is equivalently given by the $(\infty,1)$-pushout
Compare with the situation for Hochschild cohomology of $\mathcal{O}(C)$, for $C$ $\mathcal{O}$-perfect, which is given by the complex $\mathcal{O}(\mathcal{L}(C))$ of functions on the derived loop space given by the $(\infty,1)$-pullback
For suitable $C$ this factors through the infinitesimal neighbourhood of the diagonal hence is the derived self-intersection in the tangent bundle
We discuss the derived critical locus in dg-geometry over formal duals of general differential graded algebras.
Let $k$ be a field of characteristic 0.
Write $dgcAlg_k$ for the category of graded-commutative unbounded cochain dg-algebras over $k$.
For an object
we write
Let $\mathcal{O}(C)$Mod be the category of dg-modules over $\mathcal{O}(C)$ equipped with the standard model structure on dg-modules.
Write finally
for the category of commutative monoids in $\mathcal{O}(C) Mod$: the category of commutative dg-algebras under $\mathcal{O}(C)$. We regard this as a category with weak equivalences given by the underlying quasi-isomorphisms.
This category models dg-geometry over $C$ in that
Write
for the tangent complex/automorphism ∞-Lie algebra of $A$ whose underlying cochain complex is
where $Der(\mathcal{O}(C))_k$ is the module of derivations
of degree $k$ and $[d_{\mathcal{O}(C)}, -]$ is the graded commutator of derivations with the differential of $\mathcal{O}(C)$ regarded as a degree-1 derivation $d_{\mathcal{O}(C)} : \mathcal{O}(C) \to \mathcal{O}(C)$.
We say that $\mathcal{O}(C)$ is smooth if $Der(\mathcal{O}(C))$ is cofibrant as an object on $\mathcal{O}(C) Mod$.
Write
for the free $\mathcal{O}(C)$-algebra over $Der(\mathcal{O}(C))$.
We write
for its formal dual.
Every $S \in \mathcal{O}(C)$ defines a morphism
dually given by
where $\hat S : \mathcal{O}(C) \to \mathcal{O}(C)$ is the $k$-linear multiplication operator defined by $S$ and where for $v \in Der(\mathcal{O}(C))$ we set
which may be regarded as the multiplication operator given by the commutator of $k$-linear endomorphisms of $\mathcal{O}(C)$ as indicated.
The derived critical locus of a morphism $S : C \to \mathbb{A}^1$ is the homotopy pullback $C_{\{d S = 0\}}$ in $cdgAlg^{op}/C$
If $C$ is smooth in the sense that $Der(\mathcal{O}(C)) \in \mathcal{O}(C) Mod$ is cofibrant, then the derived critical locus is presented by
where on the right we have the free $\mathcal{O}(C)$-algebra over the mapping cone of $[\hat S, -]$.
By prop 1 the functor $Sym_{\mathcal{O}(C)}$ is left Quillen. Hence if $Der(\mathcal{O}(C))$ is cofibrant in $\mathcal{O}(C) Mod$ then the homotopy pushout in question may be computed as the image under $Sym_{\mathcal{O}(C)}$ of the homotopy pushout in $\mathcal{O}(C) Mod$.
By the disucssion at model structure on dg-modules, for these the homotopy cofibers are given by the ordinary mapping cone construction for chain complexes.
More in detail, write
for the mapping cone on the identity.
Then $Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to}) \mathcal{O}(C))$ is
If we extend the graded commutators in the evident way we may write the differential in $Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C))$ as
Here the second term is the differential of the BRST-complex of $\mathfrak{c}$, whereas the sum is of the type of a differential in a BRST-BV complex.
We discuss how the traditional BRST-BV formalism relates to the computation of derived critical loci as above.
The traditional approach in BRST-BV formalism starts from a somewhat different angle than the discussion here. There on
then builds a Koszul-Tate resolution of its ordinary critical locus;
then adds generators in positive degree in order to make the Koszul-tate differential a Hamiltonian vector field with respect to the extended graded Poisson bracket (“anti-bracket”);
deduces this way a BRST-complex part on $X$ (the part of the complex spanned by the “ghost”-generators).
Here the perspective is to some extent opposite to this: we assume that the BRST-complex encoding the symmetries of $S$ is already given, and then find just a single-step Koszul-type resolution, but not of an ordinary space, but of the dg-space that contains the ghost generators.
But both constructions do coincide if
the gauge symmetries close off-shell;
the $\infty$-Lie algebroid $C$ is the full BRST-complex of $S$.
(…)
Let $\mathfrak{a}$ be a Lie algebroid over a space $X$, with Chevalley-Eilenberg algebra $\mathcal{O}(\mathfrak{a})$ given by
for $f \in C^\infty(X)$, infinitesimal gauge symmetries $R^i_a \frac{\partial}{\partial x^i}$, gauge symmetry structure functions $C^{a}{}_{b c}$ and ghost generators $c^a$.
The “algebra of vector fields/derivations” $Der(\mathcal{O}(\mathfrak{a}))$ on $\mathfrak{a}$ is the automorphism ∞-Lie algebra whose underlying chain complex is
We check on generators that
and
Now let
be a function, dually a dg-algebra homomorphism
This is equivalently any function
which is gauge invariant
We have a contraction homomorphism of $\mathcal{O}(\mathfrak{a})$-modules
and may form its mapping cone,
On the free algebra of this
we have the differential given on generators by
If $\langle R_a\rangle$ is the full kernel of $\iota_{d S} : Der(C^\infty(X)) \to C^\infty(X)$ and there are no further relations, then this is the full BRST-BV complex of $S$.
The term derived critical locus for the formal dual of a BRST-BV complex and a brief indication for how to formalize it is in
For references on BRST-BV formalism see there.