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 and a function , the critical locus of is the subspace on which the de Rham differential of 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 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 is its ∞-Lie algebroid . In the context of gauge theory its function algebra is called a BRST complex.
We demonstrate that the derived critical locus of restricted to
is the object whose function algebra is essentially the BRST-BV complex for . See below for more detailed discussion.
The ambient -topos
Recall the general setup of derived geometry over a given (∞,1)-algebraic theory : we take formal duals of a small collection of -algebras over to be our test spaces and then let general derived spaces be ∞-stacks over these test spaces.
Over derived smooth loci
For derived differential geometry let CartSp be the Lawvere theory for smooth algebras, regarded as an (∞,1)-algebraic theory. Write for its (∞,1)-category of ∞-algebras over an (∞,1)-algebraic theory.
We may present this by the model structure on simplicial presheaves , left Bousfield localized at the morphism of the form
Let be a small full sub-(∞,1)-category equipped with the structure of a subcanonical (∞,1)-site.
for the (∞,1)-category of (∞,1)-sheaves over .
The following definition recalls the setup of dg-geometry over a field of characteristic 0.
for the category of commutative cochain dg-algebras over , 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 equipped with any subcanonmical sSet-site structure, write
for the (∞,1)-sheaf (∞,1)-topos over it.
The -Yoneda extension of the canonical inclusion
yields an (∞,1)-adjunction
Relative -Toposes over an object
For an (∞,1)-topos and any object, the over-(∞,1)-category
is itself an (∞,1)-topos – the over-(∞,1)-topos over – which is to be thought of as the little topos incarnation of , sitting by an etale geometric morphism over .
We consider this now in the context of dg-geometry.
Let , write for the model structure on dg-modules over .
for the category of commutative monoids in -modules.
There is a model category structure on whose fibrations and weak equivalences are those of the underlying -modules such that the free-forgetful adjunction
is a Quillen adjunction.
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
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.
preserves filtered colimits.
This follows from the general fact creates filtered colimits for closed symmetric monoidal (see here) and that is closed symmetric monoidal (see here).
To check this explicitly:
Let be a filtered diagram. We claim that there is a unique way to lift the underlying colimit to a dg-algebra cocone: for and there is by the assumption that is filtered a . Therefore in order for the cocone component to be an algebra homomorphism the product of with in has to be the image of this product in . This defines the colimiting cocone .
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 here this can be taken to be .
Let be the -topos for dg-geometry discussed above, and . Write for a cofibrant representative of the image of under .
Then the function algebra adjunction from def 1 induces a relative function algebra adjunction
Derived critical loci
In good cases the object defined this way is the formal dual of the tangent complex of the function algebra . This is the actual definition to be used in the following
In the context of dg-geometry, for an object we define
to be the free -algebra on the tangent complex of .
The derived critical locus of in is the (∞,1)-pullback
computed in .
If is -perfect (…) in that preserves this pullback, this is equivalently given by the -pushout
Compare with the situation for Hochschild cohomology of , for -perfect, which is given by the complex of functions on the derived loop space given by the -pullback
For suitable this factors through the infinitesimal neighbourhood of the diagonal hence is the derived self-intersection in the tangent bundle
Cotangent bundle in dg-geometry – the tangent complex
We discuss the derived critical locus in dg-geometry over formal duals of general differential graded algebras.
Let be a field of characteristic 0.
Write for the category of graded-commutative unbounded cochain dg-algebras over .
For an object
Let Mod be the category of dg-modules over equipped with the standard model structure on dg-modules.
for the category of commutative monoids in : the category of commutative dg-algebras under . We regard this as a category with weak equivalences given by the underlying quasi-isomorphisms.
This category models dg-geometry over in that
for the tangent complex/automorphism ∞-Lie algebra of whose underlying cochain complex is
where is the module of derivations
of degree and is the graded commutator of derivations with the differential of regarded as a degree-1 derivation .
We say that is smooth if is cofibrant as an object on .
for the free -algebra over .
for its formal dual.
Every defines a morphism
dually given by
where is the -linear multiplication operator defined by and where for we set
which may be regarded as the multiplication operator given by the commutator of -linear endomorphisms of as indicated.
Derived critical locus in an -Lie algebroid in dg-geometry
The derived critical locus of a morphism is the homotopy pullback in
If is smooth in the sense that is cofibrant, then the derived critical locus is presented by
where on the right we have the free -algebra over the mapping cone of .
By prop 1 the functor is left Quillen. Hence if is cofibrant in then the homotopy pushout in question may be computed as the image under of the homotopy pushout in .
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.
If we extend the graded commutators in the evident way we may write the differential in as
Here the second term is the differential of the BRST-complex of , whereas the sum is of the type of a differential in a BRST-BV complex.
Comparison to BRST-BV complexes
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
starts with a function on ordinary spaces,
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 (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 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 -Lie algebroid is the full BRST-complex of .
Let be a Lie algebroid over a space , with Chevalley-Eilenberg algebra given by
for , infinitesimal gauge symmetries , gauge symmetry structure functions and ghost generators .
The “algebra of vector fields/derivations” on is the automorphism ∞-Lie algebra whose underlying chain complex is
We check on generators that
be a function, dually a dg-algebra homomorphism
This is equivalently any function
which is gauge invariant
We have a contraction homomorphism of -modules
and may form its mapping cone,
On the free algebra of this
we have the differential given on generators by
If is the full kernel of and there are no further relations, then this is the full BRST-BV complex of .
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.