Schreiber derived critical locus

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.

Idea

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

$$\array{ C_{\{d S = 0\}} &\to& C \\ \downarrow && \downarrow^{\mathrlap{0}} \\ C &\stackrel{d S}{\to}& T^* C } \,.$$

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 $\mathfrak{c}$

$$\mathfrak{c}_{\{d S = 0\}} \to \mathfrak{c}$$

is the object whose function algebra is essentially the BRST-BV complex for $S$. See below for more detailed discussion.

The ambient $\infty$-topos

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.

Over derived smooth loci

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

$$C^\infty(\mathbb{R}^k) \coprod C^\infty(\mathbb{R}^l) \to C^\infty(\mathbb{R}^{k+l}) \,.$$

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

$$\mathbf{H} := Sh_{(\infty,1)}(C)$$

for the (∞,1)-category of (∞,1)-sheaves over $C$.

dg-Geometry

The following definition recalls the setup of dg-geometry over a field $k$ of characteristic 0.

Definition/Proposition

Write

$$(cdgAlg_k^-)_{proj}$$

and

$$(cdgAlg_k)_{proj} \,,$$

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

$$Map(A,B) := ([n] \mapsto Hom_{cdgAlg_k})(A, B \otimes_k \Omega^\bullet(\Delta^1)) \in sSet \,.$$

For $(cdgAlg_k^-)^{op}$ equipped with any subcanonical sSet-site structure, write

$$\mathbf{H} := Sh_{(\infty,1)}((cdgAlg_k^-)^{op})$$

for the (∞,1)-sheaf (∞,1)-topos over it.

The $(\infty,1)$-Yoneda extension of the canonical inclusion

$$cdgAlg_k^- \hookrightarrow cdgAlg_k$$

yields an (∞,1)-adjunction

$$(\mathcal{O} \dashv j) : (dgcAlg_k^{op})^\circ \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} Sh_{(\infty,1)}((dgcAlg_k)^-) \,.$$

Relative $\infty$-Toposes over an object

For $\mathbf{H}$ an (∞,1)-topos and $X \in \mathbf{H}$ any object, the over-(∞,1)-category

$$\pi_X : \mathbf{H}/X \stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}} \mathbf{H}$$

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 $\mathcal{O}(C)$.

Definition

Write

$$cdgAlg_{\mathcal{O}(C)} := CMon(\mathcal{O}(C) Mod) \simeq \mathcal{O}(C)/cdgAlg_k$$

for the category of commutative monoids in $\mathcal{O}(C)$-modules.

Proposition

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

$$cdgAlg_{\mathcal{O}(C)} \stackrel{\overset{Sym_{\mathcal{O}(C)}}{\leftarrow}}{\underset{U}{\to}} \mathcal{O}(C) Mod$$

is a Quillen adjunction.

This is

• combinatorial;

• proper.

Proof

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

1. $\mathcal{O}(C) Mod$ is a cofibrantly generated model category.

   This follows because the model structure on dg-modules (as discussed there) is itself transferred along

   $$U' : \mathcal{O}(C) Mod \to Ch^\bullet(k)$$

   from the cofibrantly generated model structure on cochain complexes.

2. $U$ 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 $A_l \to \lim_\to A_\bullet$.

3. 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 $(-)\otimes_k \Omega^\bullet_{poly}(\Delta[1])$.

Proposition

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 induces a relative function algebra adjunction

$$(cdgAlg_{\mathcal{O}(C)})^{op} \simeq cdgAlg_k^{op}/C \stackrel{\overset{\mathcal{O}/C}{\leftarrow}}{\underset{j/C}{\to}} \mathbf{H}/C \,.$$

Proof

This follows with the above, with the general properties of (∞,1)-adjunctions on slices and the discussion at dg-geometry.

Derived critical loci

Definition

Definition

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

$$\theta : \mathbb{A}^1 \to \mathbf{\flat}_{dR} \mathbf{B}\mathbb{A}^1$$

for the intrinsic curvature characteristic form.

Given a morphism

$$S : C \to \mathbb{A}^1$$

we write

$$d S : C \stackrel{S}{\to} \mathbb{A}^1 \stackrel{\theta}{\to} \mathbf{\flat}_{dR}\mathbf{B}\mathbb{A}^1$$

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

Definition

In the context of dg-geometry, for $C \in \mathcal{H}$ an object we define

$$\mathcal{O}(T^*_f C) := Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C)) \in (cdgAlg_k)^\circ$$

to be the free $\mathcal{O}(C)$-algebra on the tangent complex of $\mathcal{O}(C)$.

Definition

The derived critical locus of $S : C \to \mathbb{A}^1$ in $\mathbf{H}$ is the (∞,1)-pullback

$$\array{ C_{\{d S = 0\}} &\to& C \\ \downarrow &\swArrow_{\mathrlap{\simeq}}& \downarrow^{\mathrlap{0}} \\ C &\stackrel{d S}{\to}& T^*_f C }$$

computed in $\mathbf{H}/C$.

Note

If $C$ is $\mathcal{O}$-perfect (…) in that $\mathcal{O}$ preserves this pullback, this is equivalently given by the $(\infty,1)$-pushout

$$\array{ \mathcal{O}(C_{\{d S = 0\}}) &\leftarrow& \mathcal{O}(C) \\ \uparrow && \uparrow^{\iota_0} \\ \mathcal{O}(C) & \stackrel{\iota_{d S}}{\leftarrow} & \mathcal{O}(T^*_f C) } \,.$$

Note

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

$$\array{ \mathcal{L}C &\to& C \\ \downarrow &\swArrow_{\mathrlap{\simeq}}& \downarrow \\ C &\stackrel{}{\to}& C \times C } \,.$$

For suitable $C$ this factors through the infinitesimal neighbourhood of the diagonal hence is the derived self-intersection in the tangent bundle

$$\array{ \mathcal{L}C &\to& C \\ \downarrow &\swArrow_{\mathrlap{\simeq}}& \downarrow^{\mathrlap{0}} \\ C &\stackrel{0}{\to}& T_f C } \,.$$

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 $k$ be a field of characteristic 0.

Write $dgcAlg_k$ for the category of graded-commutative unbounded cochain dg-algebras over $k$.

For an object

$$C \in cdgAlg_k^{op}$$

we write

$$\mathcal{O}(C) \in cdgAlg_k \,.$$

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

$$cdgAlg_{\mathcal{O}(C)} := CMon(\mathcal{O}(C) Mod)$$

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

$$(cdgAlg_{\mathcal{O}(C)})^{op} \simeq cdgAlg_k^{op}/C \,.$$

Definition

Write

$$Der(A) \in \mathcal{O}(C) Mod$$

for the tangent complex/automorphism ∞-Lie algebra of $A$ whose underlying cochain complex is

$$\array{ \cdots \to Der(\mathcal{O}(C))_k \stackrel{[d_{\mathcal{O}(C)},-]}{\to} Der(\mathcal{O}(C))_{k+1} \to \cdots } \,.$$

where $Der(\mathcal{O}(C))_k$ is the module of derivations

$$v : \mathcal{O}(C)^\bullet \to \mathcal{O}(C)^{\bullet + k}$$

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

$$\mathcal{O}(T^*_f C) := Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C)) \in cdgAlg_{\mathcal{O}(C)}$$

for the free $\mathcal{O}(C)$-algebra over $Der(\mathcal{O}(C))$.

We write

$$T^*_f C \in cdgAlg^{op}_k/C$$

for its formal dual.

Remark

Every $S \in \mathcal{O}(C)$ defines a morphism

$$d S : C \to T^*_f C$$

dually given by

$$\mathcal{O}(C) \leftarrow Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C)) : Sym_{\mathcal{O}(C)} [\hat S , -] \,,$$

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

$$[\hat S, v] = v(S) \,,$$

which may be regarded as the multiplication operator given by the commutator of $k$-linear endomorphisms of $\mathcal{O}(C)$ as indicated.

Derived critical locus in an $\infty$-Lie algebroid in dg-geometry

Definition

The derived critical locus of a morphism $S : C \to \mathbb{A}^1$ is the homotopy pullback $C_{\{d S = 0\}}$ in $cdgAlg^{op}/C$

$$\array{ C_{\{d S = 0\}} &\to& C \\ \downarrow && \downarrow \\ C &\stackrel{d S}{\to}& T^*_f C } \,.$$

Proposition

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

$$\mathcal{O}(C_{\{d S = 0\}}) \simeq Sym_{\mathcal{O}(C)} Cone( Der(\mathcal{O}(C)) \stackrel{[\hat S , -]}{\to} \mathcal{O}(C)) \,,$$

where on the right we have the free $\mathcal{O}(C)$-algebra over the mapping cone of $[\hat S, -]$.

Proof

By prop 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.

$$\array{ Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C)) &\leftarrow& Cone(Der(\mathcal{O}(C)) \stackrel{Id}{\to} Der(\mathcal{O}(C))) \\ \uparrow && \uparrow \\ \mathcal{O}(C) &\stackrel{[\hat S, -]}{\leftarrow}& Der(\mathcal{O}(C)) } \,.$$

More in detail, write

$$Cone(Der(\mathcal{O}(C)) \stackrel{Id}{\to} Der(\mathcal{O}(C))) \in \mathcal{O}(C) Mod$$

for the mapping cone on the identity.

$$\array{ \cdots & Der(\mathcal{O}(C))_k &\stackrel{-[d_{\mathcal{O}(C)}, -]}{\to}& Der(\mathcal{O}(C))_k+1 \\ & \oplus &\searrow^{\pm \mathrlap{Id}}& \oplus & \cdots \\ \cdots & Der(\mathcal{O}(C))_{k-1} &\stackrel{[d_{\mathcal{O}(C)}, -]}{\to}& Der(\mathcal{O}(C))_k & \cdots } \,.$$

Then $Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to}) \mathcal{O}(C))$ is

(1)$$\array{ \cdots & Der(\mathcal{O}(C))_k &\stackrel{-[d_{\mathcal{O}(C)}, -]}{\to}& Der(\mathcal{O}(C))_k+1 \\ & \oplus &\searrow^{\pm [\hat S, -]}& \oplus & \cdots \\ \cdots & \mathcal{O}(C)_{k-1} &\stackrel{[d_{\mathcal{O}(C)}, -]}{\to}& \mathcal{O}(C)_k & \cdots } \,.$$

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

$$d = [\hat S + d_{\mathcal{O}(C)}, -] \,.$$

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.

Comparison to BRST-BV complexes

We discuss how the traditional BRST-BV formalism relates to the computation of derived critical loci as above.

General

The traditional approach in BRST-BV formalism starts from a somewhat different angle than the discussion here. There on

1. starts with a function $S$ on ordinary spaces,

2. then builds a Koszul-Tate resolution of its ordinary critical locus;

3. 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”);

4. 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

1. the gauge symmetries close off-shell;

2. the $\infty$-Lie algebroid $C$ is the full BRST-complex of $S$.

(…)

Example

Let $\mathfrak{a}$ be a Lie algebroid over a space $X$, with Chevalley-Eilenberg algebra $\mathcal{O}(\mathfrak{a})$ given by

$$d_{\mathfrak{a}} : f \mapsto c^a R^i_a \frac{\partial}{\partial x^i} f$$

$$d_{\mathfrak{a}} : c^a \mapsto \frac{1}{2} C^{a}{}_{b c} c^b \wedge c^c \,.$$

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

$$\array{ \langle \frac{\partial}{\partial c^a} \rangle & \stackrel{[d_{\mathfrak{a}}, -]}{\to} & \langle \frac{\partial}{\partial x^i} \rangle \oplus \langle c^a \frac{\partial}{\partial c^b} \rangle \\ -1 && 0 } \,.$$

We check on generators that

$$\begin{aligned} [d_{\mathfrak{a}}, \frac{\partial}{\partial c^a}] = R_a^i \frac{\partial}{\partial x^i} + C^b{}_{a c} c^c \frac{\partial}{\partial c^b} \end{aligned}$$

and

$$\begin{aligned} [d_{\mathfrak{a}}, \frac{\partial}{\partial x^i}] = c^a \frac{\partial R_a^j}{\partial x^i} \frac{\partial}{\partial x^j} \end{aligned} \,.$$

Now let

$$S : \mathfrak{a} \to \mathbb{R}$$

be a function, dually a dg-algebra homomorphism

$$\mathcal{O}(\mathfrak{a}) \leftarrow C^\infty(\mathbb{R}) : S^* \,.$$

This is equivalently any function

$$S : X \to \mathbb{R}$$

which is gauge invariant

$$d_{\mathfrak{a}} S = c^a R_a^i \frac{\partial}{\partial x^i} S = 0 \,.$$

We have a contraction homomorphism of $\mathcal{O}(\mathfrak{a})$-modules

$$\iota_{d S} : Der(\mathcal{O}(\mathfrak{a})) \to \mathcal{O}(\mathfrak{a}) \,.$$

and may form its mapping cone,

$$\array{ \langle \frac{\partial}{\partial c^a}\rangle &\stackrel{[d_{\mathfrak{a}}, -]}{\to}& \langle \frac{\partial}{\partial x^i} \rangle \oplus \langle c^a \frac{\partial}{\partial c^b}\rangle &\stackrel{\iota_{d S}}{\to}& C^\infty(X) &\stackrel{d_{\mathfrak{a}}}{\to}& \langle c^a \rangle \\ -2 && -1 && 0 && 1 } \,.$$

On the free algebra of this

$$Sym_{C^\infty(X)} \left( Der(\mathcal{O}(\mathfrak{a}))[-1] \stackrel{\iota_{d S}}{\to} C^\infty(X)\oplus \langle c^a\rangle) \right) \,.$$

we have the differential given on generators by

$$\frac{\partial}{\partial c^a} \mapsto R_a^i \frac{\partial}{\partial x^i} + C^b{}_{a c} c^c \frac{\partial}{\partial c^b}$$

$$\frac{\partial}{\partial x^i} \mapsto \frac{\partial S}{\partial x^i} + c^a \frac{\partial R_a^j}{\partial x^i} \frac{\partial}{\partial x^j}$$

$$x^i \mapsto c^a R_a^i$$

$$c^a \mapsto \frac{1}{2}C^a{}_{b c} c^b \wedge c^c$$

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 $S$.

References

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

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory – Derived critical locus (weblass=‘newWikiWord’>Derived%20critical%20locus?</span>))

For references on BRST-BV formalism see there.