nLab derived critical locus

Idea

The concept of derived critical locus is the refinement of the notion of critical locus from geometry to derived geometry.

The formal duals to derived critical loci are described by BV-BRST formalism.

Details

The following is a basic setup in dg-geometry aimed at exhibiting the (formal dual of) the BV-BRST complex as the derived critical locus of (the formal dual of) the BRST complex (the example below)

Basic dg-geometry

Let $k$ be a field of characteristic zero.

Write $dgcAlg_k$ for the category of unbounded cochain differential graded-commutative algebras (dgc-algebras) over $k$.

An object in the opposite category

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

we may regard as an affine space in dg-geometry, and hence we write

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

for the corresponding dgc-algebra.

Let $\mathcal{O}(C)$Mod be the category of dg-modules over $\mathcal{O}(C)$ equipped with the standard model structure on dg-modules.

Definition

(dgc-algebras over $\mathcal{O}(C)$)

Write

$$\begin{aligned} cdgAlg_{\mathcal{O}(C)} & \coloneqq CMon(\mathcal{O}(C) Mod) \\ & \simeq \mathcal{O}(C)/_{cdgAlg_k} \end{aligned}$$

for the category of commutative monoids in $\mathcal{O}(C)$-modules, equivalently the coslice category of $cdgAlg_k$ under $\mathcal{O}(C)$.

Proposition

There is a model category structure on $cdgAlg_{\mathcal{O}(C)}$ (def. ) whose fibrations and weak equivalences are those of the underlying $\mathcal{O}(C)$-modules such that the free-forgetful adjunction

$$cdgAlg_{\mathcal{O}(C)} \underoverset {\underset{U}{\longrightarrow}} {\overset{Sym_{\mathcal{O}(C)}}{\longleftarrow}} {\bot} \mathcal{O}(C) Mod$$

is a Quillen adjunction.

This is

1. combinatorial;

2. 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

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

   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 there) and that $A Mod$ is closed symmetric monoidal (see there).

   To check this explicitly:

   Let $A_\bullet \colon D \to cdgAlg_k$ 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 $(-)\otimes_k \Omega^\bullet_{poly}(\Delta[1])$.

Formal cotangent bundle

Given $C \in dgcAlg_k^{op}$ as above we want to consider its formal cotangent bundle $T^\ast_f c$, i.e. the infinitesimal neighbourhood around the zero section of the would-be actual cotangent bundle

Definition

Write

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

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

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

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

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

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 $d_{\mathcal{O}(C)} \colon \mathcal{O}(C) \to \mathcal{O}(C)$.

We say that $\mathcal{O}(C)$ is smooth if $Der(\mathcal{O}(C))$ is cofibrant as an object on $\mathcal{O}(C) Mod$.

Write

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

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

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

in $dgcAlg_k^{op}$ which is dually given by

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

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

Definition

(derived critical locus)

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

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

Proposition

(presentation by free dgc-algebra on mapping cone)

If $C$ is smooth in the sense that $Der(\mathcal{O}(C)) \in \mathcal{O}(C) Mod$ is cofibrant, then the derived critical locus (def. ) is presented by

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

where on the right we have the free $\mathcal{O}(C)$-algebra over the mapping cone of $[\hat S, -]$ with extension of scalars along $\mathcal{O}( C \overset{(id,0)}{\to} C \times \mathbb{A}^1 )$.

Proof

Using the pasting law we may decompose the homotopy pullback into a pasting of two homotopy pullback squares as follows:

$$\array{ C_{\{d S = 0\}} &\longrightarrow& &\longrightarrow& C \\ \downarrow &\swArrow& \downarrow &\swArrow& \downarrow^{\mathrlap{0}} \\ C &\underset{(id,0)}{\longrightarrow}& C \times \mathbb{A}^1 &\underset{}{\longrightarrow}& T^*_f C } \,.$$

First consider the square on the right:

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 corresponding to the square on the right 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.

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

More in detail, write

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

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 the mapping cone $Cone\left(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C) \right)$ 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

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

Here the second term will be the differential of the BRST-complex of $\mathfrak{c}$, whereas the sum is of the type of a differential in a BRST-BV complex.

For that to happen, however the two copies of $\mathcal{O}(C)$ in $Sym_{\mathcal{O}(C)}(\mathcal{O}(C))$ need to be identified, this is achived by the remaining homotopy pushout corresponding to the square on the left

$$\array{ \mathcal{O}(C_{\{d S = 0\}}) &\longleftarrow& Sym_{\mathcal{O}(C)} \left( Cone\left( Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C) \right) \right) \\ \uparrow && \uparrow \\ \mathcal{O}(C) &\underset{}{\longleftarrow}& Sym_{\mathcal{O}(C)}(\mathcal{O}(C)) } \,.$$

Since here the morphism on the right is the pushout of a cofibration, it is itself still a cofibration, and by assumption $Sym_{\mathcal{O}(C)}(\mathcal{O}(C))$ is cofibrant. Therefore this homotopy pushout is given by the ordinary pushout, and that yields the tensor product as in the claim.

BV-BRST complex

Traditionally the BV-BRST complex of a Lagrangian field theory is obtained by

1. choosing a Koszul-Tate complex $s_{KT}$ resolving the shell;

2. choosing a BRST complex $s_{BRST}$ exhibiting the gauge invariance

3. appealing to homological perturbation theory, for extending the sum of the two differentials to a unified BV-BRST differential

   $$s_{BV} = s_{KT} + s_{BRST} + more$$

(e.g. Henneaux 90. around (50))

Vie the concept of the derived critical locus this process is systematized: Given just $s_{BRST}$ and the Lagrangian, both $s_{KT}$ and “more” follows (if $s_{BRST}$ does capture all the relevant gauge symmetries, that is) and the appearance of the antibracket finds its conceptual explanation.

Let $\mathfrak{a}$ be a Lie algebroid over a space $X$, with Chevalley-Eilenberg algebra

$$\mathcal{O}(\mathfrak{a}) \;=\; \left( Sym_{\mathcal{O}(X)}(\langle c^a\rangle) \;,\; d_{\mathfrak{a}} \right)$$

with differential 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 functions $f \in \mathcal{O}(X)$, 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 cochain complex is

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

We check on generators that

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

and

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

Now let

$$S \;\colon\; \mathfrak{a} \longrightarrow \mathbb{R}$$

be a morphism, dually a dgc-algebra homomorphism of the form

$$\mathcal{O}(\mathfrak{a}) \longleftarrow \mathcal{O}(\mathbb{R}) \;\colon\; S^* \,.$$

This is equivalently a function

$$S \;\colon\; X \longrightarrow \mathbb{R}$$

which is gauge invariant

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

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

$$\iota_{d S} \;\colon\; Der(\mathcal{O}(\mathfrak{a})) \longrightarrow \mathcal{O}(\mathfrak{a}) \,.$$

and may form its mapping cone (1)

$$\array{ \left\langle \frac{\partial}{\partial c^a} \right\rangle &\stackrel{[d_{\mathfrak{a}}, -]}{\longrightarrow}& \left\langle c^a \frac{\partial}{\partial c^b} \right\rangle \oplus \left\langle \frac{\partial}{\partial x^i} \right\rangle \\ && &\searrow^{\mathrlap{\iota_{d S}}}& \\ && && \mathcal{O}(X) &\stackrel{d_{\mathfrak{a}}}{\to}& \left\langle c^a \right\rangle \\ -2 && -1 && 0 && 1 } \,.$$

On the free algebra of this

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

we have the differential given on generators by

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

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

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

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

and similarly after tensoring in order to identify the extra copy of $\mathcal{O}(X)$ with the base $\mathcal{O}(X)$.

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 above material is adapted from

• Urs Schreiber: derived critical locus Seminar notes, March 2011

(taking into account a correction provided by Vincent Schlegel)

aimed at providing proof for the claim in

• Kevin Costello, Owen Gwilliam, section 4.8.1 of Factorization algebras in perturbative quantum field theory (pdf)

See also

• Gabriele Vezzosi, Derived critical loci I - Basics, arxiv/1109.5213

• Tony Pantev, Bertrand Toen, M. Vaquie, G. Vezzosi, Quantization and derived moduli spaces I: shifted symplectic structures, arxiv/1111.3209

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimension 2 and 4 (pdf)