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.

Contents

Idea

Given an ordinary space CC and a function S:C𝔸 1S : C \to \mathbb{A}^1, the critical locus C {dS=0}C_{\{d S = 0\}} of SS is the subspace on which the de Rham differential of SS vanishes, the fiber product

C {dS=0} C 0 C dS T *C. \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 CC 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 CC is its ∞-Lie algebroid 𝔠C\mathfrak{c} \hookrightarrow C. In the context of gauge theory its function algebra 𝒪(𝔠)\mathcal{O}(\mathfrak{c}) is called a BRST complex.

We demonstrate that the derived critical locus of SS restricted to 𝔠\mathfrak{c}

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

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

The ambient \infty-topos

A full formalization of the following in cohesive homotopy type theory is now in cohesive (infinity,1)-topos – infinitesimal cohesion – critical locus.

Recall the general setup of derived geometry over a given (∞,1)-algebraic theory TT: we take formal duals of a small collection of \infty-algebras over TT 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=T = CartSp be the Lawvere theory for smooth algebras, regarded as an (∞,1)-algebraic theory. Write SmoothAlg 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][CartSp, sSet], left Bousfield localized at the morphism of the form

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

Let CartSpCSmoothAlg 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

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

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

dg-Geometry

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

Definition/Proposition

Write

(cdgAlg k ) proj (cdgAlg_k^-)_{proj}

and

(cdgAlg k) proj, (cdgAlg_k)_{proj} \,,

for the category of commutative cochain dg-algebras over kk, 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]Hom cdgAlg k)(A,B kΩ (Δ 1))sSet. Map(A,B) := ([n] \mapsto Hom_{cdgAlg_k})(A, B \otimes_k \Omega^\bullet(\Delta^1)) \in sSet \,.

For (cdgAlg k ) op(cdgAlg_k^-)^{op} equipped with any subcanonmical sSet-site structure, write

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

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

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

cdgAlg k cdgAlg k cdgAlg_k^- \hookrightarrow cdgAlg_k

yields an (∞,1)-adjunction

(𝒪j):(dgcAlg k op) Sh (,1)((dgcAlg k) ). (\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 H\mathbf{H} an (∞,1)-topos and XHX \in \mathbf{H} any object, the over-(∞,1)-category

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

is itself an (∞,1)-topos – the over-(∞,1)-topos over XX – which is to be thought of as the little topos incarnation of XX, sitting by an etale geometric morphism π X\pi_X over H\mathbf{H}.

We consider this now in the context of dg-geometry.

Let 𝒪(C)cdgAlg k\mathcal{O}(C) \in cdgAlg_k, write 𝒪(C)Mod\mathcal{O}(C) Mod for the model structure on dg-modules over 𝒪(C)\mathcal{O}(C).

Definition

Write

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

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

Proposition

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

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

is a Quillen adjunction.

This is

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

    from the cofibrantly generated model structure on cochain complexes.

  2. UU preserves filtered colimits.

    This follows from the general fact U:CMon(𝒞)𝒞U : CMon(\mathcal{C}) \to \mathcal{C} creates filtered colimits for 𝒞\mathcal{C} closed symmetric monoidal (see here) and that AModA Mod is closed symmetric monoidal (see here).

    To check this explicitly:

    Let A :DcdgAlg kA_\bullet : D \to cdgAlg_k be a filtered diagram. We claim that there is a unique way to lift the underlying colimit lim UA \lim_\to U A_\bullet to a dg-algebra cocone: for aA ilim UA a \in A_i \to \lim_\to U A_\bullet and bA jlim UA b \in A_j \to \lim_\to U A_\bullet there is by the assumption that DD is filtered a A iA lA jA_i \to A_l \leftarrow A_j. Therefore in order for the cocone component UA llim UA U A_l \to \lim_{\to} U A_\bullet to be an algebra homomorphism the product of aa with bb in lim UA \lim_\to U A_\bullet has to be the image of this product in A lA_l. This defines the colimiting cocone A llim A 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 kcdgAlg_k here this can be taken to be () kΩ poly (Δ[1])(-)\otimes_k \Omega^\bullet_{poly}(\Delta[1]).

Proposition

Let H\mathbf{H} be the (,1)(\infty,1)-topos for dg-geometry discussed above, and CHC \in \mathbf{H}. Write 𝒪(C)cdgAlg k\mathcal{O}(C) \in cdgAlg_k for a cofibrant representative of the image of CC under 𝒪\mathcal{O}.

Then the function algebra adjunction from def 1 induces a relative function algebra adjunction

(cdgAlg 𝒪(C)) opcdgAlg k op/Cj/C𝒪/CH/C. (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 H\mathbf{H} be a cohesive (∞,1)-topos with differential cohesion and with line object 𝔸 1\mathbb{A}^1.

Write dRB𝔸 1\mathbf{\flat}_{dR} \mathbf{B}\mathbb{A}^1 for the canonical de Rham coefficient object and and

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

for the intrinsic curvature characteristic form.

Given a morphism

S:C𝔸 1 S : C \to \mathbb{A}^1

we write

dS:CS𝔸 1θ dRB𝔸 1 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 *CCT^* C \to C for the object that represents H(C, dRB𝔸 1)\mathbf{H}(C,\mathbf{\flat}_{dR} \mathbf{B}\mathbb{A}^1) in H /C et\mathbf{H}_{/C}^{et}, see at cohesive (infinity,1)-topos – infinitesimal cohesion – structure sheaves.

In good cases the object T f *CT^*_f C defined this way is the formal dual of the tangent complex of the function algebra 𝒪(C)\mathcal{O}(C). This is the actual definition to be used in the following

Definition

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

𝒪(T f *C):=Sym 𝒪(C)Der(𝒪(C))(cdgAlg k) \mathcal{O}(T^*_f C) := Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C)) \in (cdgAlg_k)^\circ

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

Definition

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

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

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

Note

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

𝒪(C {dS=0}) 𝒪(C) ι 0 𝒪(C) ι dS 𝒪(T f *C). \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 𝒪(C)\mathcal{O}(C), for CC 𝒪\mathcal{O}-perfect, which is given by the complex 𝒪((C))\mathcal{O}(\mathcal{L}(C)) of functions on the derived loop space given by the (,1)(\infty,1)-pullback

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

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

C C 0 C 0 T fC. \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 kk be a field of characteristic 0.

Write dgcAlg kdgcAlg_k for the category of graded-commutative unbounded cochain dg-algebras over kk.

For an object

CcdgAlg k op C \in cdgAlg_k^{op}

we write

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

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

Write finally

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

for the category of commutative monoids in 𝒪(C)Mod\mathcal{O}(C) Mod: the category of commutative dg-algebras under 𝒪(C)\mathcal{O}(C). We regard this as a category with weak equivalences given by the underlying quasi-isomorphisms.

This category models dg-geometry over CC in that

(cdgAlg 𝒪(C)) opcdgAlg k op/C. (cdgAlg_{\mathcal{O}(C)})^{op} \simeq cdgAlg_k^{op}/C \,.
Definition

Write

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

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

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

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

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

of degree kk and [d 𝒪(C),][d_{\mathcal{O}(C)}, -] is the graded commutator of derivations with the differential of 𝒪(C)\mathcal{O}(C) regarded as a degree-1 derivation d 𝒪(C):𝒪(C)𝒪(C)d_{\mathcal{O}(C)} : \mathcal{O}(C) \to \mathcal{O}(C).

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

Write

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

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

We write

T f *CcdgAlg k op/C T^*_f C \in cdgAlg^{op}_k/C

for its formal dual.

Remark

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

dS:CT f *C d S : C \to T^*_f C

dually given by

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

where S^:𝒪(C)𝒪(C)\hat S : \mathcal{O}(C) \to \mathcal{O}(C) is the kk-linear multiplication operator defined by SS and where for vDer(𝒪(C))v \in Der(\mathcal{O}(C)) we set

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

which may be regarded as the multiplication operator given by the commutator of kk-linear endomorphisms of 𝒪(C)\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:C𝔸 1S : C \to \mathbb{A}^1 is the homotopy pullback C {dS=0}C_{\{d S = 0\}} in cdgAlg op/CcdgAlg^{op}/C

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

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

𝒪(C {dS=0})Sym 𝒪(C)Cone(Der(𝒪(C))[S^,]𝒪(C)), \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 𝒪(C)\mathcal{O}(C)-algebra over the mapping cone of [S^,][\hat S, -].

Proof

By prop 1 the functor Sym 𝒪(C)Sym_{\mathcal{O}(C)} is left Quillen. Hence if Der(𝒪(C))Der(\mathcal{O}(C)) is cofibrant in 𝒪(C)Mod\mathcal{O}(C) Mod then the homotopy pushout in question may be computed as the image under Sym 𝒪(C)Sym_{\mathcal{O}(C)} of the homotopy pushout in 𝒪(C)Mod\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.

Cone(Der(𝒪(C))[S^,]𝒪(C)) Cone(Der(𝒪(C))IdDer(𝒪(C))) 𝒪(C) [S^,] Der(𝒪(C)). \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(𝒪(C))IdDer(𝒪(C)))𝒪(C)Mod Cone(Der(\mathcal{O}(C)) \stackrel{Id}{\to} Der(\mathcal{O}(C))) \in \mathcal{O}(C) Mod

for the mapping cone on the identity.

Der(𝒪(C)) k [d 𝒪(C),] Der(𝒪(C)) k+1 ±Id Der(𝒪(C)) k1 [d 𝒪(C),] Der(𝒪(C)) k . \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(𝒪(C))[S^,])𝒪(C))Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to}) \mathcal{O}(C)) is

(1) Der(𝒪(C)) k [d 𝒪(C),] Der(𝒪(C)) k+1 ±[S^,] 𝒪(C) k1 [d 𝒪(C),] 𝒪(C) k . \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(𝒪(C))[S^,]𝒪(C))Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C)) as

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

Here the second term is the differential of the BRST-complex of 𝔠\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 SS 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 XX (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 SS 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 CC is the full BRST-complex of SS.

(…)

Example

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

d 𝔞:fc aR a ix if d_{\mathfrak{a}} : f \mapsto c^a R^i_a \frac{\partial}{\partial x^i} f
d 𝔞:c a12C a bcc bc c. d_{\mathfrak{a}} : c^a \mapsto \frac{1}{2} C^{a}{}_{b c} c^b \wedge c^c \,.

for fC (X)f \in C^\infty(X), infinitesimal gauge symmetries R a ix iR^i_a \frac{\partial}{\partial x^i}, gauge symmetry structure functions C a bcC^{a}{}_{b c} and ghost generators c ac^a.

The “algebra of vector fields/derivations” Der(𝒪(𝔞))Der(\mathcal{O}(\mathfrak{a})) on 𝔞\mathfrak{a} is the automorphism ∞-Lie algebra whose underlying chain complex is

c a [d 𝔞,] x ic ac b 1 0. \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

[d 𝔞,c a]=R a ix i+C b acc cc b \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

[d 𝔞,x i]=c aR a jx ix j. \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:𝔞 S : \mathfrak{a} \to \mathbb{R}

be a function, dually a dg-algebra homomorphism

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

This is equivalently any function

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

which is gauge invariant

d 𝔞S=c aR a ix iS=0. d_{\mathfrak{a}} S = c^a R_a^i \frac{\partial}{\partial x^i} S = 0 \,.

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

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

and may form its mapping cone,

c a [d 𝔞,] x ic ac b ι dS C (X) d 𝔞 c a 2 1 0 1. \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 (X)(Der(𝒪(𝔞))[1]ι dSC (X)c a)). 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

c aR a ix i+C b acc cc b \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}
x iSx i+c aR a jx ix j \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 ic aR a i x^i \mapsto c^a R_a^i
c a12C a bcc bc c c^a \mapsto \frac{1}{2}C^a{}_{b c} c^b \wedge c^c

If R a\langle R_a\rangle is the full kernel of ι dS:Der(C (X))C (X)\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 SS.

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

For references on BRST-BV formalism see there.

Revised on May 9, 2013 17:14:23 by Urs Schreiber (89.204.130.249)