nLab H-cohomology


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from point-set topology to differentiable manifolds

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infinitesimal cohesion

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Given a smooth manifold or formal smooth manifold XX, and given a differential 3-form HΩ 3(X)H \in \Omega^3(X), then forming the wedge product with HH is a nilpotent operation, and hence defines the differential of a cochain complex whose entries are the vector spaces Ω even/odd(X)\Omega^{even/odd}(X) of all odd-graded or all even-graded differential forms on XX:

Ω even(X)H()Ω odd(X)H()Ω even(X)H()Ω odd(X). \cdots \to \Omega^{even}(X) \overset{H\wedge(-)}{\longrightarrow} \Omega^{odd}(X) \overset{H\wedge(-)}{\longrightarrow} \Omega^{even}(X) \overset{H\wedge(-)}{\longrightarrow} \Omega^{odd}(X) \to \cdots \,.

The cochain cohomology of this cochain complex

H H even/odd(X);ker(Ω even/odd(X)H()Ω odd/even(X))im(Ω odd/even(X)H()Ω even/odd(X)) H^{even/odd}_{H\wedge}(X) \coloneqq; \frac{ ker \left( \Omega^{even/odd}(X) \overset{H\wedge(-)}{\longrightarrow} \Omega^{odd/even}(X) \right) }{ im \left( \Omega^{odd/even}(X) \overset{H\wedge(-)}{\longrightarrow} \Omega^{even/odd}(X) \right) }

is also called the H-cohomology of XX.

(Cavalcanti 03, p. 19)


Often it is assumed that the 3-form HH in def. is closed, because in this case H-cohomology serves as an approximation to the corresponding HH-twisted de Rham cohomology (prop. below). But of course for def. to make sense in itself, HH need not be closed.

It is immediate to consider this more generally:


(H-cohomology in graded-commutative algebra)

Let 𝒜\mathcal{A} be a graded commutative algebra in characteristic zero and let H𝒜H \in \mathcal{A} be an odd-graded element. This implies that consecutive multiplication with HH

𝒜H()𝒜H()𝒜H()𝒜 \cdots \to \mathcal{A} \overset{H \cdot(-)}{\longrightarrow} \mathcal{A} \overset{H \cdot (-)}{\longrightarrow} \mathcal{A} \overset{H \cdot (-)}{\longrightarrow} \mathcal{A} \to \cdots

is a chain complex.

The cochain cohomology of this cochain complex

H H even/odd(𝒜);ker(𝒜 even/oddH()𝒜 odd/even)im(𝒜 odd/evenH()𝒜 even/odd) H^{even/odd}_{H\cdot}(\mathcal{A}) \coloneqq; \frac{ ker \left( \mathcal{A}_{even/odd} \overset{H \cdot(-)}{\longrightarrow} \mathcal{A}_{odd/even} \right) }{ im \left( \mathcal{A}_{odd/even} \overset{H\cdot(-)}{\longrightarrow} \mathcal{A}_{even/odd} \right) }

is the H-cohomology of 𝒜\mathcal{A}.


Detection of decomposability


(H-cohomology of weakly decomposable forms is non-vanishing)

Let 𝒜\mathcal{A} be a \mathbb{N}-graded-commutative algebra. and let H𝒜H \in \mathcal{A} be of odd degree deg(H)=2n+1deg(H) = 2 n +1

If HH is weakly decomposable in that it is in the ideal generated by the elements of degree 1

H=i=1kα ideg=1β ideg=2n H \;=\; \underoverset{i = 1}{k}{\sum} \underset{deg = 1}{\underbrace{\alpha_i}} \wedge \underset{deg = 2n}{\underbrace{\beta_i}}

then its H-cohomology (def. ) does not vanish.

This statement and the following proof are due to Sackel 18.


First consider the following:

Claim: Assuming that the H-cohomology of HH did vanish, then for every sequence of natural numbers 1i 1<<i k1 \leq i_1 \lt \ldots \lt i_{\ell} \leq k, there would exist elements γ i 1i 𝒜\gamma_{i_1\cdots i_{\ell}} \in \mathcal{A} of degree deg(γ i 1i )=k3deg(\gamma_{i_1\cdots i_{\ell}}) = k-3-\ell (vanishing if >k3\ell \gt k-3) such that the products of α\alpha-s with indices not contained in this sequence

(1)α i 1i ^{1 | =k α 1α i 1^α i ^α k | 1<<k α 1α k | =0 \alpha_{\widehat{i_1\cdots i_{\ell}}} \;\coloneqq\; \left\{ \array{ 1 &\vert& \ell = k \\ \alpha_1 \wedge \cdots \wedge \widehat{\alpha_{i_1}} \wedge \cdots \wedge \widehat{\alpha_{i_{\ell}}} \wedge \cdots \wedge \alpha_k &\vert& 1 \lt \ell \lt k \\ \alpha_1 \wedge \cdots \wedge \alpha_k &\vert& \ell =0 } \right.

(hats indicate that the corresponding element is omitted) would satisfy

(2)α i 1i ^=(1) i 1++i +(Hγ i 1i + j=1 (1) j+1β i jγ i 1i j^i ). \alpha_{\widehat{i_1\cdots i_{\ell}}} \;=\; (-1)^{i_1+\cdots+i_{\ell}+\ell} \left( H \wedge \gamma_{i_1\cdots i_{\ell}} + \sum_{j=1}^{\ell}(-1)^{j+1} \beta_{i_j} \wedge \gamma_{i_1\cdots\widehat{i_j}\cdots i_{\ell}} \right) \,.

Observe that this claim immediately implies the statement by contradiction: For the case =k\ell = k the left hand side of (2) is 11 by definition (1), while its right-hand side would vanish, because the corresponding elements γ\gamma are in negative degree. This is a contradiction, implying that the assumption of vanishing H-cohomology must be wrong.

Hence it is now sufficient to prove the above claim. We proceed by induction on \ell:

For =0\ell = 0 we have

H(α 1α k)=0, H \wedge (\alpha_1 \wedge \cdots \wedge \alpha_k) = 0 \,,

and so by the assumption of vanishing HH-cohomology, there exists γ \gamma_{\emptyset} with

α ^α 1α k=Hγ . \alpha_{\hat \emptyset} \coloneqq \alpha_1 \wedge \cdots \wedge \alpha_k = H \wedge \gamma_{\emptyset} \,.

This is (2) for =0\ell = 0.

So assume now that there is 1k1 \leq \ell \leq k such that (2) holds for all < 0\ell \lt \ell_0. Then

Hα i 1i 0^ = i=1 kβ iα iα i 1i 0^ = j=1 0(1) i j+jβ i jα i 1i j^i 0^ = j=1 0(1) i 1++i 0+ 0+j+1β i j(Hγ i 1i j^i 0+ k=1 j1(1) k+1β i kγ i 1i k^i j^i 0+ k=j+1 0(1) kβ i kγ i 1i j^i k^i 0) =(1) i 1++i 0+ 0 j=1 0H(1) j+1β i jγ i 1i j^i 0, \begin{aligned} H \wedge \alpha_{\widehat{i_1\cdots i_{\ell_0}}} &= \sum_{i=1}^{k} \beta_i \wedge \alpha_i \wedge \alpha_{\widehat{i_1\cdots i_{\ell_0}}} \\ &= \sum_{j=1}^{\ell_0} (-1)^{i_j+j}\beta_{i_j} \wedge \alpha_{\widehat{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}}} \\ &= \sum_{j=1}^{\ell_0} (-1)^{i_1 + \cdots + i_{\ell_0}+\ell_0+j+1}\beta_{i_j} \wedge \left(H \wedge \gamma_{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}}+\sum_{k=1}^{j-1}(-1)^{k+1}\beta_{i_k}\wedge \gamma_{i_1\cdots \widehat{i_k} \cdots \widehat{i_j} \cdots i_{\ell_0}} + \sum_{k=j+1}^{\ell_0}(-1)^k\beta_{i_k}\wedge \gamma_{i_1\cdots \widehat{i_j} \cdots \widehat{i_k} \cdots i_{\ell_0}}\right) \\ &= (-1)^{i_1 + \cdots + i_{\ell_0}+\ell_0}\sum_{j=1}^{\ell_0} H \wedge (-1)^{j+1}\beta_{i_j} \wedge \gamma_{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}} \,, \end{aligned}

where in the third line we used the induction assumption.

By vanishing of HH-cohomology, this implies (2) for = 0\ell = \ell_0, thus finishing the proof of the claim, by induction.


Odd symplectic forms


(H-cohomology of odd symplectic form)

Let XX be a finite dimensional graded manifold in degrees 0 and 1 and equipped with a symplectic form ω\omega of odd coordinate degree (e.g. the symplectic form on a Poisson Lie algebroid when regarded as a symplectic Lie n-algebroid).

Then every H-cohomology-class (def. ) for H=ωH = \omega has a unique representative which on any local coordinate chart of Darboux coordinates (x i,p i) i{1,,n}(x^i, p_i)_{i \in \{1, \cdots, n\}} is of the form

f(x,p)dx 1dx n. f(x,p) d x^1 \wedge \cdots \wedge d x^n \,.

(Ševera 05, p. 1)


The terminology “HH-cohomology” is used in

The case of H-cohomology for H=ωH = \omega an odd symplectic form (as on a Poisson Lie algebroid regarded as a symplectic Lie n-algebroid) is considered in

  • Pavol Ševera, p. 1 of On the origin of the BV operator on odd symplectic supermanifolds, Lett Math Phys (2006) 78: 55. (arXiv:0506331)

The general non-vanishing of H-cohomology is pointed out in

Last revised on March 5, 2018 at 08:06:05. See the history of this page for a list of all contributions to it.