nLab H-cohomology

under construction

Definition

Definition

(H-cohomology)

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

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

The cochain cohomology of this cochain complex

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

is also called the H-cohomology of $X$.

(Cavalcanti 03, p. 19)

Remark

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

It is immediate to consider this more generally:

Definition

(H-cohomology in graded-commutative algebra)

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

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

is a chain complex.

The cochain cohomology of this cochain complex

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

is the H-cohomology of $\mathcal{A}$.

Properties

Detection of decomposability

Proposition

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

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

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

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

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

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

Proof

First consider the following:

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

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

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

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

Observe that this claim immediately implies the statement by contradiction: For the case $\ell = k$ the left hand side of (2) is $1$ by definition (1), while its right-hand side would vanish, because the corresponding elements $\gamma$ 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 $\ell = 0$ we have

$$H \wedge (\alpha_1 \wedge \cdots \wedge \alpha_k) = 0 \,,$$

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

$$\alpha_{\hat \emptyset} \coloneqq \alpha_1 \wedge \cdots \wedge \alpha_k = H \wedge \gamma_{\emptyset} \,.$$

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

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

$$\begin{aligned} H \wedge \alpha_{\widehat{i_1\cdots i_{\ell_0}}} &= \sum_{i=1}^{k} \beta_i \wedge \alpha_i \wedge \alpha_{\widehat{i_1\cdots i_{\ell_0}}} \\ &= \sum_{j=1}^{\ell_0} (-1)^{i_j+j}\beta_{i_j} \wedge \alpha_{\widehat{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}}} \\ &= \sum_{j=1}^{\ell_0} (-1)^{i_1 + \cdots + i_{\ell_0}+\ell_0+j+1}\beta_{i_j} \wedge \left(H \wedge \gamma_{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}}+\sum_{k=1}^{j-1}(-1)^{k+1}\beta_{i_k}\wedge \gamma_{i_1\cdots \widehat{i_k} \cdots \widehat{i_j} \cdots i_{\ell_0}} + \sum_{k=j+1}^{\ell_0}(-1)^k\beta_{i_k}\wedge \gamma_{i_1\cdots \widehat{i_j} \cdots \widehat{i_k} \cdots i_{\ell_0}}\right) \\ &= (-1)^{i_1 + \cdots + i_{\ell_0}+\ell_0}\sum_{j=1}^{\ell_0} H \wedge (-1)^{j+1}\beta_{i_j} \wedge \gamma_{i_1\cdots \widehat{i_j}\cdots i_{\ell_0}} \,, \end{aligned}$$

where in the third line we used the induction assumption.

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

Examples

Odd symplectic forms

Example

(H-cohomology of odd symplectic form)

Let $X$ be a finite dimensional graded manifold in degrees 0 and 1 and equipped with a symplectic form $\omega$ 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 $H = \omega$ has a unique representative which on any local coordinate chart of Darboux coordinates $(x^i, p_i)_{i \in \{1, \cdots, n\}}$ is of the form

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

(Ševera 05, p. 1)

References

The terminology “$H$-cohomology” is used in

• Gil Cavalcanti, New aspects of the $d d^c$-lemma (arXiv:math/0501406)

The case of H-cohomology for $H = \omega$ 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

• Kevin Sackel, MO reply Feb 22 2018