nLab H-cohomology

Contents

under construction

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebra

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

Definition

(H-cohomology)

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)

Remark

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:

Definition

(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}.

Properties

Detection of decomposability

Proposition

(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.

Proof

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.

Examples

Odd symplectic forms

Example

(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)

References

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.