nLab non-abelian de Rham cohomology

Idea

For $X$ a smooth manifold, the traditional coboundary-relation which defines the ordinary de Rham cohomology-classes $[\omega] \in H^n{dR}(X)$ of closed differential n-forms $\omega \in \Omega_{dR}^n(X)_{clsd}$

$$[\omega] = [\omega'] \;\;\;\; \Leftrightarrow \;\;\;\; \underset{ \alpha \in \Omega_{dR}^{n-1}(X) }{\exists} \omega' = \mathrm{d}\alpha+ \omega$$

is equivalent to the concordance-relation [FSS20, Prop. 6.4]

(1)$$[\omega] = [\omega'] \;\;\;\; \Leftrightarrow \;\;\;\; \underset{ \widehat{\omega} \in \Omega_{dR}^{n1}(X \times [0,1])_{clsd} }{\exists} \; \left\{ \begin{array}{l} \omega = \widehat\omega\vert_{0} \,, \\ \omega' = \widehat\omega\vert_{1} \,. \end{array} \right.$$

But the latter concordance-relation immediately generalizes to flat $L_\infty$-algebra valued differential forms

$$\Omega_{dR}\big( X; \mathfrak{a} \big)_{clsd} \;\; \coloneqq \;\; Hom_{dgAlg}\big( CE(\mathfrak{a}) ,\, \Omega^\bullet_{dR}(X) \big)$$

with coefficients in any $L_\infty$-algebra $\mathfrak{a}$, which reduces to the ordinary case for $\mathfrak{a} \equiv b^{n-1} \mathbb{R}$ the line Lie $n$-algebra.

Therefore it makes sense to define [FSS20, Def. 6.3]:

Definition

The non-abelian de Rham cohomology of a smooth manifold $X$ with coefficients in a $L_\infty$-algebra $\mathfrak{a}$ is the set of concordance classes of flat $\mathfrak{a}$-valued differential forms on $X$:

(2)$$H_{dR}\big( X ;\, \mathfrak{a} \big) \;\; \coloneqq \;\; \Omega_{dR}\big( X ;\, \mathfrak{a} \big) _{clsd} \big/ \mathrm{concordance} \,.$$

Examples

Example

With (1) it follows that the ordinary de Rham cohomology in degree $n$ is equivalently non-abelian de Rham cohomology with coefficients in the line Lie n-algebra $b^{n-1}\mathfrak{u}(1)$:

$$H_{dR}\big( X ;\, b^{n-1}\mathfrak{u}(1) \big) \;\; \simeq \;\; H^n_{dR}\big( X ;\, b^{n-1}\mathbb{R} \big) \,.$$

Example

In higher gauge theories of Maxwell-type, nonabelian de Rham cohomology of a Cauchy surface with coefficients in an L-infinity algebra characteristic of the theory’s Gauss law reflects the total flux of the higher gauge fields.

See at geometry of physics – flux quantization the section Total flux in Nonabelian de Rham cohomology.

Properties

Recipient of non-abelian character map

For $\mathcal{A}$ (the homotopy type of) a topological space which is nilpotent (for instance: simply connected) and of rational finite type (all its rational cohomology-groups are finite-dimensional $\mathbb{Q}$-vector spaces) one may regard the homotopy classes of maps into $\mathcal{A}$ as the nonabelian cohomology classified by $\mathcal{A}$ (the non-abelian cohomology in degree=1 with coefficients in the loop space $\infty$-group $\Omega \mathcal{A}$ ):

(3)$$H\big( X ;\, \mathcal{A} \big) \;\; \coloneqq \;\; \pi_0 \, Maps\big( X ,\, \mathcal{A} \big) \;\; \simeq \;\; \pi_0 \, Maps\big( X ,\, B \Omega \mathcal{A} \big) \;\; \equiv \;\; H^1\big( X ;\, \Omega \mathcal{A} \big) \,.$$

For example, in the case that

$$\mathcal{A} \,\equiv\, K(n,A)$$

is an Eilenberg-MacLane space for a discrete abelian group $A$, this reduces to ordinary cohomology:

$$H\big( X ;\, K(n,A) \big) \;\; \simeq \;\; H^n(X;\, A) \,,$$

or if

$$\mathcal{A} \;\equiv\; KU_0 \,\simeq\, B U \times \mathbb{Z}$$

is the classifying space KU$_0$ for complex topological K-theory, then this reduces to to complex topological K-theory:

$$H\big( X ;\, KU_0 \big) \;\; \simeq \;\; K(X) \,.$$

Generally, if $\mathcal{E}$ is an Omega-spectrum of spaces, then

$$H\big( X ;\, E_n \big) \;\; \simeq \;\; E^n(X)$$

coincides with the Whitehead-generalized $E$-cohomology.

Now the rationalization-unit $\eta^{\mathbb{Q}}_{\mathcal{A}} \,\colon\, \mathcal{A} \to \mathcal{A}_{\mathbb{Q}}$ followed by suitable extension of scalars along $\mathbb{Q} \to \mathbb{R}$ induces cohomology operations in the non-abelian cohomology (3), to what may be called non-abelian rational cohomology, and non-abelian real cohomology with coefficients in $\mathcal{A}$

(4)$$H\big( -;\, \mathcal{A} \big) \longrightarrow H\big( -;\, \mathcal{A}_{\mathbb{Q}} \big) \longrightarrow H\big( -;\, \mathcal{A}_{\mathbb{R}} \big) \;\; \simeq \;\; H_{dR}\big( - ;\, \mathfrak{l}\mathcal{A} \big) \,,$$

and, shown on the right, a non-abelian version of the de Rham theorem — given essentially by the fundamental theorem of dg-algebraic rational homotopy theory — identifies this non-abelian real cohomology with coefficients in $\mathcal{A}$ with the non-abelian de Rham cohomology (2) with coefficients in the real-Whitehead $L_\infty$-algebra $\mathfrak{l}\mathcal{A}$ of $\mathcal{A}$.

For the case that $\mathcal{A} = KU_0$ the cohomology operation (4) coincides with the Chern character on complex topological K-theory, and generally for $\mathcal{A} = \mathcal{E}_n$ a term in an Omega-spectrum it coincides with the Chern-Dold character map on Whitehead-generalized cohomology (Prop. 7.2).

Therefore, it makes sense to refer to (4) generally as the character map on nonabelian cohomology taking values in non-abelian de Rham cohomology (FSS20, Part IV).

Related entries

• non-abelian cohomology

• differential non-abelian cohomology

References

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Part III of: The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalized, World Scientific (2023) [arXiv:2009.11909, doi:10.1142/13422]