nLab
nonabelian Hodge theory

under construction

Idea

Nonabelian Hodge theory generalizes aspects of Hodge theory from abelian cohomology (abelian sheaf cohomology) to nonabelian cohomology.

Nonabelian Hodge theorem

Notice or recall (for instance from generalized universal bundle and action groupoid) the following equivalent description of sections of associated bundles:

for $G$ a group with action $\rho$ on an object $V$ witnessed by the action groupoid sequence

$$V\to V//G\to BG$$V \to V//G \to \mathbf{B}G

the $\rho$-associated bundle $E\to X$ to a $G$-principal bundle $P\to X$ classified by an anafunctor $X\stackrel{\simeq }{\leftarrow }Y\to BG$ is the pullback

$$\begin{array}{ccc}E& \to & V//G\\ \downarrow & & \downarrow \\ Y& \to & BG\end{array}.$$\array{ E &\to& V//G \\ \downarrow && \downarrow \\ Y &\to& \mathbf{B}G } \,.

Since this is a pullback diagram by definition, a glance at a pasting diagram of the form

$$\begin{array}{ccccc}& & E& \to & V//G\\ & \nearrow & \downarrow & & \downarrow \\ Y& \stackrel{=}{\to }& Y& \to & BG\end{array}$$\array{ && E &\to& V//G \\ & \nearrow & \downarrow && \downarrow \\ Y &\stackrel{=}{\to}& Y &\to& \mathbf{B}G }

shows that sections

$$\begin{array}{ccc}& & E\\ & {}^{\sigma }\nearrow & \downarrow \\ Y& \stackrel{=}{\to }& Y\end{array}$$\array{ && E \\ & {}^{\sigma}\nearrow & \downarrow \\ Y &\stackrel{=}{\to}& Y }

are in bijection with maps $Y\to V//G$ that make

$$\begin{array}{ccc}Y& \to & V//G\\ {\downarrow }^{=}& & \downarrow \\ Y& \to & BG\end{array}$$\array{ Y &\to& V//G \\ \downarrow^= && \downarrow \\ Y &\to& \mathbf{B}G }

commute.

In the special case that $X$ is a connected manifold and $G$ a discrete group we can without restriction take $Y=\hat{X}//{\pi }_1(X)$ be the action groupoid of the universal cover by the homotopy group, so that the classifying map $Y\to BG$ is the same as a group homomorphism

$$\rho :{\pi }_1(X)\to G.$$\rho : \pi_1(X) \to G \,.

In that case the above says that a section of the associated bundle is a $\rho$-equivariant map

$$\varphi :\hat{X}\to V.$$\phi : \hat X \to V \,.

This is the way these sections are formulated usually in the literature. The above description has the advantage that it works more generally in nonabelian cohomology for principal bundles generalized to principal ∞-bundles.

Next consider furthermore the special case that $V=G/K$ is the coset homogeneous space of $G$ quotiented by a subgroup $K$. Then if $G$ is a Lie group or algebraic group consider moreover a choice of $G$-invariant metric on the quotient $G/K$. Also consider a Riemannian manifold structure on $X$.

Then

Definition

The energy of a section $\sigma$ of an associated $G/K$-bundle as above is the real number

$$E(\varphi ):={\int }_X\mid d\varphi {\mid }^2.$$E(\phi) := \int_X |d \phi|^2 \,.

Here

• $\varphi$ is the $\rho$-equivariant map describing the section as above,

• the norm is taken with respect to the chocen invariant metric on $G/K$

• and the integral is taken with respect to the Riemannian metric on $X$.

Definition

Such a $\varphi$ is called harmonic of it is a critical point of $E(-)$.

Theorem

(Corlette, generalizing Eells-Sampson)

If $\rho :{\pi }_1(X)\to G$ is a representation with

• $G$ a reductive algebraic group

• $K$ is a maximal compact subgroup

• $\rho ({\pi }_1(X))$ is

  • Zariski-dense in $G$

  • or its Zariski-closure is itself reductive

then there exists a harmonic section $\varphi$ in the above sense.

Proof

A version of the proof is reproduced on p.8 of

• Carlos Simpson, The Hodge filtration on nonabelian cohomology (arXiv)

Related concepts

• Hodge structure

• noncommutative Hodge theory

References

Corlette’s nonabelian Hodge theorem is in

• K. Corlette, Flat $G$-bundles with canonical metric J. Diff Geometry 28 (1988)

Work by Carlos Simpson:

• Carlos Simpson, The Hodge filtration on nonabelian cohomology (arXiv:9604005)

• Carlos Simpson, Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology (arXiv:9712020)

• Carlos Simpson, Algebraic aspects of higher nonabelian Hodge theory (arXiv:9902067)

• Carlos Simpson, Tony Pantev, Ludmil Katzarkov, Nonabelian mixed Hodge structures (arXiv)

The nonabelian Hodge theorem is generalized to twisted bundles in

• Alberto García Raboso, Twisted Nonabelian Hodge theorem (pdf slides)