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 theorems

Nonabelian harmonic sections

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

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

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

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

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

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

shows that sections

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

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

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

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 $Y \to \mathbf{B}G$ is the same as a group homomorphism

$$\rho : \pi_1(X) \to G \,.$$

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

$$\phi : \hat X \to V \,.$$

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

$$E(\phi) := \int_X |d \phi|^2 \,.$$

Here

• $\phi$ 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 $\phi$ is called harmonic if 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 $\phi$ in the above sense.

This is due to (Corlett 88). A version of the proof is reproduced in Simpson 96, p. 8

Kähler case: Equivalence between Local systems and Higgs bundles

The nonabelian Hodge theorem due to (Simpson 92) establishes, for $X$ a compact Kähler manifold, an equivalence between (irreducible) flat vector bundles on $X$ and (stable) Higgs bundles with vanishing first Chern class.

Relation to the abelian Hodge theorem

The sense in which the nonabelian Hodge theorem of (Simpson 92) generalizes the abelian Hodge theorem is the following (Simpson 92, Introduction).

The abelian cohomology group $H^1(X,\mathbb{C}_{disc})$ classifies flat complex line bundles whose underlying line bundle is trivial, hence closed differential 1-forms modulo 0-forms. The abelian Hodge theorem gives for this hence the decomposition

$$H^1(X,\mathbb{C}_{disc}) \simeq H^1(X, \mathcal{O}_X) \oplus H^0(X, \Omega^1_X) \,.$$

It is this kind of relation which is generalized by the nonabelian Hodge theorem. Here one starts instead with the nonabelian cohomology set $H^1(X, GL_n(\mathbb{C})_{disc})$ which classifies flat rank-$n$ vector bundles on $X$, for $n \in \mathbb{N}$. The equivalence to Higgs bundles gives now a decomposition of these structures into a holomorphic vector bundle classified by $H^1(X, GL_n(\mathcal{O}_X))$ and a differential 1-form with values in endomorphisms of that, subject to some conditions.

Statement

A quick review of the theorem in (Simpson 92) is for instance in (Raboso 15, section 1.2). An elegant abstract reformulation in terms of differential cohesion/D-geometry, following (Simpson 96) is in (Raboso 15, section 3.3):

Analogous to how the de Rham stack $\int_{inf} X = X_{dR}$ of $X$ is the (homotopy) quotient of $X$ by the first order infinitesimal neighbourhood of the diagonal in $X \times X$, so there is a space (stack) $X_{Dol}$ which is the formal completion of the 0-section of the tangent bundle of $X$ (Simpson 96).

Now a flat vector bundle on $X$ is essentially just a vector bundle on the de Rham stack $X_{dR}$, and a Higgs bundle is essentially just a vector bundle on $X_{Dol}$. Therefore in this language the nonabelian Hodge theorem reads (for $G$ a linear algebraic group over $\mathbb{C}$)

$$\mathbf{H}(X_{dR}, \mathbf{B}G) \simeq \mathbf{H}(X_{Dol}, \mathbf{B}G)^{ss,0} \,,$$

where the superscript on the right denotes restriction to semistable Higgs bundles with vanishing first Chern class (see Raboso 15, Theorem 3.3).

Generalizations to twisted bundles

A generalization of the nonabelian Hodge theorem of (Simpson 92) to twisted bundles in discussed in (Raboso 15).

Relation to geometric Langlands

Nonabelian Hodge theory is closely related to the geometric Langlands correspondence.

Related concepts

• Hodge structure

• noncommutative Hodge theory

References

• wikipedia nonabelian Hodge correspondence

Lecture notes on nonabelian Hodge theory include:

• Ron Donagi, Tony Pantev, Lectures on the geometric Langlands

  conjecture and non-abelian Hodge theory_, 2009 (pdf)

• Alberto García Raboso, Steven Rayan, Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles, and complex variations of Hodge structure, Fields Inst. Monogr. 34 (2015), 131–171 (arXiv) (Springer)

Corlette’s nonabelian Hodge theorem can be found in:

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

Works by Carlos Simpson on nonabelian Hodge theory include:

• Carlos Simpson, Higgs bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. (1992), no. 75, 5{95. MR 1179076 (94d:32027) (numdam)

• Carlos Simpson, The Hodge filtration on nonabelian cohomology, Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217{281. MR

  1492538 (99g:14028) (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 of (Simpson 92) is generalized to twisted bundles in:

• Alberto García Raboso, A twisted nonabelian Hodge correspondence, PhD thesis 2014 (arXiv:1501.05872, pdf slides)

Gothen’s paper starts with a survey on nonabelian Hodge correspondence,

• Peter B. Gothen, Higgs bundles and the real symplectic group, arXiv:1102.4175

• C. C. Liu, S. Rayan, Y. Tanaka, The Kapustin–Witten equations and nonabelian Hodge theory Eur. J. Math. 8 (Suppl 1) 23–41 (2022) doi