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nonabelian Hodge theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

under construction

Contents

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 GG a group with action ρ\rho on an object VV witnessed by the action groupoid sequence

VV//GBG V \to V//G \to \mathbf{B}G

the ρ\rho-associated bundle EXE \to X to a GG-principal bundle PXP \to X classified by an anafunctor XYBGX \stackrel{\simeq}{\leftarrow} Y \to \mathbf{B}G is the pullback

E V//G Y BG. \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

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

shows that sections

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

are in bijection with maps YV//GY \to V//G that make

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

commute.

In the special case that XX is a connected manifold and GG a discrete group we can without restriction take Y=X^//π 1(X)Y = \hat X//\pi_1(X) be the action groupoid of the universal cover by the homotopy group, so that the classifying map YBGY \to \mathbf{B}G is the same as a group homomorphism

ρ:π 1(X)G. \rho : \pi_1(X) \to G \,.

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

ϕ:X^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/KV = G/K is the coset homogeneous space of GG quotiented by a subgroup KK. Then if GG is a Lie group or algebraic group consider moreover a choice of GG-invariant metric on the quotient G/KG/K. Also consider a Riemannian manifold structure on XX.

Then

Definition

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

E(ϕ):= X|dϕ| 2. E(\phi) := \int_X |d \phi|^2 \,.

Here

  • ϕ\phi is the ρ\rho-equivariant map describing the section as above,

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

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

Definition

Such a ϕ\phi is called harmonic if it is a critical point of E()E(-).

Theorem

(Corlette, generalizing Eells-Sampson)

If ρ:π 1(X)G\rho : \pi_1(X) \to G is a representation with

then there exists a harmonic section ϕ\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 XX a compact Kähler manifold, an equivalence between (irreducible) flat vector bundles on XX 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, disc)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, disc)H 1(X,𝒪 X)H 0(X,Ω X 1). 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() disc)H^1(X, GL_n(\mathbb{C})_{disc}) which classifies flat rank-nn vector bundles on XX, for nn \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(𝒪 X))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 14, section 1.2). An elegant abstract reformulation in terms of differential cohesion/D-geometry, following (Simpson 96) is in (Raboso 14, section 4.2.1):

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

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

H(X dR,BG)H(X Dol,BG) ss,0, \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 14, theorem 4.2).

Generalizations to twisted bundles

A generalization of the nonabelian Hodge theorem of (Simpson 92) to twisted bundles in discussed in (Raboso 14).

Relation to geometric Langlands

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

References

Lecture notes on geometric Langlands duality and nonabelian Hodge theory include

Corlette’s nonabelian Hodge theorem is in

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

Work by Carlos Simpson on nonabelian Hodge theory includes

The nonabelian Hodge theorem of (Simpson 92) is generalized to twisted bundles in

Revised on January 28, 2014 12:52:22 by Urs Schreiber (89.204.137.194)