Higher solutions of Hitchin’s self-dualityequations

Sebastian HellerUniversitat Hamburg

Tokyo, December 13th, 2017

Almost a definition

A higher solution of Hitchin’s self-duality equation is a solution ofthe self-duality equations on a open dense subset X of a compactRiemann surface X which behaves nicely at the boundary ∂X .

Self-duality equations

We restrict to the easiest situation beyond the trivial abelian case:

Hitchin’s self-duality equations are

F∇ = −[Φ,Φ∗]

∂∇Φ = 0

where

I V = C2 → X the trivial rank 2 vector bundle over a compactRiemann surface (usually of genus g ≥ 2);

I ∇ = d + A a unitary connection, A ∈ Ω1(X ; su(2));

I Higgs field Φ ∈ Γ(X ;K ⊗ sl(2,C)).

A classical example - Uniformization

Let g be a metric of curvature −4 on X , S be a spin bundle of X .S inherits a hermitian metric with unitary connection ∇S .

For V = S ⊕ S∗ and Φ =

(0 01 0

)∈ H0(X ,KEnd(S ⊕ S∗))

(∇S ⊕ (∇S)∗,Φ)

is a solution of the self-duality equations.

The moduli space of solutions MSD

I G = g : X → SU(2) acts on solutions of the selfdfualityequation via

(∇,Φ).g = (∇.g , g−1Φg)

I irreducible solutions: no invariant subbundle

Theorem (Hitchin 87)

The moduli space of irreducible solutions

MSD = (∇,Φ) | irred. sol. of the SD eqns /G

is a smooth manifold of (real) dimension 12g −12 (g = genus(X )).

The hyper-Kahler structure on MSD

A hyper-Kahler structure on a manifold is a Riemannian metricwhich is Kahler for 3 anti-commuting complex structures I , J,K .

The Riemannian metric

g((A, Φ), (A, Φ)) = 2i

∫X

tr((A(0,1))∗ ∧ A(0,1) + Φ ∧ Φ∗)

is well-defined on the quotient MSD .

Higgs pairs

DefinitionA Higgs pair (∂,Φ) consists of a holomorphic structure ∂ on avectorbundle V → X and a holomorphic (traceless)endomorphism-valued 1-form Φ ∈ H0(X ;KEnd(V )).

Forgetful map:

(∇,Φ) sol. of the SD eqn. 7−→ (∂∇,Φ) a Higgs pair

Stable Higgs pairs

Definition (in our situation)

A Higgs pair is stable if it does not admit invariant line subbundlesof non-negative degree.

Example

If (∇,Φ) is a irreducible solution of the SD eqn. then (∂∇,Φ) isstable.

Theorem (Hitchin)

A stable Higgs pair gives rise to a (irreducible) solution of the SDequations, unique up to the action of G.

I

The complexified gauge group GC = g : X → SL(2,C) acts onthe space of stable Higgs pairs, and the quotient

MDol = Higgspairs/GC

is a complex manifold of dimension 6g-6.

MDol =MSD

(as real manifolds) equips MSD with the complex structure I .

Example

The complex submanifold Φ = 0 is the moduli space of stablebundles.

J

Every (irreducible) solution (∇,Φ) of the self-duality equationsgives rise to a (irreducible) flat connection

∇+ Φ + Φ∗.

Theorem (Donaldson)

Every irreducible flat connection gives rise to a (unique G-orbit of)solution(s) to the self-duality equations.

This equips MSD with the complex structure J from the modulispace of irreducible flat connections MdR .

Harmonic maps and the self-duality equations

For a solution (∇,Φ) of the self-duality equations, and a parallelframe Ψ of the connection ∇+ Φ + Φ∗, the map

Ψ∗Ψ: X → H = A ∈ SL(2,C) | A∗ = A

is harmonic, where H consists of two copies of the hyperbolic3-space H3 with respect to the quadratic form −Det.

RemarkDonaldson used this fact and the harmonic map flow to prove hisresult.

Example

For the solution corresponding to the uniformization of X , theharmonic map is the developing map of the hyperbolic metric.

The Hitchin system

Theorem (Hitchin)

MSD with g and I , J,K = IJ is a hyper-Kahler manifold.

More structure:

I (MSD , I , ω2 + iω3) gets into a integrable system via

det : MSD → H0(X ;K 2); [∇,Φ] 7→ det(Φ),

the famous Hitchin system.

Fibers, over quadratic differentials q with simple zeros, are affinePrym varieties for the spectral curve X√q → X .

The twistor space

Every hyper-Kahler manifold M can be encoded in complexgeometric data:

I P := M × S2

I I(p,x ,y ,z) := (xIp + yJp + zKp)⊕ I(x ,y ,z) is integrable;

I P → CP1 = S2 is holomorphic; its fibers are M equippedwith corresponding complex structure;

I real involution ρ : (p, λ) 7→ (p,−λ−1);

M can be recovered as (a component of) the space of ρ-invariantholomorphic sections: λ 7→ (p, λ).

P carries additional structure which encodes g .

Deligne’s description of the twistor space

Idea: interpolate between Higgs pair (I ) and flat connection (J)via the associated family of flat connections:

λ ∈ C∗ 7→ ∇λ := ∇+ λ−1Φ + λΦ∗.

Definition (Deligne; Simpson)

A λ-connection (for λ ∈ C) consists of a holomorphic structure ∂and a first order linear differential operator D satisfying

I D(fs) = λ∂0f ⊗ s + f Ds for functions f and sections s;

I D∂ + ∂D = 0.

Example

I λ = 0: (∂,Φ = D) is a Higgs pair;

I λ = 1: ∂ + D is a flat connection;

I λ 6= 0 : ∂ + 1λD is a flat connection.

The Hodge moduli space

I GC acts on the space of λ-connections;

I the moduli space

MHod = (λ, ∂,D) | λ ∈ C/GC

of stable/irreducible λ-connections is a complex manifold ofdimension 6g-5;

I MHod → C; its fibers are biholomorphic to MDol (if λ = 0)or to MdR (if λ 6= 0).

Deligne gluing

I X denotes the complex conjugate Riemann surface

For λ 6= 0 define

Ψ(λ, ∂,D) = ( 1λ ,

1λD,

1λ ∂),

the later being a 1λ -connection on X .

Ψ is compatible with the GC-action and gives rise to anisomorphism

Ψ: (MHod(X ))|C∗ → (MHod(X ))|C∗

covering the map λ 7→ 1λ .

DefinitionThe Deligne-Hitchin moduli space is

MDH =MHod(X ) ∪ΨMHod(X )→ CP1.

MDH as a twistor space

Natural automorphisms of MDH are given by:

I for any t ∈ C∗ : t.(λ, ∂,D) = (tλ, ∂, tD);

I for t = −1 we denote the automorphism by N;

I C ([λ, ∂,D]X ) = [λ, ∂, D]X ; N and C commute and give riseto an antiholomorphic involution covering λ 7→ −λ−1.

Theorem (Deligne, Simpson)

MDH → CP1 is isomorphic to the twistorspace of MSD such that

I ρ = CN

I real sections corresponding to solutions of the self-dualityequations are given by

λ 7→ [λ, ∂∇ + λΦ∗, λ∂∇ + Φ]X .

Benefits

We have

I a complex analytic description of the twistor space of MSD ;

I not used the knowledge about solutions of the self-dualityequations.

Natural question:Can we use (the complex geometry of) MDH to reconstruct allsolutions of the self-duality equations?

Question (Simpson 95)

Is every ρ-invariant section of MDH → CP1 given by a solution ofthe self-duality equations?

The answer

Theorem (L.Heller, H.)

There exists sections of MDH → CP1 which are invariant underthe real involution ρ but are not given by solutions of theself-duality equation.

These sections are called higher solutions to the self-dualityequations.

More sections

Example (Harmonic maps to f : X → SU(2) = S3)

I consider ∇ = f ∗∇spin and Φ = 14 (f −1df − i ∗ f −1df );

I they satisfyF∇ = [Φ,Φ∗]

and∂∇Φ = 0

if and only if f is harmonic;

Thens(λ) = [λ, ∂∇ − λΦ∗, λ∂∇ + Φ]X

is a section of MDH which is invariant under C .

Further examples: (equivariant) harmonic maps into AdS3 andinto the space of oriented circles in the 2-sphere.

Harmonic maps of Riemann surfaces

into hyperbolic space H3 the sphere S3

and its quotients

existence by harmonic map flow harmonic map flowdevelops singularities

solutions of the via Hitchin systemself-duality equations for Tori only!

uniqueness of harmonic maps nontrivial deformationsin each homotopy class of harmonic maps

associated families Deligne-Hitchin Reconstruction bytwistor space loop group factorization

Harmonic maps from 2-tori to the 3-sphere

I abelian fundamental group;

I generic flat connection splits ∇ = ∇+ ⊕∇−;

(gauge classes of) families of flat connections on tori can bedescribed easily:

ΣD

//

λ

MdR(C∗)

C

[∇λ] //MdR(SL(2,C))

Spectral data

Σ is a compact Riemann surface after adding points over 0,∞.

D is an odd map into the affine bundle of flat line bundleconnections over the Jacobian with a first order pole at 0 (andsimilar behavior at ∞).

RemarkActually, D should be considered as a odd map into the C∗Deligne-Hitchin moduli space.

D satisfies a reality condition, all informations of D are containedin the forgetful map into the Jacobian.

To reconstruct the harmonic map one needs the so-called eigenlinebundle which fixes the remaining gauge-freedom.

CMC tori

Figure: By Nick Schmitt.

Experimental moduli space of embedded CMC surfaces