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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