Topology of the moduli space of Higgs bundles over a ... · Hitchin-Kobayashi correspondence The Hitchin-Kobayashi correspondence is a general principle which asserts that the moduli

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Topology of the moduli space of Higgs bundlesover a compact Riemann surface

Graeme Wilkin (National University of Singapore)

中国科学技术大学合肥, 中国

25 September, 2014

Graeme Wilkin (National University of Singapore) Topology of moduli spaces of Higgs bundles

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Goal

The moduli space of Higgs bundles was defined by Hitchin in 1987.In Hitchin’s words “...the moduli space of all solutions turns out tobe a manifold with an extremely rich geometric structure”.In this talk I will describe some of the features of this moduli space.

Definition of Higgs bundles and the moduli space of HiggsbundlesProperties of the moduli spaceSurvey of results on the cohomology of the moduli space(Hitchin, Gothen, Hausel, Hausel-Rodriguez-Villegas,Garcia-Prada-Heinloth-Schmitt)Morse theory for the Yang-Mills-Higgs functional and newresults about the equivariant cohomology of the space ofsemistable Higgs bundles (joint with G. Daskalopoulos and R.Wentworth).


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Holomorphic bundles

Let X be a compact Riemann surface and let p : E→ X be asmooth complex vector bundle.Smooth classification of bundles. E is determined up to C∞

isomorphism by its rank n and first Chern class c1(E) ∈ H2(X,Z).E is a holomorphic bundle if all of the transition functions areholomorphic gαβ : Uα ∩ Uβ → GL(n,C)The classification of holomorphic bundles up to isomorphism ismore complicated and the moduli space of holomorphic bundles isan interesting object in gauge theory and algebraic geometry.


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Gauge-theoretic description of holomorphic bundlesEach trivialisation p−1(Uα) ∼= Uα × Cn has a natural ∂-operator

∂ : Ω0(Uα × Cn)→ Ω0,1(Uα × Cn)

If the transition functions are all holomorphic then this operator iswell-defined on overlapping trivialisations

∂(gαβ(z)s(z)) = gαβ(z) ∂(s(z))

and so it extends to a C-linear operator ∂E : Ω0(E)→ Ω0,1(E).This operator satisfies the Leibniz rule

∂E(f s) = (∂f)s + f ∂Es, f ∈ Ω0(X), s ∈ Ω0(E)

and we can extend it to a C-linear operator Ωp,q(E)→ Ωp,q+1(E).We automatically have ∂2

E = 0 since Ω0,2(E) = 0 when the basemanifold has complex dimension one.Therefore, holomorphic bundles −→ ∂ − operators.


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Gauge-theoretic description of holomorphic bundles (cont.)What about the converse? Do ∂-operators determine holomorphicbundles?Theorem. [Newlander-Nirenberg] Let E→ X be a complex vectorbundle over a complex manifold X. An operator∂E : Ω0(E)→ Ω0,1(E) satisfying

the Leibniz rule, and∂2

E = 0

defines a holomorphic structure on E.

Therefore, holomorphic bundles bijective←→ ∂ − operators.When X is a compact Riemann surface (and so the condition∂2

E = 0 is trivial) then the space of all such ∂ operators is an affinespace locally modelled on Ω0,1(End(E)).

A0,1 := ∂E0 +Ω0,1(End(E))Graeme Wilkin (National University of Singapore) Topology of moduli spaces of Higgs bundles

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Gauge-theoretic description of holomorphic bundles (cont.)Now fix a Hermitian metric H on the fibres of E. Given aholomorphic structure ∂E on E, there is a unique Chern connectioncompatible with the metric

DH,E : Ω0(E)→ Ω1(E), such that the (0, 1) part is ∂E

We write

DH,E = ∂E + ∂H,E where ∂H,E : Ω0(E)→ Ω1,0(E)

Let A := dE0 +Ω1(ad(E)) denote the space of all connectionscompatible with the metric.For each Hermitian metric, the Chern connection constructiondefines a homeomorphism

A0,1 ∼=−→ A


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Gauge-theoretic description of holomorphic bundles (cont.)Using the Chern connection construction, we can define thecurvature of a holomorphic structure F(∂E,H) = D2

H,E and hencethe Yang-Mills functional

YM(∂E,H) = ∥F(∂E,H)∥2

The Yang-Mills equations are the Euler-Lagrange equations for thecritical points of this functional

D∗H,E F(∂E,H) = 0

Remark. The Yang-Mills functional and the Yang-Mills equationsdepend on the metric H. Some natural questions are:

Given a holomorphic structure ∂E, do there exist “good”metrics for which ∂E is a solution to the Yang-Mills equations?Is ∂E is a minimiser for the Yang-Mills functional?


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How to classify holomorphic bundles up to equivalence?When E is a line bundle, the classification of holomorphicstructures up to isomorphism is a classical result due to Abel andJacobi.The holomorphic line bundles on a compact Riemann surface areclassified up to isomorphism by the points in the Jacobian

Jac(X) := Cg/Λ

where Λ ∼= Z2g is the period lattice of X.

In the gauge-theoretic setting of connections, this theorem saysthat

Jac(X) ∼= A0,1/GC(metric)∼= A/GC

When rank(E) > 1 then the problem is more complicated. Theisomorphism classes do not form a “good” moduli space (it is noteven Hausdorff).


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How to classify holomorphic bundles up to equivalence?Mumford’s Geometric Invariant Theory shows how to construct acoarse moduli space by restricting to the subset of stable orsemistable bundles.Definition. A holomorphic bundle E→ X is stable (resp.semistable) if and only if for every proper non-zero holomorphicsub-bundle F ⊂ E we have

deg(F)rank(F) <

deg(E)rank(E)

(resp. deg(F)

rank(F) ≤deg(E)rank(E)

)A bundle is polystable iff it is the direct sum of stable bundles ofthe same slope (degree-rank ratio).Now we can define the moduli space of stable/semistable bundles

Mst(E) := A0,1st /GC, Mss(E) := A0,1

ss //GC

where the double quotient // indicates that we identify orbitswhose closures intersect.


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Hitchin-Kobayashi correspondence

The Hitchin-Kobayashi correspondence is a general principle whichasserts that the moduli space should have an interpretation as asymplectic quotient.Given a fixed Hermitian metric, the minimum of the Yang-Millsfunctional is achieved by connections D with curvature FD = c · id,where c is a topological invariant of E.

Theorem. [Narasimhan-Seshadri, Donaldson, Uhlenbeck-Yau]

Mss(E) := A0,1polyst/G

C ∼= FD = c · id/G

where G is the group of gauge transformations that preserve theHermitian metric on E.Equivalently, a bundle is polystable iff there exists a uniqueHermitian metric such that the bundle minimises the Yang-Millsfunctional.


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Higgs bundles

Higgs bundles are a generalisation of holomorphic bundles(“holomorphic bundles with extra structure”).The moduli space has many interesting features as a consequenceof this extra structure.(Nonabelian Hodge theory, Mirror symmetry, Langlands program)Definition. [Hitchin (1987), Simpson (1988)] A Higgs bundle is apair (∂E, ϕ) consisting of a holomorphic bundle ∂E and a sectionϕ ∈ Ω1,0(End(E)) such that ∂E ϕ = 0 and ϕ ∧ ϕ = 0.(Note that ϕ ∧ ϕ = 0 is automatic when dimC X = 1.)Let B denote the space of all Higgs bundles with underlyingsmooth bundle E→ X.Unlike the space of holomorphic bundles (which is affine), thespace of Higgs bundles is singular (more later).


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Constructing the moduli space of Higgs bundles

Definition. A Higgs bundle (∂E, ϕ) is stable (resp. semistable) ifand only if for every proper non-zero Higgs sub-bundle F ⊂ E wehave

deg(F)rank(F) <

deg(E)rank(E)

(resp. deg(F)

rank(F) ≤deg(E)rank(E)

)A Higgs bundle is polystable iff it is the direct sum of stable Higgsbundles with the same slope.Now we can define

MHiggsss (E) := Bss//GC = Bpolyst/GC, MHiggs

st (E) := Bst/GC

where the double quotient // indicates that we identify orbitswhose closures intersect.


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Hitchin-Kobayashi correspondence

Given a fixed Hermitian metric on E, the Yang-Mills-Higgsfunctional is

YMH(∂E, ϕ) := ∥F(∂E) + [ϕ, ϕ∗]∥2

Let YMHmin denote the space of Higgs pairs minimising YMH.Hitchin (1987) (dimC X = 1) and Simpson (1988) (dimC X ≥ 1)proved a Hitchin-Kobayashi correspondence for Higgs pairs.Theorem. [Hitchin, Simpson]A Higgs bundle (∂E, ϕ) is polystable iff there exists a uniqueHermitian metric such that (∂E, ϕ) ∈ YMHmin.Equivalently

MHiggsss (E) := Bpolyst/GC ∼= YMHmin /G


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What does the space of Higgs bundles look like?

For a complex vector bundle E over a compact Riemann surface X,the space of all holomorphic structures is an affine space locallymodelled on Ω0,1(End(E)).The situation is more complicated for Higgs bundles, since thecondition ∂Eϕ = 0 is not of constant rank.There is a map from the space of Higgs bundles to the affine spaceof holomorphic bundles

B → A0,1

(∂E, ϕ) 7→ ∂E

For each holomorphic structure ∂E, the fibre of this map isH1,0(End(E)) = ϕ | ∂Eϕ = 0 ∼= H0(End(E)⊗ K).The dimension of the fibre depends on ∂E. In particular, we seethat B has singularities for rank(E) ≥ 2.


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Structure of the moduli space of Higgs bundlesWhen X is a compact Riemann surface, the moduli space of stableHiggs bundles is a smooth manifold.This is not necessarily true for semistable or polystable Higgsbundles, since singularities may occur at E = F1 ⊕ F2 (bundlessuch as this have larger isotropy group).

When the degree and rank of E are coprime then (∂E, ϕ) is stableiff it is semistable (and hence MHiggs

ss is smooth in this case).

Theorem. [Hitchin (dimC X = 1), Fujiki (dimC X ≥ 1)]MHiggs

ss is a hyperkahler quotient.

There is an action of C∗ on MHiggsss

t · [(∂E, ϕ)] = [(∂E, tϕ)]The fixed points of this action are called variations of Hodgestructure (due to Simpson). In low rank these can be classified viathe theory of holomorphic chains.


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Structure of the moduli space of Higgs bundles (cont.)

The action of U(1) ⊂ C∗ is Hamiltonian. The associated momentmap is µ[(∂E, ϕ)] =

√−1∥ϕ∥2L2 .

Theorem. [Hitchin, 1987]When the degree and rank are coprime then the function f = |µ| isproper and Morse-Bott.

Hitchin (1987, rank 2) and Gothen (1995, rank 3), (2002,G = SU(2, 1) or U(2, 1)) classified the critical sets and computedthe Morse index and the cohomology of each critical set.As a consequence, when the degree and rank are coprime, they cancompute the Betti numbers of the moduli space via Morse-Botttheory.


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Hitchin’s calculation for rank(E) = 2, det E = L

At the non-minimal critical sets of f = ∥ϕ∥2L2 the bundle splitsE = L1 ⊕ L2 and the Higgs field simplifies to ϕ ∈ H0(L∗

1L2 ⊗ K).The result is a special case of a holomorphic chain

L1ϕ−→ L2

where deg L2 < deg L1 ≤ deg L2 + 2g− 2.(Note that L1 ⊕ L2 is unstable as a holomorphic bundle, but thepair (L1 ⊕ L2, ϕ) is stable as a Higgs bundle.)These critical sets are classified by ℓ = deg L1. If we fix thedeterminant of E = L1 ⊕ L2, d = deg(E) then the critical set isisomorphic to Sd−2ℓ+2g−2X, the symmetric product of X.


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Hitchin’s calculation for rank(E) = 2, det E = L (cont.)

The minimal critical set for f = ∥ϕ∥2 is at ϕ = 0, which gives usMss(E), the moduli space of stable holomorphic bundles.Hitchin proved that when rank(E) = 2, deg(E) = 1 then

Pt(MHiggsss (E)) = Pt(Mss(E)) +

g−1∑ℓ=1

tλℓPt(S1−2ℓ+2g−2X)

Gothen obtained similar formulas for rank(E) = 3, deg(E) = 1 andalso when the structure group is U(2, 1) or SU(2, 1).The idea of the calculation is the same, but classifying the criticalsets and computing the Morse index is more complicated.Recently, Garcia-Prada, Heinloth and Schmitt computed thePoincare polynomial for rank(E) = 4 and coprime degree.


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Counting rational points (Hausel)

Another approach to computing cohomology of a projective varietyis to count the rational points on a variety defined over a finitefield Fq.Harder and Narasimhan (1975) did this for moduli spaces of stablebundles with coprime rank and degree.One can then compute the cohomology using the Weil conjectures(proved by Deligne).For (quasiprojective) moduli spaces of Higgs bundles, this programwas carried out by Hausel (2006). He developed a new theory“arithmetic harmonic analysis” to compute the number of rationalpoints on a holomorphic symplectic quotient.This leads to conjectures on the Betti numbers and, moregenerally, on the mixed Hodge polynomials for moduli spaces ofHiggs bundles. These conjectures have been verified in low rank(Hausel-Villegas-Rodriguez, 2008).


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Back to holomorphic bundles (Atiyah and Bott)A famous result of Atiyah and Bott shows that we can inductivelycompute the (equivariant) cohomology of the moduli space ofsemistable holomorphic bundles using the Morse theory of theYang-Mills functional.Idea. Fix a Hermitian metric on E. The Yang-Mills functionalf (∂E) = YM(∂E,H) is an energy function on the space ofholomorphic bundles.The Hitchin-Kobayashi correspondence shows that (modulo gauge)the minimum can be identified with the moduli space of semistablebundles.Atiyah and Bott (1983) classified the non-minimal critical sets andshowed that there is an algebraic stratification byHarder-Narasimhan type with “good” properties.Daskalopoulos (1992) and Rade (1992) later showed that thisstratification coincides with the Morse stratification by the gradientflow of the Yang-Mills functional.


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Atiyah and Bott’s calculationBasic idea. (All calculations in G-equivariant cohomology)

PGt (A0,1

ss ) = cohomology of total space− contributions from critical sets

More precisely.

PGt (A0,1

ss (E )) = Pt(BG)−∑

crit. sets Ck

tλkPGt (Ck)

The same methods also show that the inclusionA0,1

ss (E ) → A0,1(E ) induces a surjection in equivariant cohomologyH∗G(A0,1(E )) H∗

G(A0,1ss (E ))

This last fact is known as Kirwan surjectivity and gives adescription of the generators of H∗

G(A0,1ss (E )).

Later work by Jeffrey-Kirwan (1998) and Earl-Kirwan (2004) givesa complete description of the intersection pairings and thegenerators/relations for the cohomology ring.


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Atiyah and Bott for Higgs bundlesCan we carry out this program for Higgs bundles?Atiyah and Bott’s calculation does not require that the modulispace is smooth, since the Morse theory only sees the moduli spaceas the lowest critical set.Therefore, a calculation for Higgs bundles in the spirit of Atiyahand Bott would give us

new results on the Betti numbers when the degree and rankare not coprimeinsight into the question of Kirwan surjectivity for Higgsbundles.

Problem. As we saw before, the space of Higgs bundles issingular. How to do Morse theory on this space?We are trading the difficulties caused by the singularities in themoduli space for a new set of difficulties caused by the singularitiesin the space of Higgs bundles.


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Stratification of the space of Higgs bundlesQuestion. Fix a Hermitian metric on E. Is there a Morsestratification of the space of Higgs bundles given by theYang-Mills-Higgs functional?

YMH(∂E, ϕ) = ∥F(∂E) + [ϕ, ϕ∗]∥2

Theorem.[W. (2008)]For any initial condition (∂0, ϕ0) ∈ B, the gradient flow of theYang-Mills-Higgs functional converges to a critical point of YMHand therefore there is a Morse stratification of B.Moreover, the Morse stratification coincides with theHarder-Narasimhan stratification from algebraic geometry.The limit of the flow is isomorphic to the graded object of theHarder-Narasimhan-Seshadri double filtration of (∂0, ϕ0).

Jiayu Li and Xi Zhang have recently generalised this result tocompact Kahler manifolds of any dimension.


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Results on the topology of the moduli space

We choose an ordering on the strata and denote the Morsestratification by

B =

∞∪k=0

Bk

Theorem. [Daskalopoulos, Weitsman, Wentworth, W. (2011)] LetE→ X be a complex vector bundle of rank 2 and degree eithereven or odd. Then

PGt (Bss) = PG

t (B)−∞∑

k=0

tλkPGt (Bk) +

g−1∑ℓ=1

tλℓPt(Sdeg(E)−2ℓ+2g−2X)

Moreover, the Kirwan map H∗G(B)→ H∗

G(Bss) is surjective forstructure group G = GL(2,C).The Kirwan map is not surjective for structure group G = SL(2,C).


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Results on the topology of the moduli space (cont.)

The idea of the proof is to first use the gradient flow to reduce thecalculation to the critical sets.We then interpret the singularities at each critical set via thetheory of holomorphic chains.When rank(E) = 2 then the holomorphic chain is just a symmetricproduct SdkX (fixed determinant) or SdkX× Jac(X) (non-fixeddeterminant).Basic idea.

PGt (Bss) = cohomology of total space

− contributions from critical sets+ correction terms from singularities

The correction terms arise due to a failure of YMH to beequivariantly perfect on the singular space B.


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Results on the topology of the moduli space (cont.)The next theorem interprets the failure of Kirwan surjectivity interms of the action of the mapping class group on the modulispace.Theorem. [Hitchin, Donaldson (1987)] Let E→ X have rank 2,degree zero and fixed determinant. Then

MHiggsss (E) ∼= Hom(π1(X),SL(2,C))//SL(2,C)

Therefore there is an induced action of the mapping class groupMod(X) on MHiggs

ss (E).Theorem. [Daskalopoulos, Wentworth, W. (2010)] When E hasrank 2, degree zero and fixed determinant then the Torelli groupacts non-trivially on the cohomology of MHiggs

ss (E) and hence thesame is true for Hom(π1(X),SL(2,C))//SL(2,C).

The proof involves showing that (a) the Torelli group acts triviallyon the image of the Kirwan map, and (b) the Torelli group actsnon-trivially on each correction term in the Morse theorycalculation.Graeme Wilkin (National University of Singapore) Topology of moduli spaces of Higgs bundles

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Results on the topology of the moduli space (cont.)

More recently, we have extended these results to U(2, 1) andSU(2, 1) bundles, generalising Gothen’s results. The singularitiesare more complicated and we require some new ideas to overcomethis, but the basic idea of computing correction terms still holds.Theorem. [Wentworth, W. (2013)]Let B denote the space of U(2, 1), PU(2, 1) or SU(2, 1) Higgsbundles of any degree. Then

we can compute PGt (Bss) via the Morse theory of the

Yang-Mills-Higgs functional,the Kirwan map is surjective for U(2, 1) and PU(2, 1) Higgsbundles, andthe Kirwan map is not surjective for SU(2, 1) Higgs bundleswith Toledo invariant τ ≤ 4

3(g− 1). Moreover, the Torelligroup acts non-trivially on the equivariant cohomology ofthese spaces iff τ < 4

3(g− 1).


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Interpretation via Morse complex

Morse-Smale-Witten theory obtains more information about thecohomology ring structure using gradient flow lines for the Morsefunction.The Morse complex consists of the cohomology of the critical setstogether with differentials determined by the gradient flow lines.The Morse-Smale-Witten theory shows that the cohomology of thecomplex Hℓ = ker dℓ/ im dℓ−1 is the cohomology of the manifold.For Higgs bundles we can write down a conjectural Morse complexwith extra terms corresponding to the correction terms from ourprevious calculation.The differentials from (approximate) flow lines then correspond tocancellation of the correction terms. The Morse complex thencomputes the cohomology of the total space B.


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Relationship to Nakajima quiver varieties

There are many analogies between Higgs bundles and Nakajimaquiver varieties. These were originally developed to constructmoduli of instantons on ALE hyerkahler 4-manifolds(Kronheimer-Nakajima (1990)).Later, Nakajima (2001) constructs representations of quantumaffine algebras on spaces of topological invariants of quivervarieties.For certain special cases of low rank, the same idea of correctionterms gives the correct Betti numbers for the moduli space(previously computed by Nakajima (2004) and Hausel (2006)).Making these calculations rigorous would give a proof of Kirwansurjectivity for these moduli spaces.A more recent result (W. 2013) shows that for quivers there is arelationship between approximate flow lines and Nakajima’s Heckecorrespondence.


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Relationship to Nakajima quiver varieties (cont.)

Questions.Is it possible to carry out the Morse theory for Nakajimaquiver varieties? Other quivers with relations?Can we construct representations of quantum algebras on aMorse complex? What about other algebras?What about Higgs bundles of higher rank?


Documents

Topology of the moduli space of Higgs bundles over a ... · Hitchin-Kobayashi correspondence The Hitchin-Kobayashi correspondence is a general principle which asserts that the moduli