The reverse Yang-Mills-Higgs heat flow in a neighbourhood ... · Yang-Mills-Higgs flow lines Definition. A YMH flow line is a continuous map x: R!B given by t 7!xt = (@ A t;ϕt)

.

.

. ..

.

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The reverse Yang-Mills-Higgs heat flow in aneighbourhood of a critical point

Graeme Wilkin (National University of Singapore)

New perspectives on Higgs bundlesSimons Centre for Geometry and Physics

16 June, 2016

arxiv:1605.05970

Graeme Wilkin (National University of Singapore) Reverse YMH heat flow

.. The space of Higgs bundles

Let X be a compact Riemann surface and let E → X be asmooth complex Hermitian vector bundle.Let GC be the complex gauge group and G ⊂ GC the subgrouppreserving the Hermitian structure.Let A0,1 be the space of holomorphic structures∂A : Ω0(E)→ Ω0,1(E) (satisfying Leibniz rule and (∂A)

2 = 0).Define the space of Higgs bundles

B := (∂A, ϕ) ∈ A0,1 × Ω0(End(E)⊗ K ) | ∂Aϕ = 0, ϕ ∧ ϕ = 0

The Yang-Mills-Higgs functional YMH : B → R is

YMH(∂A, ϕ) = ∥FA + [ϕ, ϕ∗]∥2L2 =

∫X|FA + [ϕ, ϕ∗]|2 dvol

A pair (∂A, ϕ) minimises YMH iff i ∗ (FA + [ϕ, ϕ∗]) = λ · id, whereλ = 2π

vol(X) ·deg(E)rank(E) =

2πvol(X) · slope(E).


.. The moduli space of Higgs bundles

Let Bmin ⊂ B denote the set of YMH minimisers.A Higgs bundle (∂A, ϕ) is stable (resp. semistable) if and only if

slope(F ) < slope(E) (resp. ≤)

for every proper, non-zero, ϕ-invariant subbundle F ⊂ E .(∂, ϕ) is polystable⇔ direct sum of stable Higgs bundles of thesame slopeThe moduli space of semistable Higgs bundles is

MHiggsss := Bss//GC ∼= Bpolyst ./GC.

Theorem. (Hitchin ’87, Simpson ’88)

MHiggsss

∼= Bmin/G

Equivalently (∂A, ϕ) is polystable iff there exists g ∈ GC suchthat g · (∂A, ϕ) ∈ Bmin.


.. The Yang-Mills-Higgs heat flow

Idea of Simpson’s proof.Given an initial condition (∂A0 , ϕ0) ∈ B, solve for aone-parameter family ht of metrics satisfying a nonlinear heatequation

dhdt

+∆A0h = (nonlinear terms in h)

Equivalently, solve for a 1-parameter family gt ∈ GC such that

dgdt

g−1t = −i ∗ (FA + [ϕ, ϕ∗]), g0 = id .

and set ht = g∗t gt . Then the one parameter family

(∂At , ϕt) = gt · (∂A0 , ϕ0) solves the YMH heat flow

∂A∂t

= i ∂At ∗ (FAt + [ϕt , ϕ∗t ])

∂ϕ

∂t= i[ϕt , ∗(FAt + [ϕt , ϕ

∗t ])]


.. The Yang-Mills-Higgs heat flow (cont.)

Simpson showed that if the initial condition is polystable, thenthe heat flow converges (on the space of metrics) to a metricminimising YMH.In general, critical points of YMH correspond to a direct sumdecomposition (E1, ϕ1)⊕ · · · ⊕ (Eℓ, ϕℓ) of YMH-minimisers.Theorem. (W., ’08)For any initial condition (∂A, ϕ) ∈ B

the YMH heat flow converges in the smooth topology to aunique critical point,this critical point is isomorphic to the graded object of theHarder-Narasimhan-Seshadri filtration of (∂A, ϕ), andthere is a continuous deformation retract from eachHarder-Narasimhan stratum to the associated critical set.

A priori the flow depends on the complex structure on X , butthe above theorem is valid for any complex structure.Question. What about flow lines between critical sets?


.. Yang-Mills-Higgs flow lines

Definition. A YMH flow line is a continuous map x : R→ Bgiven by

t 7→ xt = (∂At , ϕt)

which satisfies the YMH heat flow equation for all t ∈ R.A flow line connects two critical points xu and xℓ iff

xu = limt→−∞

xt , xℓ = limt→+∞

xt

Existing results of Swoboda (’12) and Janner-Swoboda (’15)construct flow lines for a perturbed Yang-Mills flow.Unfortunately the perturbations destroy the algebraic structureof the flow.Naito, Kozono, Maeda (’90) constructed the unstable manifoldfor the Yang-Mills flow using a contraction mapping technique.This method uses the manifold structure of the space ofconnections and does not say anything about the isomorphismclass of a point in the unstable manifold.


.. Yang-Mills-Higgs flow lines (cont.)

We would also like to study flow lines on the space of Higgsbundles (which is singular). In addition, we would like tounderstand the isomorphism classes of critical pointsconnected by flow lines.Therefore one needs to be careful about perturbing the flow ordefining projections using local coordinates.

Goal.1. Develop a method for constructing flow lines that (a) workson a singular space, and (b) allows us to classify theisomorphism classes.2. Use this construction to give an algebraic criterion for twocritical points to be connected by a YMH flow line.3. Refine this algebraic criterion further, to give a geometricdescription of the flow lines between two critical sets in terms ofsecant varieties of the underlying Riemann surface.


.. Example: Rank 2 Yang-Mills flow

Example. To illustrate the idea, consider the Yang-Mills flowwhen rank(E) = 2. Let xu = Lu

1 ⊕ Lu2 and xℓ = Lℓ

1 ⊕ Lℓ2 be critical

points and suppose that deg Lu1 > deg Lℓ

1 > deg Lℓ2 > deg Lu

2.(Therefore YM(xu) > YM(xℓ).)Theorem 1. If the upwards flow with initial condition ∂A on Econverges to xu then E admits an extension

0→ Lu2 → E → Lu

1 → 0

Conversely, if E admits such an extension then it is isomorphicto an initial condition which flows up to xu.Theorem 2. xu and xℓ are connected by a flow line iff thereexists ∂A on E admitting extensions as in the diagram below.

0 // Lu2

// EOO∼=

// Lu1

// 0

0 Lℓ2

oo Eoo Lℓ1

oo 0oo


.. Example: Rank 2 Yang-Mills flow (cont.)

How to determine whether an extension 0→ Lu2 → E → Lu

1 → 0flows down to a given critical point?The embedding X → PH1(L∗

1L2) determines the limit of thedownwards flow.Suppose deg Lu

1 − deg Lu2 > 2n. Let

Secn(X ) :=∪

p1,...,pn∈X

spanp1, . . . , pn ⊂ PH1((Lu1)

∗Lu2)

be the nth secant variety of X in PH1((Lu1)

∗Lu2).

Theorem 3. If e ∈ spanp1, . . . , pn ⊂ Secn(X ) \ Secn−1(X ),then 0→ Lu

2 → E → Lu1 → 0 with extension class e flows down

toL1[−p1 − · · · − pn]⊕ L2[p1 + · · ·+ pn]

This describes unbroken flow lines between critical sets.


.. The scattering construction

An important problem in scattering theory is to study aninteracting system on a scale which is large compared to thescale of the interaction itself.For example, in the system below a particle passes through aforce field which is very weak outside some compact set K .We would like to relate the asymptotic behaviour of the particleas t → −∞ to the asymptotic behaviour as t → +∞.

K

−∞← t

t → +∞


.. Structure of critical points

The methods used to solve the scattering problem can also beapplied to the problem of constructing flow lines for YMH.At a critical point x = (∂A, ϕ), the Higgs bundle splits as a directsum of YMH minima (E1, ϕ1)⊕ · · · ⊕ (Eℓ, ϕℓ) ordered so thatslope(Ej) < slope(Ek ) if j < k .The moment map β = ∗(FA + [ϕ, ϕ∗]) is then determined by theslope of the subbundles Ej .

iβ = diag(λ1, . . . , λn), λj =2π · slope(Ej)

vol(X ).

The slice Sx at a critical point x consists of all nearby Higgspairs y such that y − x is orthogonal to the GC-orbit through x .If x = (∂A, ϕ) and y = x + (a, φ) ∈ Sx , then

∂∗Aa− ∗[ϕ, ∗φ] = 0, ∂Aφ+ [a, ϕ] + [a, φ] = 0 (slice equations)


.. Structure of critical points (cont.)

Given x critical, all the points in the orbit G · x are also critical.The unstable set of x (denoted W−

x ) consists of all points ysuch that the reverse flow exists and converges to x in the C∞

topology.For y0 ∈ Sx , define the linearised flow by yt = e−iβt · y0.The unstable set of the linearised flow is a subset of the slicecalled the negative slice and denoted S−

x .

S−x := y0 ∈ Sx : lim

t→−∞e−iβt · y0 = x

The analytic part of the theorem is that S−x and W−

x arerelated by the action of GC.The algebraic part of the theorem involves analysing theHarder-Narasimhan types of isomorphism classes in S−

x .


.. The action of the linearised flow

Example. Suppose the critical point x corresponds to a directsum (E1, ϕ1)⊕ (E2, ϕ2).

iβ =

(λ1 00 λ2

), λ1 < λ2

If y0 ∈ Sx has the form

y0−x = (a, φ) =((

0 a0 0

),

(0 φ0 0

))∈ Ω0,1(End(E))⊕Ω1,0(End(E))

then the linearised flow acts by yt − x = e(λ1−λ2)t(y0 − x).In this case the point y0 ∈ Sx corresponds to a Higgs extension

0→ (E1, ϕ1)→ (E , ϕ)→ (E2, ϕ2)→ 0

The slice equations reduce to the condition that (a, φ) isharmonic.The linearized flow acts by scalar multiplication on theextension class.


.. The action of the linearised flow (cont.)

More generally, if a critical point x has the form

(E1, ϕ1)⊕ · · · ⊕ (Eℓ, ϕℓ)

with slope(Ej) < slope(Ek ) iff j < k , and a Higgs pair y ∈ Sxcorresponds to a filtration of Higgs bundles

(F1, ϕ1) ⊂ · · · ⊂ (Fℓ, ϕℓ)

such that (Fj , ϕj)/(Fj−1, ϕj−1) ∼= (Ej , ϕj), then the linearised flow“collapses” the filtration and converges to x as t →∞.


.. The unstable set via the scattering method

The scattering method is to flow upwards for time t using thelinearised flow (action of eiβt ), and then downwards for time tusing the YMH heat flow (action of gt ∈ GC).The goal is then to show that this process converges as t →∞.

b

b

bb

x

eiβt · y0

y0ft · y0

S−x

ft

gt (YMH flow)linearisedflow

Theorem. (W.)ht = f ∗t ft converges to asmooth metric h∞.There exists a smooth gaugetransformation f∞ such thatf ∗∞f∞ = h∞ and f∞ · y0 is in theunstable set of x .


.. Convergence of the scattering methodIdea of proof.0. Given a self-adjoint endomorphismh of E , define

σ(h) = tr(h) + tr(h−1)− 2 rank(E)

1. Prove the uniform bound

σ(ft(y)∗ft(y)) ≤ C∥y−x∥2C0 ∀y ∈ S−x

2. Use Donaldson’s distance de-creasing formula for the metric flow toshow that

σ((ft2(y0)−1ft1(y0))

∗ft2(y0)−1ft1(y0))

≤ σ(ft1−t2(yt2)∗ft1−t2(yt2))

≤ C∥yt2 − x∥2 ≤ Ce−Kt2

b

b

b

b

b b

b

x

yt1 = eiβt1 · y0

y0

yt2

ft1(y0) · y0

ft2(y0) · y0

ft1−t2(yt2) · yt2

S−x


.. Convergence of the scattering method (cont.)

Therefore the sequence of metrics ht = ft(y0)∗ft(y0) has the

Cauchy property with respect to σ

σ(ht1h−1t2 ) ≤ Ce−KT for all t1, t2 ≥ T

Can then show that it converges in the C0 norm.The smoothness of the limit metric h∞ then follows from thesmoothing properties of the heat equation (derivatives of thecurvature are bounded).Given any smooth gauge transformation f ∈ GC such thatf ∗f = h∞, can construct a solution to the reverse YMH heat flowfrom f · y0 which converges in the C∞ topology to a critical pointg · x ∈ G · x .The G-equivariance of the flow then implies that there is a flowline from g−1 · f · y0 to x .Every point in the negative slice S−

x is complex gaugeequivalent to a point in the unstable set W−

x .Graeme Wilkin (National University of Singapore) Reverse YMH heat flow

.. Convergence of the scattering method (cont.)

Can then reverse the process to show that every point y0 in theunstable set W−

x is complex gauge equivalent to a point in thenegative slice S−

x .

b

b

bb

x

gt · y0

ft · y0

y0

S−x

W−x

ft

gt (YMH flow)

linearisedflow

Again, we can show thatht = f ∗t ft converges to asmooth metric h∞.There exists a smooth gaugetransformation f∞ such thatf ∗∞f∞ = h∞ and f∞ · y0 is in thenegative slice S−

x .

Every point in W−x is complex gauge equivalent to a point in S−

x .


.. Classification of points in the unstable set

Therefore, the isomorphism classes of points in the unstableset W−

x are completely classified by the isomorphism classesof points in the negative slice S−

x .The isomorphism classes in the negative slice can be classifiedin terms of filtrations.Definition. Let x ∈ B be a critical point of YMH correspondingto a direct sum decomposition (E1, ϕ1)⊕ · · · ⊕ (Eℓ, ϕℓ) orderedso that slope(Ej) < slope(Ek ) for all j < k . A Higgs pair y ∈ Badmits a filtration upwards compatible with x iff y admits afiltration (F1, φ1) ⊂ · · · ⊂ (Fℓ, φℓ) such that(Fj , φj)/(Fj−1, φj−1) ∼= (Ej , ϕj).

Theorem. (W.) Let x be a critical point of YMH. Then eachpoint y ∈W−

x is complex gauge equivalent to a Higgs bundleadmitting a filtration upwards compatible with x .Conversely, if y ∈ B admits a filtration upwards compatible withx then it is complex gauge equivalent to a point in W−

x .Graeme Wilkin (National University of Singapore) Reverse YMH heat flow

..

An algebraic criterion for critical points to beconnected by flow lines

Corollary. (W.)Two critical points xu, xℓ ∈ B with YMH(xu) > YMH(xℓ) areconnected by a YMH flow line if and only if there exists y ∈ Badmitting a filtration upwards compatible with xu, and such thatthe graded object of the Harder-Narasimhan-Seshadri doublefiltration of y is isomorphic to the lower critical point xℓ.

Simple case. Consider a critical point x corresponding to adirect sum of stable Higgs bundles (E1, ϕ1)⊕ (E2, ϕ2).Then points in the negative slice correspond to Higgsextensions with harmonic extension class

0→ (E1, ϕ1)→ (E , ϕ)→ (E2, ϕ2)→ 0

The previous theorem says that a Higgs bundle is isomorphic toa point in W−

x if and only if it is isomorphic to such an extension.


.. A geometric criterion for convergence

Question. What are the possible Harder-Narasimhan types ofsuch extensions? Can we describe these using the geometry ofextension classes?First consider the simple case of the rank 2 Yang-Mills flow(E1,E2 are line bundles, slope(E1) < slope(E2) and ϕ = 0).A point in the negative slice corresponds to an extension

0→ E1 → E → E2 → 0

If E ′ is the maximal semistable subbundle of E thenslope(E ′) > slope(E1), so Hom(E ′,E1) = 0, therefore E ′ is asubsheaf of E2.

E ′

~~~~~~~~

0 // E1 // E // E2 // 0


.. First step: An extension of line bundles (cont.)

Conversely, if E ′ is a locally free subsheaf of E2 then a result ofNarasimhan and Ramanan shows that E ′ lifts to a subsheaf ofE iff the extension class e ∈ H1(E∗

2 E1) pulls back to0 ∈ H1((E ′)∗E1).

E ′

~~~~

0 // E1 // E // E2 // 0

Therefore, we can ask the following questions.Question 1. Given a subsheaf E ′ ⊂ E2, which extensionclasses e ∈ H1(E∗

2 E1) are in the kernel of the pullback mapH1(E∗

2 E1)→ H1((E ′)∗E1)?Question 2. Given an extension class e ∈ H1(E∗

2 E1), for whichsubsheaves E ′ ⊂ E2 is e ∈ ker(H1(E∗

2 E1)→ H1((E ′)∗E1))?Which of these subsheaves lifts to the maximal semistablesubbundle of E?


.. First step: An extension of line bundles

When is E ′ the maximal semistable subbundle of E?Suppose E ′ is destabilising (deg(E ′) > 1

2(deg(E1) + deg(E2))).E ′ is a locally free subsheaf of E2, therefore E ′ = E2[−D] forsome effective divisor D =

∑nipi on X .

Then (following Lange-Narasimhan) e is in the linear span ofD ⊂ X via the embedding X → PH1(E∗

2 E1).(Recall deg E∗

2 E1 < deg(E ′)∗E1 < −12(deg(E ′)− deg E1) < 0.)

Take minimal subset Dmin of D such that e is in linear span ofDmin. Then max. semistable subbundle of E is E2[−Dmin].Inequality on deg(E ′)⇒ nondegeneracy of 2|D|th secant varietyTherefore

Extension classes for (possibly broken) Yang-Mills flowlines from E1 ⊕ E2 to E ′

1 ⊕ E2[−D] are determined by asubspace of the |D|th secant variety of X in PH1(E∗

2 E1).The unbroken flow lines correspond to an open subset ofthis subspace.


.. Higher rank case

Now consider an extension E of a line bundle by a rank nsemistable bundle E2. After tensoring the whole setup by a linebundle, we can reduce to the case of an extension of O by E2.The same argument as before shows that the maximalsemistable subbundle of E is a subsheaf of E2.

E ′

~~~~

0 // O // E // E2 // 0

Conversely, a locally free subsheaf E ′ ⊂ E2 lifts to a subsheaf ofE iff the extension class is in the kernel of H1(E∗

2 )→ H1((E ′)∗).If rank(E ′) = rank(E2) then E ′ is a Hecke modification of E2.Therefore we can construct Hecke modifications usingYang-Mills flow lines.


.. Constructing Hecke modifications

More precisely, using the same idea as before (and results ofChoe-Hitching on secant varieties in PH1(E∗

2 )), we can provethe following result.

Theorem. (W.)Let O ⊕ E2 be a critical point of the Yang-Mills functional. Thena semistable bundle E ′ is a Hecke modification of E2 if and onlyif there is a (possibly broken) flow line from O ⊕ E2 to a criticalpoint isomorphic to L⊕ E ′ for some line bundle L.The unbroken flow lines are determined by secant varieties ofPE∗

2 in PH1(X ,E∗2 )∼= PH0(PE∗

2 , π∗KX ⊗OPE∗

2(1))∗.

Describing the unbroken flow lines explicitly as before requiresa bound on the Segre invariant of E2 so that the secantvarieties are nondegenerate.


.. YMH flow lines with nonzero Higgs field

Now we would like to carry out an analogous process when theHiggs field is non-zero.One complication is that a Hecke modification of theholomorphic bundle may introduce poles into the Higgs field.To preserve the holomorphicity of the Higgs field ϕ, we needthe Hecke modification to be compatible with ϕ.More precisely, given v ∈ E∗

p and µ ∈ C, want to impose thecondition that v(ϕ(s)) = µv(s). Equivalently, ϕ(s)− µs ∈ ker vand so ϕ preserves the locally free subsheaf E ′ = ker v .

0 // E ′ //

ϕ′

E v //

ϕ

Cp //

µ

0

0 // E ′ ⊗ K // E ⊗ K v // Cp // 0

When the discriminant of ϕ has simple zeros then Witten showsthat the space of Hecke modifications is the spectral curve.


.. YMH flow lines with nonzero Higgs field (cont.)

Which Hecke modifications correspond to extensions?Now we can introduce the Higgs field into the previousargument for the Yang-Mills flow.Suppose first that deg E ′ = deg E2 − 1.Let (E ′, ϕ′

2) be a Hecke modification of (E2, ϕ2) over p ∈ X .As before, if the Higgs extension class pulls back to zero then(E ′, ϕ′

2) is a subsheaf of (E , ϕ).

(E ′, ϕ′2)

zzt tttt

0 // (O, ϕ1) // (E , ϕ) // (E2, ϕ2) // 0

Conversely, we would like to construct a Higgs extension classwhich pulls back to zero. Can do this if ϕ1(p) matches theeigenvalue of the Hecke modification.Can find such ϕ1 since the canonical system is basepoint free.


.. YMH flow lines with nonzero Higgs field (cont.)

Therefore we see that the spectral curve determines the flowlines between generic points of adjacent critical sets.Theorem. (W.)Let (E2, ϕ2) be a stable Higgs bundle with deg E2 > 2.Then (E ′, ϕ′

2) with deg E ′ = deg E2 − 1 is a Hecke modificationof (E2, ϕ2) if and only if there exists ϕ1 ∈ H0(K ) and a YMH flowline between (E2, ϕ2)⊕ (O, ϕ1) and (E ′, ϕ′

2)⊕ (L, ϕL) for someline bundle L.

(E ′, ϕ′2)

zzt tttt

0 // (O, ϕ1) // (E , ϕ) // (E2, ϕ2) // 0

If the discriminant of ϕ2 has simple zeros then the flow lines aredetermined by the spectral curve inside the projectivisation ofthe space of Higgs extensions, where ϕ1 is allowed to vary.


.. The Hecke correspondence via YMH flow lines

In the case deg E ′ = deg E − 1, let Cℓ be the lower critical setand Cu be the upper critical set. Suppose deg E is coprime torank E .There are canonical projections Cℓ →Mss

Higgs(E′) and

Cu →MstHiggs(E).

Let Fℓ,u denote the space of flow lines connecting Cℓ and Cu.Let Pℓ,u ⊂ Cℓ × Cu denote the space of pairs of critical pointsconnected by a flow line.

Fℓ,u

Pℓ,u

!!CCC

CCCC

Cℓ Cu


.. The Hecke correspondence via YMH flow lines (cont.)

Corollary. (W.) The induced correspondence Hℓ,u is the Heckecorrespondence.

Fℓ,u

Pℓ,u

yyssssss

sssss

%%KKKKK

KKKKKK

Cℓ

Hℓ,u

yytttttt

tttt

%%JJJJJ

JJJJJ

Cu

Mss

Higgs(E′) Mst

Higgs(E)


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The reverse Yang-Mills-Higgs heat flow in a neighbourhood ... · Yang-Mills-Higgs flow lines Definition. A YMH flow line is a continuous map x: R!B given by t 7!xt = (@ A t;ϕt)