qiongling li (qgm-caltech) · qiongling li (qgm-caltech)(joint with brian collier, uiuc)...

Asymptotics of certain families of Higgs bundles

Qiongling Li (QGM-Caltech)

(joint with Brian Collier, UIUC)

AMS-EMS-SPM Porto Meeting– Higgs Bundles and Character Varieties

June, 2015

Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) (AMS-EMS-SPM Porto Meeting– Higgs Bundles and Character Varieties)Asymptotics of certain families of Higgs bundles June, 2015 1 / 11

Set-up

S be a closed surface of genus g ≥ 2.

Σ be a Riemann surface structure on S .

G be a real or complex reductive Lie group (usually SL(n,C) or PSL(n,R)).

G-Character variety

RepG = Hom+(π1(S),G )/G .

Teich(S) = {ρ : π1(S)→ PSL(2,R)|ρ is discrete and faithful}/PSL(2,R) has twoisomorphic connected components.

Composing with the the unique irreducible representation PSL(2,R)→ PSL(n,R)we obtain a distinguished component of RepPSL(n,R), called Hitchin component.

Set-up

G-Character variety

Set-up

G-Character variety

Set-up

G-Character variety

Set-up

G-Character variety

Set-up

G-Character variety

Set-up

Theorem(Hitchin)

Hitn(S) ∼=n⊕

H0(Σ,K j),

where K is the canonical line bundle over Σ.

The proof of this theorem uses Higgs bundle techniques.The Higgs bundleparametrization of Hitn(S) is as follows:

Bundle: E = Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

Higgs field: φ =

0 n−12 q2 . . . qn−1 qn

1 0 n−32 q2 . . . qn−2 qn−1

. . .. . .

n−32 q2

1 0 n−12 q2

Set-up

Theorem(Hitchin)

Hitn(S) ∼=n⊕

H0(Σ,K j),

The proof of this theorem uses Higgs bundle techniques.

The Higgs bundleparametrization of Hitn(S) is as follows:

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

Higgs field: φ =

0 n−12 q2 . . . qn−1 qn

1 0 n−32 q2 . . . qn−2 qn−1

. . .. . .

n−32 q2

1 0 n−12 q2

Set-up

Theorem(Hitchin)

Hitn(S) ∼=n⊕

H0(Σ,K j),

The proof of this theorem uses Higgs bundle techniques.The Higgs bundleparametrization of Hitn(S) is as follows:

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

Higgs field: φ =

0 n−12 q2 . . . qn−1 qn

1 0 n−32 q2 . . . qn−2 qn−1

. . .. . .

n−32 q2

1 0 n−12 q2

Set-up

Theorem (Hitchin n = 2, Simpson in the general case)

Let (E , φ) be a stable SL(n,C)-Higgs bundle, then there exists a unique metric hon E , solving the Hitchin equation

+ [φ, φ∗h ] = 0

∇0,1

Ahφ = 0

where• FAh

—curvature of the Chern connection Ah,• φ∗h—the hermitian adjoint of φ.Conversely, if (Ah, φ) solves Hitchin equation, then the Higgs bundle (E , φ) ispolystable.

If (Ah, φ) solves Hitchin equation, then Ah + φ+ φ∗h is a flat SL(n,C)-connection.

Set-up

Let (E , φ) be a stable SL(n,C)-Higgs bundle, then there exists a unique metric hon E , solving the Hitchin equation{

FAh+ [φ, φ∗h ] = 0

∇0,1

Ahφ = 0

where• FAh

Set-up

FAh+ [φ, φ∗h ] = 0

∇0,1

Ahφ = 0

where• FAh

—curvature of the Chern connection Ah,• φ∗h—the hermitian adjoint of φ.

Conversely, if (Ah, φ) solves Hitchin equation, then the Higgs bundle (E , φ) ispolystable.

Set-up

FAh+ [φ, φ∗h ] = 0

∇0,1

Ahφ = 0

where• FAh

Motivation

Question

Given a ray of (E , φ) parametrized by (qt2, · · · , qtn), as t goes to ∞,

(1) how is the solution metric ht?(2) the flat connection ∇t?(3) the parallel transport operator Tγ(t) along a path γ?

This question is also asked in the paper by Katzarkov, Noll, Pandit, and Simpson,referred as “Hitchin WKB problem”.

• To answer the above questions, the first step is to solve Hitchin equationasymptotically. This is in general impossible.

• However, we manage to understand the two cases:(1) t(0, · · · , qn)(2) t(0, · · · , qn−1, 0).

Motivation

Question

Given a ray of (E , φ) parametrized by (qt2, · · · , qtn), as t goes to ∞,(1) how is the solution metric ht?

(2) the flat connection ∇t?(3) the parallel transport operator Tγ(t) along a path γ?

Motivation

Question

Given a ray of (E , φ) parametrized by (qt2, · · · , qtn), as t goes to ∞,(1) how is the solution metric ht?(2) the flat connection ∇t?

(3) the parallel transport operator Tγ(t) along a path γ?

Motivation

Question

Given a ray of (E , φ) parametrized by (qt2, · · · , qtn), as t goes to ∞,(1) how is the solution metric ht?(2) the flat connection ∇t?(3) the parallel transport operator Tγ(t) along a path γ?

Motivation

Question

Motivation

Question

Why these two cases?

Theorem (Baraglia n, Collier n − 1)

For (0, · · · , qn) and (0, · · · , qn−1, 0), the metric solving Hitchin equation isdiagonal on the line bundles

Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

Since the metric is diagonal, for (0, · · · , qn), the equations

+ [φ, φ∗h ] = 0

∇0,1

Ahφ = 0

Simplify to b n2c coupled equationsF

A1+ t2h2

1qn ∧ q̄n − h−1

1h2 = 0

+ h−1j−1hj − h−1

jhj+1 = 0 1 < j <

+ h−1n2−1h n

2− h−2

Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

+ [φ, φ∗h ] = 0

∇0,1

Ahφ = 0

Simplify to b n2c coupled equations

A1+ t2h2

1qn ∧ q̄n − h−1

1h2 = 0

+ h−1j−1hj − h−1

jhj+1 = 0 1 < j <

+ h−1n2−1h n

2− h−2

Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

+ [φ, φ∗h ] = 0

∇0,1

Ahφ = 0

Simplify to b n2c coupled equationsF

A1+ t2h2

1qn ∧ q̄n − h−1

1h2 = 0

+ h−1j−1hj − h−1

jhj+1 = 0 1 < j <

+ h−1n2−1h n

2− h−2

Metric Asymptotics

Theorem (Collier-L)

For (0, · · · , tqn) ∈n⊕

H0(K j), at any point p ∈ Σ away from the zeros of qn, as

t→∞, the metric hj(t) on Kn+1−2j

2 admits the expansion

hj(t) = (t|qn|)−n+1−2j

(1 + O

))for all j

The analogous result is also true for (0, · · · , tqn−1, 0).

• Note that the Hitchin equation is highly nontrivial. The solutions are globallydepending on the parameters qn. Here, our results show that asymptotically, thesolutions to Hitchin system only depend on the local values of qn.

• Main Tool: the maximum principle (numerous times)

Metric Asymptotics

Theorem (Collier-L)

For (0, · · · , tqn) ∈n⊕

hj(t) = (t|qn|)−n+1−2j

(1 + O

))for all j

Metric Asymptotics

Theorem (Collier-L)

For (0, · · · , tqn) ∈n⊕

hj(t) = (t|qn|)−n+1−2j

(1 + O

))for all j

Parallel Transport Asymptotics

Given a path γ : [0, L]→ Σ̃, let Tγ(t) : Eγ(0)→Eγ(L) be parallel transportoperators along γ for flat connections ∇t .

Theorem (Collier-L)

If the path γ is “relatively far” away from zeros of qn, then as t→∞,

Tγ(t) =(Id + O

1n µ1

e−t1n µn

where S is constant unitary, and µk = 2Re(∫

2πkin z).

The analogous result is also true for (0, · · · , tqn−1, 0).• This theorem relys on very technical analysis on the error estimates of theHermitian metric solution and the special structure of Toda lattice.

• Both theorems are proved in (0, q3) case by J.Loftin.

Theorem (Collier-L)

Tγ(t) =(Id + O

1n µ1

e−t1n µn

2πkin z).

Theorem (Collier-L)

Tγ(t) =(Id + O

1n µ1

e−t1n µn

2πkin z).

Theorem (Collier-L)

Tγ(t) =(Id + O

1n µ1

e−t1n µn

2πkin z).

• This theorem relys on very technical analysis on the error estimates of theHermitian metric solution and the special structure of Toda lattice.

Theorem (Collier-L)

Tγ(t) =(Id + O

1n µ1

e−t1n µn

2πkin z).

Theorem (Collier-L)

Tγ(t) =(Id + O

1n µ1

e−t1n µn

2πkin z).

Work in Progress (Quiver bundles)

Instead of thinking Higgs bundle

E = Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

0 qn1 0

. . .. . .

: E → E ⊗ K

We consider the following Quiver bundles (studied by Alvarez-Consul andGarcia-Prada.)

E = L1 ⊕ L2 ⊕ · · · ⊕ Ln

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

where Lj and L is holomorphic line bundle, φj : Lj→Lj+1 ⊗ L is nonzero for all j .

E = Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

0 qn1 0

. . .. . .

: E → E ⊗ K

E = L1 ⊕ L2 ⊕ · · · ⊕ Ln

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

E = Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

0 qn1 0

. . .. . .

: E → E ⊗ K

E = L1 ⊕ L2 ⊕ · · · ⊕ Ln

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

E = Kn−12 ⊕ K

n−32 ⊕ · · · ⊕ K− n−3

2 ⊕ K− n−12

0 qn1 0

. . .. . .

: E → E ⊗ K

E = L1 ⊕ L2 ⊕ · · · ⊕ Ln

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

For the Quiver bundles.E = L1 ⊕ L2 ⊕ · · · ⊕ Lk

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

where Li and L is holomorphic line bundle, φj : Lj→Lj+1 ⊗ L is nonzero for all j .

• In this case, the quiver bundle equation coincides with Hitchin equation.

• We observe that our methods can be generalized to this case when the quiverbundle (E , φ) has the reality symmetry. This is an onging work with B. Collier.

Application: Understanding asymptotics of maximal representation in Sp(4,R).

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

0 φnφ1 0

. . .. . .

φn−1 0

: E → E ⊗ L

Thank You!

qiongling li (qgm-caltech) · qiongling li (qgm-caltech)(joint with brian collier, uiuc)...

Documents