Asymptotics of certain families of Higgs bundles Qiongling Li - - PowerPoint PPT Presentation

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

Let S be a closed surface of genus g ≥ 2.

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 2 / 11

Set-up

Let S be a closed surface of genus g ≥ 2. Σ be a Riemann surface structure on S.

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 2 / 11

Set-up

Let 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)).

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 2 / 11

Set-up

Let 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.

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 2 / 11

Set-up

Let 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 two isomorphic connected components.

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 2 / 11

Set-up

Let 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 two isomorphic 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.

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 2 / 11

Set-up

Theorem(Hitchin)

Hitn(S) ∼ =

n

• j=2

H0(Σ, K j), where K is the canonical line bundle over Σ.

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 3 / 11

Set-up

Theorem(Hitchin)

Hitn(S) ∼ =

n

• j=2

H0(Σ, K j), where K is the canonical line bundle over Σ. The proof of this theorem uses Higgs bundle techniques.

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 3 / 11

Set-up

Theorem(Hitchin)

Hitn(S) ∼ =

n

• j=2

H0(Σ, K j), where K is the canonical line bundle over Σ. The proof of this theorem uses Higgs bundle techniques.The Higgs bundle parametrization of Hitn(S) is as follows: Bundle: E = K

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2

Higgs field: φ =         

n−1 2 q2

. . . qn−1 qn 1

n−3 2 q2

. . . qn−2 qn−1 ... ...

n−3 2 q2

1

n−1 2 q2

1         

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 3 / 11

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 h

• n E, solving the Hitchin equation

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 4 / 11

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 h

• n E, solving the Hitchin equation

FAh + [φ, φ∗h] = 0 ∇

0,1

Ahφ = 0

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 4 / 11

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 h

• n E, solving the Hitchin equation

FAh + [φ, φ∗h] = 0 ∇

0,1

Ahφ = 0

where

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

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 4 / 11

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 h

• n E, solving the Hitchin equation

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

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 4 / 11

Motivation

Question

Given a ray of (E, φ) parametrized by (qt

2, · · · , qt n), as t goes to ∞,

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 5 / 11

Motivation

Question

Given a ray of (E, φ) parametrized by (qt

2, · · · , qt n), as t goes to ∞,

(1) how is the solution metric ht?

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 5 / 11

Motivation

Question

Given a ray of (E, φ) parametrized by (qt

2, · · · , qt n), as t goes to ∞,

(1) how is the solution metric ht? (2) the flat connection ∇t?

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 5 / 11

Motivation

Question

Given a ray of (E, φ) parametrized by (qt

2, · · · , qt n), 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?

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 5 / 11

Motivation

Question

Given a ray of (E, φ) parametrized by (qt

2, · · · , qt n), 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 equation
• asymptotically. This is in general impossible.

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 5 / 11

Motivation

Question

Given a ray of (E, φ) parametrized by (qt

2, · · · , qt n), 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 equation
• asymptotically. This is in general impossible.
• However, we manage to understand the two cases:

(1) t(0, · · · , qn) (2) t(0, · · · , qn−1, 0).

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 5 / 11

Why these two cases?

Theorem (Baraglia n, Collier n − 1)

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

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2 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 6 / 11

Why these two cases?

Theorem (Baraglia n, Collier n − 1)

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

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2

Since the metric is diagonal, for (0, · · · , qn), the equations (⋆) FAh + [φ, φ∗h] = 0 ∇

0,1

Ahφ = 0

Simplify to ⌊ n

2⌋ coupled equations

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 6 / 11

Why these two cases?

Theorem (Baraglia n, Collier n − 1)

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

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2

Since the metric is diagonal, for (0, · · · , qn), the equations (⋆) FAh + [φ, φ∗h] = 0 ∇

0,1

Ahφ = 0

Simplify to ⌊ n

2⌋ coupled equations

         FA1 + t2h2

1 qn ∧ ¯

qn − h−1

1 h2 = 0

FAj + h−1

j−1hj − h−1

j

hj+1 = 0 1 < j < n 2 FA n

2 + h−1 n 2 −1h n 2 − h−2 n 2

= 0

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 6 / 11

Metric Asymptotics

Theorem (Collier-L)

For (0, · · · , tqn) ∈

n

• j=2

H0(K j), at any point p ∈ Σ away from the zeros of qn, as t→∞, the metric hj(t) on K

n+1−2j 2

admits the expansion hj(t) = (t|qn|)− n+1−2j

n

• 1 + O
• t− 2

n

• for all j

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

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 7 / 11

Metric Asymptotics

Theorem (Collier-L)

For (0, · · · , tqn) ∈

n

• j=2

H0(K j), at any point p ∈ Σ away from the zeros of qn, as t→∞, the metric hj(t) on K

n+1−2j 2

admits the expansion hj(t) = (t|qn|)− n+1−2j

n

• 1 + O
• t− 2

n

• 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 globally

depending on the parameters qn. Here, our results show that asymptotically, the solutions to Hitchin system only depend on the local values of qn.

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 7 / 11

Metric Asymptotics

Theorem (Collier-L)

For (0, · · · , tqn) ∈

n

• j=2

H0(K j), at any point p ∈ Σ away from the zeros of qn, as t→∞, the metric hj(t) on K

n+1−2j 2

admits the expansion hj(t) = (t|qn|)− n+1−2j

n

• 1 + O
• t− 2

n

• 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 globally

depending on the parameters qn. Here, our results show that asymptotically, the solutions to Hitchin system only depend on the local values of qn.

• Main Tool: the maximum principle (numerous times)

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 7 / 11

Parallel Transport Asymptotics

Given a path γ : [0, L] → Σ, let Tγ(t) : Eγ(0)→Eγ(L) be parallel transport

• perators along γ for flat connections ∇t.

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 8 / 11

Parallel Transport Asymptotics

Given a path γ : [0, L] → Σ, let Tγ(t) : Eγ(0)→Eγ(L) be parallel transport

• perators along γ for flat connections ∇t.

Theorem (Collier-L)

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

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 8 / 11

Parallel Transport Asymptotics

Given a path γ : [0, L] → Σ, let Tγ(t) : Eγ(0)→Eγ(L) be parallel transport

• perators 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
• t− 1

2n

• S

    e−t

1 n µ1

... e−t

1 n µn

    S−1

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 8 / 11

Parallel Transport Asymptotics

Given a path γ : [0, L] → Σ, let Tγ(t) : Eγ(0)→Eγ(L) be parallel transport

• perators 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
• t− 1

2n

• S

    e−t

1 n µ1

... e−t

1 n µn

    S−1 where S is constant unitary, and µk = 2Re

• γ e

2πki n z

• .

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

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 8 / 11

Parallel Transport Asymptotics

Given a path γ : [0, L] → Σ, let Tγ(t) : Eγ(0)→Eγ(L) be parallel transport

• perators 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
• t− 1

2n

• S

    e−t

1 n µ1

... e−t

1 n µn

    S−1 where S is constant unitary, and µk = 2Re

• γ e

2πki n z

• .

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

• This theorem relys on very technical analysis on the error estimates of the

Hermitian metric solution and the special structure of Toda lattice.

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 8 / 11

Parallel Transport Asymptotics

Given a path γ : [0, L] → Σ, let Tγ(t) : Eγ(0)→Eγ(L) be parallel transport

• perators 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
• t− 1

2n

• S

    e−t

1 n µ1

... e−t

1 n µn

    S−1 where S is constant unitary, and µk = 2Re

• γ e

2πki n z

• .

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

• This theorem relys on very technical analysis on the error estimates of the

Hermitian metric solution and the special structure of Toda lattice.

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

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 8 / 11

Work in Progress (Quiver bundles)

Instead of thinking Higgs bundle E = K

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2

φ =      qn 1 ... ... 1      : E → E ⊗ K

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 9 / 11

Work in Progress (Quiver bundles)

Instead of thinking Higgs bundle E = K

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2

φ =      qn 1 ... ... 1      : E → E ⊗ K We consider the following Quiver bundles (studied by Alvarez-Consul and Garcia-Prada.) E = L1 ⊕ L2 ⊕ · · · ⊕ Ln

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 9 / 11

Work in Progress (Quiver bundles)

Instead of thinking Higgs bundle E = K

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2

φ =      qn 1 ... ... 1      : E → E ⊗ K We consider the following Quiver bundles (studied by Alvarez-Consul and Garcia-Prada.) E = L1 ⊕ L2 ⊕ · · · ⊕ Ln φ =      φn φ1 ... ... φn−1      : E → E ⊗ L

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 9 / 11

Work in Progress (Quiver bundles)

Instead of thinking Higgs bundle E = K

n−1 2

⊕ K

n−3 2

⊕ · · · ⊕ K − n−3

2

⊕ K − n−1

2

φ =      qn 1 ... ... 1      : E → E ⊗ K We consider the following Quiver bundles (studied by Alvarez-Consul and Garcia-Prada.) E = L1 ⊕ L2 ⊕ · · · ⊕ Ln φ =      φn φ1 ... ... φn−1      : E → E ⊗ L where Lj and L is holomorphic line bundle, φj : Lj→Lj+1 ⊗ L is nonzero for all j.

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 9 / 11

Work in Progress (Quiver bundles)

For the Quiver bundles. E = L1 ⊕ L2 ⊕ · · · ⊕ Lk φ =      φn φ1 ... ... φn−1      : E → E ⊗ L where Li and L is holomorphic line bundle, φj : Lj→Lj+1 ⊗ L is nonzero for all j.

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 10 / 11

Work in Progress (Quiver bundles)

For the Quiver bundles. E = L1 ⊕ L2 ⊕ · · · ⊕ Lk φ =      φn φ1 ... ... φn−1      : 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.

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 10 / 11

Work in Progress (Quiver bundles)

For the Quiver bundles. E = L1 ⊕ L2 ⊕ · · · ⊕ Lk φ =      φn φ1 ... ... φn−1      : 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 quiver

bundle (E, φ) has the reality symmetry. This is an onging work with B. Collier.

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 10 / 11

Work in Progress (Quiver bundles)

For the Quiver bundles. E = L1 ⊕ L2 ⊕ · · · ⊕ Lk φ =      φn φ1 ... ... φn−1      : 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 quiver

bundle (E, φ) has the reality symmetry. This is an onging work with B. Collier. Application: Understanding asymptotics of maximal representation in Sp(4, R).

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 10 / 11

Thank You!

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 11 / 11