qiongling li (qgm-caltech) · qiongling li (qgm-caltech)(joint with brian collier, uiuc)...
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![Page 1: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/1.jpg)
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
![Page 2: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/2.jpg)
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 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.
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
![Page 3: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/3.jpg)
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 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.
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
![Page 4: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/4.jpg)
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 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.
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
![Page 5: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/5.jpg)
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 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.
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
![Page 6: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/6.jpg)
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 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.
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
![Page 7: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/7.jpg)
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 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.
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
![Page 8: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/8.jpg)
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 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
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 3 / 11
![Page 9: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/9.jpg)
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 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
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 3 / 11
![Page 10: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/10.jpg)
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 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
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 3 / 11
![Page 11: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/11.jpg)
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
{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.
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
![Page 12: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/12.jpg)
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{
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.
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
![Page 13: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/13.jpg)
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{
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.
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
![Page 14: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/14.jpg)
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{
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.
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
![Page 15: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/15.jpg)
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).
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
![Page 16: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/16.jpg)
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).
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
![Page 17: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/17.jpg)
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).
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
![Page 18: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/18.jpg)
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).
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
![Page 19: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/19.jpg)
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).
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
![Page 20: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/20.jpg)
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).
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
![Page 21: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/21.jpg)
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
(?)
{FAh
+ [φ, φ∗h ] = 0
∇0,1
Ahφ = 0
Simplify to b n2c coupled equationsF
A1+ t2h2
1qn ∧ q̄n − h−1
1h2 = 0
FAj
+ h−1j−1hj − h−1
jhj+1 = 0 1 < j <
n
2
FA n2
+ h−1n2−1h n
2− h−2
n2
= 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
![Page 22: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/22.jpg)
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
(?)
{FAh
+ [φ, φ∗h ] = 0
∇0,1
Ahφ = 0
Simplify to b n2c coupled equations
F
A1+ t2h2
1qn ∧ q̄n − h−1
1h2 = 0
FAj
+ h−1j−1hj − h−1
jhj+1 = 0 1 < j <
n
2
FA n2
+ h−1n2−1h n
2− h−2
n2
= 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
![Page 23: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/23.jpg)
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
(?)
{FAh
+ [φ, φ∗h ] = 0
∇0,1
Ahφ = 0
Simplify to b n2c coupled equationsF
A1+ t2h2
1qn ∧ q̄n − h−1
1h2 = 0
FAj
+ h−1j−1hj − h−1
jhj+1 = 0 1 < j <
n
2
FA n2
+ h−1n2−1h n
2− h−2
n2
= 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
![Page 24: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/24.jpg)
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 Kn+1−2j
2 admits the expansion
hj(t) = (t|qn|)−n+1−2j
n
(1 + O
(t−
2n
))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)
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
![Page 25: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/25.jpg)
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 Kn+1−2j
2 admits the expansion
hj(t) = (t|qn|)−n+1−2j
n
(1 + O
(t−
2n
))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)
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
![Page 26: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/26.jpg)
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 Kn+1−2j
2 admits the expansion
hj(t) = (t|qn|)−n+1−2j
n
(1 + O
(t−
2n
))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)
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
![Page 27: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/27.jpg)
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
(t−
12n
))S
e−t
1n µ1
. . .
e−t1n µn
S−1
where S is constant unitary, and µk = 2Re(∫
γe
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.
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
![Page 28: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/28.jpg)
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
(t−
12n
))S
e−t
1n µ1
. . .
e−t1n µn
S−1
where S is constant unitary, and µk = 2Re(∫
γe
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.
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
![Page 29: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/29.jpg)
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
(t−
12n
))S
e−t
1n µ1
. . .
e−t1n µn
S−1
where S is constant unitary, and µk = 2Re(∫
γe
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.
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
![Page 30: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/30.jpg)
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
(t−
12n
))S
e−t
1n µ1
. . .
e−t1n µn
S−1
where S is constant unitary, and µk = 2Re(∫
γe
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.
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
![Page 31: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/31.jpg)
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
(t−
12n
))S
e−t
1n µ1
. . .
e−t1n µn
S−1
where S is constant unitary, and µk = 2Re(∫
γe
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.
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
![Page 32: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/32.jpg)
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
(t−
12n
))S
e−t
1n µ1
. . .
e−t1n µn
S−1
where S is constant unitary, and µk = 2Re(∫
γe
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.
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
![Page 33: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/33.jpg)
Work in Progress (Quiver bundles)
Instead of thinking Higgs bundle
E = Kn−12 ⊕ K
n−32 ⊕ · · · ⊕ K− n−3
2 ⊕ K− n−12
φ =
0 qn1 0
. . .. . .
1 0
: E → E ⊗ K
We consider the following Quiver bundles (studied by Alvarez-Consul andGarcia-Prada.)
E = L1 ⊕ L2 ⊕ · · · ⊕ Ln
φ =
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 .
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
![Page 34: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/34.jpg)
Work in Progress (Quiver bundles)
Instead of thinking Higgs bundle
E = Kn−12 ⊕ K
n−32 ⊕ · · · ⊕ K− n−3
2 ⊕ K− n−12
φ =
0 qn1 0
. . .. . .
1 0
: E → E ⊗ K
We consider the following Quiver bundles (studied by Alvarez-Consul andGarcia-Prada.)
E = L1 ⊕ L2 ⊕ · · · ⊕ Ln
φ =
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 .
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
![Page 35: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/35.jpg)
Work in Progress (Quiver bundles)
Instead of thinking Higgs bundle
E = Kn−12 ⊕ K
n−32 ⊕ · · · ⊕ K− n−3
2 ⊕ K− n−12
φ =
0 qn1 0
. . .. . .
1 0
: E → E ⊗ K
We consider the following Quiver bundles (studied by Alvarez-Consul andGarcia-Prada.)
E = L1 ⊕ L2 ⊕ · · · ⊕ Ln
φ =
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 .
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
![Page 36: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/36.jpg)
Work in Progress (Quiver bundles)
Instead of thinking Higgs bundle
E = Kn−12 ⊕ K
n−32 ⊕ · · · ⊕ K− n−3
2 ⊕ K− n−12
φ =
0 qn1 0
. . .. . .
1 0
: E → E ⊗ K
We consider the following Quiver bundles (studied by Alvarez-Consul andGarcia-Prada.)
E = L1 ⊕ L2 ⊕ · · · ⊕ Ln
φ =
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 .
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
![Page 37: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/37.jpg)
Work in Progress (Quiver bundles)
For the Quiver bundles.E = L1 ⊕ L2 ⊕ · · · ⊕ Lk
φ =
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).
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
![Page 38: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/38.jpg)
Work in Progress (Quiver bundles)
For the Quiver bundles.E = L1 ⊕ L2 ⊕ · · · ⊕ Lk
φ =
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).
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
![Page 39: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/39.jpg)
Work in Progress (Quiver bundles)
For the Quiver bundles.E = L1 ⊕ L2 ⊕ · · · ⊕ Lk
φ =
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).
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
![Page 40: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/40.jpg)
Work in Progress (Quiver bundles)
For the Quiver bundles.E = L1 ⊕ L2 ⊕ · · · ⊕ Lk
φ =
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).
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
![Page 41: Qiongling Li (QGM-Caltech) · Qiongling Li (QGM-Caltech)(joint with Brian Collier, UIUC) Asymptotics of certain families of Higgs bundles(AMS-EMS-SPM Porto Meeting{ Higgs Bundles](https://reader033.vdocuments.us/reader033/viewer/2022050422/5f9144bc7c5fcb7c727cade4/html5/thumbnails/41.jpg)
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