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 Rep G = Hom + ( π 1 ( 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 Rep G = Hom + ( π 1 ( S ) , G ) / G . Teich ( S ) = { ρ : π 1 ( 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 Rep G = Hom + ( π 1 ( S ) , G ) / G . Teich ( S ) = { ρ : π 1 ( 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 Rep PSL ( 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) n � Hit n ( S ) ∼ H 0 (Σ , K j ) , = j =2 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) n � Hit n ( S ) ∼ H 0 (Σ , K j ) , = j =2 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) n � Hit n ( S ) ∼ H 0 (Σ , K j ) , = j =2 where K is the canonical line bundle over Σ. The proof of this theorem uses Higgs bundle techniques.The Higgs bundle parametrization of Hit n ( S ) is as follows: n − 1 n − 3 ⊕ · · · ⊕ K − n − 3 ⊕ K − n − 1 Bundle: E = K ⊕ K 2 2 2 2 n − 1 0 2 q 2 . . . q n − 1 q n n − 3 1 0 2 q 2 . . . q n − 2 q n − 1 ... ... Higgs field: φ = n − 3 2 q 2 n − 1 1 0 2 q 2 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
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 on 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 on E , solving the Hitchin equation � F A h + [ φ, φ ∗ h ] = 0 0 , 1 ∇ A h φ = 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 on E , solving the Hitchin equation � F A h + [ φ, φ ∗ h ] = 0 0 , 1 ∇ A h φ = 0 where • F A h —curvature of the Chern connection A h , • 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 on E , solving the Hitchin equation � F A h + [ φ, φ ∗ h ] = 0 0 , 1 ∇ A h φ = 0 where • F A h —curvature of the Chern connection A h , • φ ∗ h —the hermitian adjoint of φ . Conversely, if ( A h , φ ) solves Hitchin equation, then the Higgs bundle ( E , φ ) is polystable. If ( A h , φ ) solves Hitchin equation, then A h + 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 ( q t 2 , · · · , q t 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 ( q t 2 , · · · , q t n ), as t goes to ∞ , (1) how is the solution metric h 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 ( q t 2 , · · · , q t n ), as t goes to ∞ , (1) how is the solution metric h t ? (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 ( q t 2 , · · · , q t n ), as t goes to ∞ , (1) how is the solution metric h t ? (2) the flat connection ∇ t ? (3) the parallel transport operator T γ ( 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 ( q t 2 , · · · , q t n ), as t goes to ∞ , (1) how is the solution metric h t ? (2) the flat connection ∇ t ? (3) the parallel transport operator T γ ( 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 ( q t 2 , · · · , q t n ), as t goes to ∞ , (1) how is the solution metric h t ? (2) the flat connection ∇ t ? (3) the parallel transport operator T γ ( 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 , · · · , q n ) (2) t (0 , · · · , q n − 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
Recommend
More recommend
