H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES Higgs bundles, spectral data and applications Laura P. Schaposnik University of Illinois AMS-EMS-SPM International Meeting 2015 Porto, 10 - 13 June 2015
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE PLAN Higgs bundles Based on: 1 The Hitchin fibration. arXiv:1301.1981 Spectral data approach. arXiv:1111.2550 Hyperkähler structure. Real slices of Higgs bundles 2 And work w/ A triple of involutions. D. Baraglia Branes and Langlands duality. 1309.1195 ( B , A , A ) -branes and low rank isogenies. 1506.00372 Monodromy of ( B , A , A ) -branes 3 & /w ( B , A , A ) -branes and split real forms. S. Bradlow Braids and polyhedrons. 1506.XXXXX Character varieties.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE H ITCHIN FIBRATION C ONSIDER A COMPACT R IEMANN SURFACE Σ OF g ≥ 2 AND K := T ∗ Σ . A Higgs bundle is a pair ( E , Φ) for: E a holomorphic vector bundle, and Φ : E → E ⊗ K holomorphic map.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE H ITCHIN FIBRATION C ONSIDER A COMPACT R IEMANN SURFACE Σ OF g ≥ 2 AND K := T ∗ Σ . A Higgs bundle is a pair ( E , Φ) for: E a holomorphic vector bundle, and Φ : E → E ⊗ K holomorphic map. For G c ⊂ GL ( n , C ) , define G c -Higgs bundles by adding conditions on ( E , Φ) .
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE H ITCHIN FIBRATION C ONSIDER A COMPACT R IEMANN SURFACE Σ OF g ≥ 2 AND K := T ∗ Σ . A Higgs bundle is a pair ( E , Φ) for: E a holomorphic vector bundle, and Φ : E → E ⊗ K holomorphic map. For G c ⊂ GL ( n , C ) , define G c -Higgs bundles by adding conditions on ( E , Φ) . ( E , Φ) is an SL ( n , C ) -Higgs bundle if Λ n E ∼ = O , and Tr (Φ) = 0.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE H ITCHIN FIBRATION C ONSIDER A COMPACT R IEMANN SURFACE Σ OF g ≥ 2 AND K := T ∗ Σ . Stability can be defined through A Higgs bundle is a pair ( E , Φ) for: Φ -invariant subbundles of E . E a holomorphic vector bundle, and Φ : E → E ⊗ K holomorphic map. For G c ⊂ GL ( n , C ) , define G c -Higgs bundles by adding conditions on ( E , Φ) . ( E , Φ) is an SL ( n , C ) -Higgs bundle if Λ n E ∼ = O , and Tr (Φ) = 0. For G real form of G c , the definition can be extended to G -Higgs bundles.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE H ITCHIN FIBRATION C ONSIDER A COMPACT R IEMANN SURFACE Σ OF g ≥ 2 AND K := T ∗ Σ . Stability can be defined through A Higgs bundle is a pair ( E , Φ) for: Φ -invariant subbundles of E . E a holomorphic vector bundle, and the moduli space of Φ : E → E ⊗ K holomorphic map. M G S- equivalence classes of polystable For G c ⊂ GL ( n , C ) , define G -Higgs bundles G c -Higgs bundles by adding conditions on ( E , Φ) . ( E , Φ) is an SL ( n , C ) -Higgs bundle if Λ n E ∼ = O , and Tr (Φ) = 0. For G real form of G c , the definition can be extended to G -Higgs bundles.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE H ITCHIN FIBRATION C ONSIDER A COMPACT R IEMANN SURFACE Σ OF g ≥ 2 AND K := T ∗ Σ . Stability can be defined through A Higgs bundle is a pair ( E , Φ) for: Φ -invariant subbundles of E . E a holomorphic vector bundle, and the moduli space of Φ : E → E ⊗ K holomorphic map. M G S- equivalence classes of polystable For G c ⊂ GL ( n , C ) , define G -Higgs bundles G c -Higgs bundles by adding conditions on ( E , Φ) . For classical Higgs bundles the Hitchin fibration is ( E , Φ) is an SL ( n , C ) -Higgs bundle if Λ n E ∼ = O , and Tr (Φ) = 0. n H 0 (Σ , K i ) � h : M G c A G c = → i = 1 For G real form of G c , the ( Tr (Φ) , . . . , Tr (Φ n )) ( E , Φ) �→ definition can be extended to G -Higgs bundles.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES T HE H ITCHIN FIBRATION C ONSIDER A COMPACT R IEMANN SURFACE Σ OF g ≥ 2 AND K := T ∗ Σ . Stability can be defined through A Higgs bundle is a pair ( E , Φ) for: Φ -invariant subbundles of E . E a holomorphic vector bundle, and the moduli space of Φ : E → E ⊗ K holomorphic map. M G S- equivalence classes of polystable For G c ⊂ GL ( n , C ) , define G -Higgs bundles G c -Higgs bundles by adding conditions on ( E , Φ) . For classical Higgs bundles the Hitchin fibration is ( E , Φ) is an SL ( n , C ) -Higgs bundle if Λ n E ∼ = O , and Tr (Φ) = 0. n H 0 (Σ , K i ) � h : M G c A G c = → i = 1 For G real form of G c , the ( Tr (Φ) , . . . , Tr (Φ n )) ( E , Φ) �→ definition can be extended to G -Higgs bundles.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES S PECTRAL DATA APPROACH H OW TO UNDERSTAND THE H ITCHIN MAP h : M G → A G For classical Higgs bundles the Hitchin map sends ( E , Φ) to the coeffi- cients a i = Tr (Φ i ) of the characteristic polynomial,
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES S PECTRAL DATA APPROACH H OW TO UNDERSTAND THE H ITCHIN MAP h : M G → A G For classical Higgs bundles the Hitchin map sends ( E , Φ) to the coeffi- cients a i = Tr (Φ i ) of the characteristic polynomial, and the polynomial det (Φ − η I ) = η n + a 1 η n − 1 + . . . a n − 1 η + a n = 0 defines the spectral surge π : S → Σ in the total space of K .
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES S PECTRAL DATA APPROACH H OW TO UNDERSTAND THE H ITCHIN MAP h : M G → A G For classical Higgs bundles the Hitchin map sends ( E , Φ) to the coeffi- cients a i = Tr (Φ i ) of the characteristic polynomial, and the polynomial det (Φ − η I ) = η n + a 1 η n − 1 + . . . a n − 1 η + a n = 0 defines the spectral surge π : S → Σ in the total space of K . The spectral data associated to a Higgs bundle is the curve S together with a line bundle on it satisfying certain conditions.
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES S PECTRAL DATA APPROACH H OW TO UNDERSTAND THE H ITCHIN MAP h : M G → A G For classical Higgs bundles the Hitchin map sends ( E , Φ) to the coeffi- cients a i = Tr (Φ i ) of the characteristic polynomial, and the polynomial det (Φ − η I ) = η n + a 1 η n − 1 + . . . a n − 1 η + a n = 0 defines the spectral surge π : S → Σ in the total space of K . The spectral data associated to a Higgs bundle is the curve S together with a line bundle on it satisfying certain conditions. GL ( n , C ) -Higgs bundles. Smooth fibres are Jac ( S ) .
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES S PECTRAL DATA APPROACH H OW TO UNDERSTAND THE H ITCHIN MAP h : M G → A G For classical Higgs bundles the Hitchin map sends ( E , Φ) to the coeffi- cients a i = Tr (Φ i ) of the characteristic polynomial, and the polynomial det (Φ − η I ) = η n + a 1 η n − 1 + . . . a n − 1 η + a n = 0 defines the spectral surge π : S → Σ in the total space of K . The spectral data associated to a Higgs bundle is the curve S together with a line bundle on it satisfying certain conditions. GL ( n , C ) -Higgs bundles. SL ( n , C ) -Higgs bundles. Smooth fibres are Jac ( S ) . Smooth fibres are Prym ( S , Σ) .
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES S PECTRAL DATA APPROACH H OW TO UNDERSTAND THE H ITCHIN MAP h : M G → A G For classical Higgs bundles the Hitchin map sends ( E , Φ) to the coeffi- cients a i = Tr (Φ i ) of the characteristic polynomial, and the polynomial det (Φ − η I ) = η n + a 1 η n − 1 + . . . a n − 1 η + a n = 0 defines the spectral surge π : S → Σ in the total space of K . The spectral data associated to a Higgs bundle is the curve S together with a line bundle on it satisfying certain conditions. GL ( n , C ) -Higgs bundles. SL ( n , C ) -Higgs bundles. Smooth fibres are Jac ( S ) . Smooth fibres are Prym ( S , Σ) . We say a fibre is smooth if it is over a point defining a smooth S .
H IGGS BUNDLES R EAL SLICES OF H IGGS BUNDLES M ONODROMY OF ( B , A , A ) - BRANES S PECTRAL DATA APPROACH H OW TO UNDERSTAND THE H ITCHIN MAP h : M G → A G For classical Higgs bundles the Hitchin map sends ( E , Φ) to the coeffi- cients a i = Tr (Φ i ) of the characteristic polynomial, and the polynomial det (Φ − η I ) = η n + a 1 η n − 1 + . . . a n − 1 η + a n = 0 defines the spectral surge π : S → Σ in the total space of K . The spectral data associated to a Higgs bundle is the curve S together with a line bundle on it satisfying certain conditions. GL ( n , C ) -Higgs bundles. SL ( n , C ) -Higgs bundles. Smooth fibres are Jac ( S ) . Smooth fibres are Prym ( S , Σ) . We say a fibre is smooth if it is over a point defining a smooth S . Important: S smooth ⇒ char (Φ) irreducible
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