Ends of moduli spaces of Higgs bundles Hartmut Weiss (CAU Kiel) - PowerPoint PPT Presentation

Ends of moduli spaces of Higgs bundles Hartmut Weiss (CAU Kiel) joint with Rafe Mazzeo, Jan Swoboda and Frederik Witt Special Session Higgs Bundles and Character Varieties Joint International Meeting of the AMS, EMS and SPM Porto, June 2015

Hitchin’s equation Setting X compact Riemann surface, π : E → X complex rank-2 vector bundle ◮ Auxiliary data: g compatible Riemannian metric on X , h hermitian metric on E ◮ Fixed determinant case: A 0 fixed unitary connection on E , consider unitary connections of the form α ∈ Ω 1 ( su ( E )) A = A 0 + α, and trace-free Higgs-field Φ ∈ Ω 1 , 0 ( sl ( E )) ◮ Hitchin’s equation ¯ F ⊥ A + [Φ ∧ Φ ∗ ] = 0 , ∂ A Φ = 0 where F ⊥ A is the trace-free part of the curvature

Hitchin’s equation Setting X compact Riemann surface, π : E → X complex rank-2 vector bundle ◮ Auxiliary data: g compatible Riemannian metric on X , h hermitian metric on E ◮ Fixed determinant case: A 0 fixed unitary connection on E , consider unitary connections of the form α ∈ Ω 1 ( su ( E )) A = A 0 + α, and trace-free Higgs-field Φ ∈ Ω 1 , 0 ( sl ( E )) ◮ Hitchin’s equation ¯ F ⊥ A + [Φ ∧ Φ ∗ ] = 0 , ∂ A Φ = 0 where F ⊥ A is the trace-free part of the curvature

Hitchin’s equation Setting X compact Riemann surface, π : E → X complex rank-2 vector bundle ◮ Auxiliary data: g compatible Riemannian metric on X , h hermitian metric on E ◮ Fixed determinant case: A 0 fixed unitary connection on E , consider unitary connections of the form α ∈ Ω 1 ( su ( E )) A = A 0 + α, and trace-free Higgs-field Φ ∈ Ω 1 , 0 ( sl ( E )) ◮ Hitchin’s equation ¯ F ⊥ A + [Φ ∧ Φ ∗ ] = 0 , ∂ A Φ = 0 where F ⊥ A is the trace-free part of the curvature

Hitchin’s equation Setting X compact Riemann surface, π : E → X complex rank-2 vector bundle ◮ Auxiliary data: g compatible Riemannian metric on X , h hermitian metric on E ◮ Fixed determinant case: A 0 fixed unitary connection on E , consider unitary connections of the form α ∈ Ω 1 ( su ( E )) A = A 0 + α, and trace-free Higgs-field Φ ∈ Ω 1 , 0 ( sl ( E )) ◮ Hitchin’s equation ¯ F ⊥ A + [Φ ∧ Φ ∗ ] = 0 , ∂ A Φ = 0 where F ⊥ A is the trace-free part of the curvature

Hitchin’s equation Basic question Consider sequence ( A n , Φ n ) of solutions ◮ � Φ n � L 2 ≤ C < ∞ : Uhlenbeck compactness = ⇒ ( A n , Φ n ) subconverges to solution ( A ∞ , Φ ∞ ) ◮ � Φ n � L 2 → ∞ : ( A n , Φ n ) exiting end of the moduli space Question What is the degeneration behavior of a diverging sequence of solutions? Ultimate goals: ◮ Describe asymptotics of Hyperkahler metric ◮ Compute space of L 2 -harmonic forms

Hitchin’s equation Basic question Consider sequence ( A n , Φ n ) of solutions ◮ � Φ n � L 2 ≤ C < ∞ : Uhlenbeck compactness = ⇒ ( A n , Φ n ) subconverges to solution ( A ∞ , Φ ∞ ) ◮ � Φ n � L 2 → ∞ : ( A n , Φ n ) exiting end of the moduli space Question What is the degeneration behavior of a diverging sequence of solutions? Ultimate goals: ◮ Describe asymptotics of Hyperkahler metric ◮ Compute space of L 2 -harmonic forms

Hitchin’s equation Basic question Consider sequence ( A n , Φ n ) of solutions ◮ � Φ n � L 2 ≤ C < ∞ : Uhlenbeck compactness = ⇒ ( A n , Φ n ) subconverges to solution ( A ∞ , Φ ∞ ) ◮ � Φ n � L 2 → ∞ : ( A n , Φ n ) exiting end of the moduli space Question What is the degeneration behavior of a diverging sequence of solutions? Ultimate goals: ◮ Describe asymptotics of Hyperkahler metric ◮ Compute space of L 2 -harmonic forms

Hitchin’s equation Basic question Consider sequence ( A n , Φ n ) of solutions ◮ � Φ n � L 2 ≤ C < ∞ : Uhlenbeck compactness = ⇒ ( A n , Φ n ) subconverges to solution ( A ∞ , Φ ∞ ) ◮ � Φ n � L 2 → ∞ : ( A n , Φ n ) exiting end of the moduli space Question What is the degeneration behavior of a diverging sequence of solutions? Ultimate goals: ◮ Describe asymptotics of Hyperkahler metric ◮ Compute space of L 2 -harmonic forms

Hitchin’s equation Basic question Consider sequence ( A n , Φ n ) of solutions ◮ � Φ n � L 2 ≤ C < ∞ : Uhlenbeck compactness = ⇒ ( A n , Φ n ) subconverges to solution ( A ∞ , Φ ∞ ) ◮ � Φ n � L 2 → ∞ : ( A n , Φ n ) exiting end of the moduli space Question What is the degeneration behavior of a diverging sequence of solutions? Ultimate goals: ◮ Describe asymptotics of Hyperkahler metric ◮ Compute space of L 2 -harmonic forms

Hitchin’s equation Basic question Consider sequence ( A n , Φ n ) of solutions ◮ � Φ n � L 2 ≤ C < ∞ : Uhlenbeck compactness = ⇒ ( A n , Φ n ) subconverges to solution ( A ∞ , Φ ∞ ) ◮ � Φ n � L 2 → ∞ : ( A n , Φ n ) exiting end of the moduli space Question What is the degeneration behavior of a diverging sequence of solutions? Ultimate goals: ◮ Describe asymptotics of Hyperkahler metric ◮ Compute space of L 2 -harmonic forms

Limiting configurations The limiting fiducial solution Consider trivial rank-2 vector bundle over C and the Higgs field � 0 � 1 ⇒ det Φ = − zdz 2 . Φ = dz , = z 0 Goal: Find hermitian metric H ∞ on C × such that ∂ ( H − 1 ¯ [Φ ∧ Φ ∗ H ∞ ] = 0 . ∞ ∂ H ∞ ) = 0 , Ansatz: Rotationally symmetric � α ( r ) � b ( r ) H ∞ = ¯ b ( r ) β ( r ) with α, β real valued, α > 0 and αβ − | b | 2 = 1.

Limiting configurations The limiting fiducial solution Consider trivial rank-2 vector bundle over C and the Higgs field � 0 � 1 ⇒ det Φ = − zdz 2 . Φ = dz , = z 0 Goal: Find hermitian metric H ∞ on C × such that ∂ ( H − 1 ¯ [Φ ∧ Φ ∗ H ∞ ] = 0 . ∞ ∂ H ∞ ) = 0 , Ansatz: Rotationally symmetric � α ( r ) � b ( r ) H ∞ = ¯ b ( r ) β ( r ) with α, β real valued, α > 0 and αβ − | b | 2 = 1.

Limiting configurations The limiting fiducial solution Consider trivial rank-2 vector bundle over C and the Higgs field � 0 � 1 ⇒ det Φ = − zdz 2 . Φ = dz , = z 0 Goal: Find hermitian metric H ∞ on C × such that ∂ ( H − 1 ¯ [Φ ∧ Φ ∗ H ∞ ] = 0 . ∞ ∂ H ∞ ) = 0 , Ansatz: Rotationally symmetric � α ( r ) � b ( r ) H ∞ = ¯ b ( r ) β ( r ) with α, β real valued, α > 0 and αβ − | b | 2 = 1.

Limiting configurations The limiting fiducial solution Consider trivial rank-2 vector bundle over C and the Higgs field � 0 � 1 ⇒ det Φ = − zdz 2 . Φ = dz , = z 0 Goal: Find hermitian metric H ∞ on C × such that ∂ ( H − 1 ¯ [Φ ∧ Φ ∗ H ∞ ] = 0 . ∞ ∂ H ∞ ) = 0 , Ansatz: Rotationally symmetric � α ( r ) � b ( r ) H ∞ = ¯ b ( r ) β ( r ) with α, β real valued, α > 0 and αβ − | b | 2 = 1.

Limiting configurations The (limiting) fiducial solution Short Calculation = ⇒ The unique solution is given by � r 1 / 2 0 � H ∞ = r − 1 / 2 0 and the corresponding pair r 1 / 2 ∞ := 1 � 1 � � dz z − d ¯ � � � 0 z 0 A fid Φ fid , ∞ := dz zr − 1 / 2 8 0 − 1 ¯ 0 z solves the decoupled equation ¯ [Φ ∞ ∧ (Φ ∞ ) ∗ ] = 0 , F A ∞ = 0 , ∂ A ∞ Φ ∞ = 0 on C × .

Limiting configurations The (limiting) fiducial solution Short Calculation = ⇒ The unique solution is given by � r 1 / 2 0 � H ∞ = r − 1 / 2 0 and the corresponding pair r 1 / 2 ∞ := 1 � 1 � � dz z − d ¯ � � � 0 z 0 A fid Φ fid , ∞ := dz zr − 1 / 2 8 0 − 1 ¯ 0 z solves the decoupled equation ¯ [Φ ∞ ∧ (Φ ∞ ) ∗ ] = 0 , F A ∞ = 0 , ∂ A ∞ Φ ∞ = 0 on C × .

Limiting configurations The (limiting) fiducial solution Short Calculation = ⇒ The unique solution is given by � r 1 / 2 0 � H ∞ = r − 1 / 2 0 and the corresponding pair r 1 / 2 ∞ := 1 � 1 � � dz z − d ¯ � � � 0 z 0 A fid Φ fid , ∞ := dz zr − 1 / 2 8 0 − 1 ¯ 0 z solves the decoupled equation ¯ [Φ ∞ ∧ (Φ ∞ ) ∗ ] = 0 , F A ∞ = 0 , ∂ A ∞ Φ ∞ = 0 on C × .

Limiting configurations Globally X ) with simple zeroes. Let X × = X \ q − 1 (0). Fix q ∈ H 0 ( K 2 A limiting configuration associated with q is a pair ( A ∞ , Φ ∞ ) on X × such that ◮ ( A ∞ , Φ ∞ ) solves ¯ F ⊥ [Φ ∞ ∧ Φ ∗ A ∞ = 0 , ∞ ] = 0 , ∂ A ∞ Φ ∞ = 0 ◮ det Φ ∞ = q ◮ ( A ∞ , Φ ∞ ) = ( A fid ∞ , Φ fid ∞ ) near each p ∈ q − 1 (0) after fixed choice of holomorphic coordinate and unitary frame.

Limiting configurations Globally X ) with simple zeroes. Let X × = X \ q − 1 (0). Fix q ∈ H 0 ( K 2 A limiting configuration associated with q is a pair ( A ∞ , Φ ∞ ) on X × such that ◮ ( A ∞ , Φ ∞ ) solves ¯ F ⊥ [Φ ∞ ∧ Φ ∗ A ∞ = 0 , ∞ ] = 0 , ∂ A ∞ Φ ∞ = 0 ◮ det Φ ∞ = q ◮ ( A ∞ , Φ ∞ ) = ( A fid ∞ , Φ fid ∞ ) near each p ∈ q − 1 (0) after fixed choice of holomorphic coordinate and unitary frame.

Limiting configurations Globally X ) with simple zeroes. Let X × = X \ q − 1 (0). Fix q ∈ H 0 ( K 2 A limiting configuration associated with q is a pair ( A ∞ , Φ ∞ ) on X × such that ◮ ( A ∞ , Φ ∞ ) solves ¯ F ⊥ [Φ ∞ ∧ Φ ∗ A ∞ = 0 , ∞ ] = 0 , ∂ A ∞ Φ ∞ = 0 ◮ det Φ ∞ = q ◮ ( A ∞ , Φ ∞ ) = ( A fid ∞ , Φ fid ∞ ) near each p ∈ q − 1 (0) after fixed choice of holomorphic coordinate and unitary frame.