Symmetry, Curves and Monopoles H.W. Braden EMPG Edinburgh, November 2011 Curve results with T.P. Northover. Monopole Results in collaboration with V.Z. Enolski, A.D’Avanzo. H.W. Braden Symmetry, Curves and Monopoles
Overview Zero Curvature / Lax Equations Spectral Curve 풞 ⊂ 풮 − → ↑ ↓ Reconstruction ← − Baker-Akhiezer Function H.W. Braden Symmetry, Curves and Monopoles
Overview Zero Curvature / Lax Equations Spectral Curve 풞 ⊂ 풮 − → ↑ ↓ Reconstruction ← − Baker-Akhiezer Function t U + C ∈ Jac ( 풞 ) H.W. Braden Symmetry, Curves and Monopoles
Overview Zero Curvature / Lax Equations Spectral Curve 풞 ⊂ 풮 − → ↑ ↓ Reconstruction ← − Baker-Akhiezer Function t U + C ∈ Jac ( 풞 ) ▶ BPS Monopoles ▶ Sigma Model reductions in AdS/CFT ▶ KP, KdV solitons ▶ Harmonic Maps ▶ SW Theory/Integrable Systems H.W. Braden Symmetry, Curves and Monopoles
Overview Zero Curvature / Lax Equations Spectral Curve 풞 ⊂ 풮 − → ↑ ↓ Reconstruction ← − Baker-Akhiezer Function t U + C ∈ Jac ( 풞 ) Difficulties: ▶ Transcendental constraints. ℒ 2 trivial H.W. Braden Symmetry, Curves and Monopoles
Overview Zero Curvature / Lax Equations Spectral Curve 풞 ⊂ 풮 − → ↑ ↓ Reconstruction ← − Baker-Akhiezer Function t U + C ∈ Jac ( 풞 ) Difficulties: ▶ Transcendental constraints. ℒ 2 trivial ▶ Flows and Theta Divisor. H 0 ( 풞 , ℒ ) = 0 H.W. Braden Symmetry, Curves and Monopoles
Overview Zero Curvature / Lax Equations Spectral Curve 풞 ⊂ 풮 − → ↑ ↓ Reconstruction ← − Baker-Akhiezer Function t U + C ∈ Jac ( 풞 ) Difficulties: ▶ Transcendental constraints. ℒ 2 trivial ▶ Flows and Theta Divisor. H 0 ( 풞 , ℒ ) = 0 휃 ( t U + C ∣ 휏 ) H.W. Braden Symmetry, Curves and Monopoles
Setting BPS Monopoles ▶ Reduction of F = ∗ F L = − 1 2 Tr F ij F ij + Tr D i Φ D i Φ . 3 ▶ B i = 1 휖 ijk F jk = D i Φ ∑ 2 j , k =1 ▶ A monopole of charge n � √ − 1 ∼ 1 − n √ � 2 r + O ( r − 2 ) , x 2 1 + x 2 2 + x 2 2 Tr Φ( r ) 2 r = � 3 � � r →∞ ▶ Monopoles ↔ Nahm Data ↔ Hitchin Data H.W. Braden Symmetry, Curves and Monopoles
Setting BPS Monopoles: Nahm Data for charge n SU (2) monopoles Three n × n matrices T i ( s ) with s ∈ [0 , 2] satisfying the following: 3 dT i ds = 1 ∑ N1 Nahm’s equation 휖 ijk [ T j , T k ] . 2 j , k =1 N2 T i ( s ) is regular for s ∈ (0 , 2) and has simple poles at s = 0 , 2. Residues form su (2) irreducible n -dimensional representation. N3 T i ( s ) = − T † T i ( s ) = T t i ( s ), i (2 − s ). H.W. Braden Symmetry, Curves and Monopoles
Setting BPS Monopoles: Nahm Data for charge n SU (2) monopoles Three n × n matrices T i ( s ) with s ∈ [0 , 2] satisfying the following: 3 dT i ds = 1 ∑ N1 Nahm’s equation 휖 ijk [ T j , T k ] . 2 j , k =1 N2 T i ( s ) is regular for s ∈ (0 , 2) and has simple poles at s = 0 , 2. Residues form su (2) irreducible n -dimensional representation. N3 T i ( s ) = − T † T i ( s ) = T t i ( s ), i (2 − s ). A ( 휁 ) = T 1 + iT 2 − 2 iT 3 휁 + ( T 1 − iT 2 ) 휁 2 M ( 휁 ) = − iT 3 + ( T 1 − iT 2 ) 휁 3 dT i ds = 1 ⇒ [ d ∑ Nahm’s eqn. 휖 ijk [ T j , T k ] ⇐ ds + M , A ] = 0 . 2 j , k =1 H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves 풞 ⊂ 풮 ▶ [ d ds + M ( 휁 ) , A ( 휁 )] = 0 , 풞 : 0 = det( 휂 1 n + A ( 휁 )) := P ( 휂, 휁 ) P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves 풞 ⊂ 풮 ▶ [ d ds + M ( 휁 ) , A ( 휁 )] = 0 , 풞 : 0 = det( 휂 1 n + A ( 휁 )) := P ( 휂, 휁 ) P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) ▶ Where does 풞 lie? 풞 ⊂ 풮 H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves 풞 ⊂ 풮 ▶ [ d ds + M ( 휁 ) , A ( 휁 )] = 0 , 풞 : 0 = det( 휂 1 n + A ( 휁 )) := P ( 휂, 휁 ) P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) ▶ Where does 풞 lie? 풞 ⊂ 풮 ( 휂, 휁 ) → 휂 d ▶ 풞 monopole ⊂ T ℙ 1 := 풮 d 휁 ∈ T ℙ 1 Minitwistor description H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves 풞 ⊂ 풮 ▶ [ d ds + M ( 휁 ) , A ( 휁 )] = 0 , 풞 : 0 = det( 휂 1 n + A ( 휁 )) := P ( 휂, 휁 ) P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) ▶ Where does 풞 lie? 풞 ⊂ 풮 ( 휂, 휁 ) → 휂 d ▶ 풞 monopole ⊂ T ℙ 1 := 풮 d 휁 ∈ T ℙ 1 Minitwistor description ▶ 풞 휎 − model ⊂ ℙ 2 := 풮 ▶ 풮 = T ∗ Σ Hitchin Systems on a Riemann surface Σ ▶ 풮 = K 3 ▶ 풮 a Poisson surface ▶ separation of variables ↔ Hilb [ N ] ( 풮 ) ▶ X the total space of an appropriate line bundle ℒ over 풮 ↔ noncompact CY H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves 풞 ⊂ 풮 ▶ [ d ds + M ( 휁 ) , A ( 휁 )] = 0 , 풞 : 0 = det( 휂 1 n + A ( 휁 )) := P ( 휂, 휁 ) P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) ▶ Where does 풞 lie? 풞 ⊂ 풮 ( 휂, 휁 ) → 휂 d ▶ 풞 monopole ⊂ T ℙ 1 := 풮 d 휁 ∈ T ℙ 1 Minitwistor description ▶ 풞 휎 − model ⊂ ℙ 2 := 풮 ▶ 풮 = T ∗ Σ Hitchin Systems on a Riemann surface Σ ▶ 풮 = K 3 ▶ 풮 a Poisson surface ▶ separation of variables ↔ Hilb [ N ] ( 풮 ) ▶ X the total space of an appropriate line bundle ℒ over 풮 ↔ noncompact CY ▶ Symmetry: 풞 ⊂ ℙ a , b , c [ X , Y , Z ] ∼ [ 휆 a X , 휆 b Y , 휆 c Z ] , 휆 ∈ ℂ ∗ H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: data ▶ Homology basis { 훾 i } 2 g i =1 = { 픞 i , 픟 i } g i =1 ▶ algorithm for branched covers of ℙ 1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries ▶ Holomorphic differentials du i ( i = 1 , . . . , g ) H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: data ▶ Homology basis { 훾 i } 2 g i =1 = { 픞 i , 픟 i } g i =1 ▶ algorithm for branched covers of ℙ 1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries ▶ Holomorphic differentials du i ( i = 1 , . . . , g ) ▶ Period Matrix 휏 = ℬ풜 − 1 where (∮ ) 픞 i du j ( 풜 ) Π := = ∮ ℬ 픟 i du j ▶ Principle (Kontsevich, Zagier): Whenever you meet a new number, and have decided (or convinced yourself) that it is transcendental, try to figure out whether it is a period H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: data ▶ Homology basis { 훾 i } 2 g i =1 = { 픞 i , 픟 i } g i =1 ▶ algorithm for branched covers of ℙ 1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries ▶ Holomorphic differentials du i ( i = 1 , . . . , g ) ▶ Period Matrix 휏 = ℬ풜 − 1 where (∮ ) 픞 i du j ( 풜 ) Π := = ∮ ℬ 픟 i du j ▶ Principle (Kontsevich, Zagier): Whenever you meet a new number, and have decided (or convinced yourself) that it is transcendental, try to figure out whether it is a period ▶ normalized holomorphic differentials 휔 i , ∮ ∮ 픞 i 휔 j = 훿 ij 픟 i 휔 j = 휏 ij ▶ 풞 often has an antiholomorphic involution/real structure H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: data ▶ Homology basis { 훾 i } 2 g i =1 = { 픞 i , 픟 i } g i =1 ▶ algorithm for branched covers of ℙ 1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries ▶ Holomorphic differentials du i ( i = 1 , . . . , g ) ▶ Period Matrix 휏 = ℬ풜 − 1 where (∮ ) 픞 i du j ( 풜 ) Π := = ∮ ℬ 픟 i du j ▶ Principle (Kontsevich, Zagier): Whenever you meet a new number, and have decided (or convinced yourself) that it is transcendental, try to figure out whether it is a period ▶ normalized holomorphic differentials 휔 i , ∮ ∮ 픞 i 휔 j = 훿 ij 픟 i 휔 j = 휏 ij ▶ 풞 often has an antiholomorphic involution/real structure ▶ reality constrains the form of the period matrix. ▶ there may be between 0 and g + 1 ovals of fixed points of the antiholomorphic involution. ▶ Imposing reality can be one of the hardest steps. H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: Flows 휃 ( t U + C ∣ 휏 ) H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: Flows 휃 ( t U + C ∣ 휏 ) ▶ Meromorphic differentials describe flows ⇐ ⇒ U H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: Flows 휃 ( t U + C ∣ 휏 ) ▶ Meromorphic differentials describe flows ⇐ ⇒ U ▶ 휃 ( e ∣ 휏 ) = 0 ⇐ ⇒ e ∈ Θ ⊂ Jac 풞 ( g − 1 ∫ P ) ∑ ▶ e ≡ 휙 Q P i + K Q , 휙 Q ( P ) := 휔 Q i =1 ( g − 1 ) ∑ = dim H 1 ( 풞 , ℒ ∑ g − 1 i = 1 P i ) = dim H 0 ( 풞 , ℒ ∑ g − 1 mult e 휃 = i P i i = 1 P i ) i = 1 H.W. Braden Symmetry, Curves and Monopoles
Spectral Curves: Flows 휃 ( t U + C ∣ 휏 ) ▶ Meromorphic differentials describe flows ⇐ ⇒ U ▶ 휃 ( e ∣ 휏 ) = 0 ⇐ ⇒ e ∈ Θ ⊂ Jac 풞 ( g − 1 ∫ P ) ∑ ▶ e ≡ 휙 Q P i + K Q , 휙 Q ( P ) := 휔 Q i =1 ( g − 1 ) ∑ = dim H 1 ( 풞 , ℒ ∑ g − 1 i = 1 P i ) = dim H 0 ( 풞 , ℒ ∑ g − 1 mult e 휃 = i P i i = 1 P i ) i = 1 ▶ − K Q = 휙 ∗ (Δ − ( g − 1) Q ) = 휙 Q (Δ), deg Δ = g − 1 , 2Δ ≡ 풦 풞 H.W. Braden Symmetry, Curves and Monopoles
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