The Geometry of Monopoles: New and Old IV H.W. Braden Varna, June - PowerPoint PPT Presentation

The Geometry of Monopoles: New and Old IV H.W. Braden Varna, June 2011 Curve results with T.P. Northover. Monopole Results in collaboration with V.Z. Enolski, A.D’Avanzo. H.W. Braden The Geometry of Monopoles: New and Old IV

Recall ▶ The only spectral curves of a BPS monopole of the form √ Γ( 1 6 )Γ( 1 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 have b = ± 5 1 3 ) 3 = − 1 2, 휒 2 . 1 1 6 6 휋 2 These correspond to tetrahedrally symmetric monopoles. H.W. Braden The Geometry of Monopoles: New and Old IV

Recall ▶ The only spectral curves of a BPS monopole of the form √ Γ( 1 6 )Γ( 1 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 have b = ± 5 1 3 ) 3 = − 1 2, 휒 2 . 1 1 6 6 휋 2 These correspond to tetrahedrally symmetric monopoles. ▶ SO (3) induces an action on T ℙ 1 via PSU (2) Invariant curves yield symmetric monopoles. ( p ) q ∣ p ∣ 2 + ∣ q ∣ 2 = 1 , ∈ PSU (2) , − ¯ q p ¯ 휁 → ¯ p 휁 − ¯ 휂 q q 휁 + p , 휂 → ( q 휁 + p ) 2 H.W. Braden The Geometry of Monopoles: New and Old IV

Recall ▶ The only spectral curves of a BPS monopole of the form √ Γ( 1 6 )Γ( 1 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 have b = ± 5 1 3 ) 3 = − 1 2, 휒 2 . 1 1 6 6 휋 2 These correspond to tetrahedrally symmetric monopoles. ▶ SO (3) induces an action on T ℙ 1 via PSU (2) Invariant curves yield symmetric monopoles. ( p ) q ∣ p ∣ 2 + ∣ q ∣ 2 = 1 , ∈ PSU (2) , − ¯ q p ¯ 휁 → ¯ p 휁 − ¯ 휂 q q 휁 + p , 휂 → ( q 휁 + p ) 2 ▶ Space-time symmetries yield geodesic submanifolds of the moduli space. If 1-dimensional then orbits of geodesic scattering H.W. Braden The Geometry of Monopoles: New and Old IV

Recall ▶ The only spectral curves of a BPS monopole of the form √ Γ( 1 6 )Γ( 1 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 have b = ± 5 1 3 ) 3 = − 1 2, 휒 2 . 1 1 6 6 휋 2 These correspond to tetrahedrally symmetric monopoles. ▶ SO (3) induces an action on T ℙ 1 via PSU (2) Invariant curves yield symmetric monopoles. ( p ) q ∣ p ∣ 2 + ∣ q ∣ 2 = 1 , ∈ PSU (2) , − ¯ q p ¯ 휁 → ¯ p 휁 − ¯ 휂 q q 휁 + p , 휂 → ( q 휁 + p ) 2 ▶ Space-time symmetries yield geodesic submanifolds of the moduli space. If 1-dimensional then orbits of geodesic scattering ▶ Consider cyclic space-time symmetry H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Spectral Curves ▶ 휔 = exp(2 휋 i / n ), p = 휔 1 / 2 q = 0 ( 휂, 휁 ) → ( 휔휂, 휔휁 ) 휂 i 휁 j invariant for i + j ≡ 0 mod n 휂 n + a 1 휂 n − 1 휁 + a 2 휂 n − 2 휁 2 + . . . + a n 휁 n + 훽휁 2 n + 훾 = 0 H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Spectral Curves ▶ 휔 = exp(2 휋 i / n ), p = 휔 1 / 2 q = 0 ( 휂, 휁 ) → ( 휔휂, 휔휁 ) 휂 i 휁 j invariant for i + j ≡ 0 mod n 휂 n + a 1 휂 n − 1 휁 + a 2 휂 n − 2 휁 2 + . . . + a n 휁 n + 훽휁 2 n + 훾 = 0 ▶ Impose reality conditions and centre a 1 = 0 휂 n + a 2 휂 n − 2 휁 2 + . . . + a n 휁 n + 훽휁 2 n + ( − 1) n ¯ 훽 = 0 , a i ∈ R By an overall rotation we may choose 훽 real H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Spectral Curves ▶ 휔 = exp(2 휋 i / n ), p = 휔 1 / 2 q = 0 ( 휂, 휁 ) → ( 휔휂, 휔휁 ) 휂 i 휁 j invariant for i + j ≡ 0 mod n 휂 n + a 1 휂 n − 1 휁 + a 2 휂 n − 2 휁 2 + . . . + a n 휁 n + 훽휁 2 n + 훾 = 0 ▶ Impose reality conditions and centre a 1 = 0 휂 n + a 2 휂 n − 2 휁 2 + . . . + a n 휁 n + 훽휁 2 n + ( − 1) n ¯ 훽 = 0 , a i ∈ R By an overall rotation we may choose 훽 real ▶ x = 휂/휁 , 휈 = 휁 n 훽 , x n + a 2 x n − 2 + . . . + a n + 휈 + ( − 1) n ∣ 훽 ∣ 2 = 0 휈 H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Spectral Curves ▶ 휔 = exp(2 휋 i / n ), p = 휔 1 / 2 q = 0 ( 휂, 휁 ) → ( 휔휂, 휔휁 ) 휂 i 휁 j invariant for i + j ≡ 0 mod n 휂 n + a 1 휂 n − 1 휁 + a 2 휂 n − 2 휁 2 + . . . + a n 휁 n + 훽휁 2 n + 훾 = 0 ▶ Impose reality conditions and centre a 1 = 0 휂 n + a 2 휂 n − 2 휁 2 + . . . + a n 휁 n + 훽휁 2 n + ( − 1) n ¯ 훽 = 0 , a i ∈ R By an overall rotation we may choose 훽 real ▶ x = 휂/휁 , 휈 = 휁 n 훽 , x n + a 2 x n − 2 + . . . + a n + 휈 + ( − 1) n ∣ 훽 ∣ 2 = 0 휈 ▶ Affine Toda Spectral Curve y = 휈 − ( − 1) n ∣ 훽 ∣ 2 휈 y 2 = ( x n + a 2 x n − 2 + . . . + a n ) 2 − 4( − 1) n ∣ 훽 ∣ 2 H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Overview ▶ 풞 monopole is an unbranched n : 1 cover 풞 Toda g monopole = ( n − 1) 2 , g Toda = ( n − 1) H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Overview ▶ 풞 monopole is an unbranched n : 1 cover 풞 Toda g monopole = ( n − 1) 2 , g Toda = ( n − 1) ▶ Sutcliffe: Ansatz for Nahm’s equations for cyclic monopoles in terms of Affine Toda equation. Cyclic Nahm eqns. ⊃ Affine Toda eqns. H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Overview ▶ 풞 monopole is an unbranched n : 1 cover 풞 Toda g monopole = ( n − 1) 2 , g Toda = ( n − 1) ▶ Sutcliffe: Ansatz for Nahm’s equations for cyclic monopoles in terms of Affine Toda equation. Cyclic Nahm eqns. ⊃ Affine Toda eqns. ▶ Cyclic Nahm eqns. ≡ Affine Toda eqns. H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Overview ▶ 풞 monopole is an unbranched n : 1 cover 풞 Toda g monopole = ( n − 1) 2 , g Toda = ( n − 1) ▶ Sutcliffe: Ansatz for Nahm’s equations for cyclic monopoles in terms of Affine Toda equation. Cyclic Nahm eqns. ⊃ Affine Toda eqns. ▶ Cyclic Nahm eqns. ≡ Affine Toda eqns. ▶ Cyclic monopoles ≡ (particular) Affine Toda solns. H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Overview ▶ 풞 monopole is an unbranched n : 1 cover 풞 Toda g monopole = ( n − 1) 2 , g Toda = ( n − 1) ▶ Sutcliffe: Ansatz for Nahm’s equations for cyclic monopoles in terms of Affine Toda equation. Cyclic Nahm eqns. ⊃ Affine Toda eqns. ▶ Cyclic Nahm eqns. ≡ Affine Toda eqns. ▶ Cyclic monopoles ≡ (particular) Affine Toda solns. ▶ Implementation in terms of curves, period matrices, theta functions etc. H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclic Nahm eqns. ≡ Affine Toda eqns. Sutcliffe Ansatz T 1 + iT 2 = ( T 1 − iT 2 ) T ⎛ ⎞ e ( q 1 − q 2 ) / 2 0 0 . . . 0 ⎜ ⎟ e ( q 2 − q 3 ) / 2 0 0 . . . 0 ⎜ ⎟ ⎜ ⎟ . . ... . . = ⎜ ⎟ . . ⎜ ⎟ ⎝ ⎠ e ( q n − 1 − q n ) / 2 0 0 0 . . . e ( q n − q 1 ) / 2 0 0 . . . 0 T 3 = − i 2 Diag ( p 1 , p 2 , . . . , p n ) d ds ( T 1 + iT 2 ) = i [ T 3 , T 1 + iT 2 ] ⇒ p i − p i +1 = ˙ q i − ˙ q i +1 ds T 3 = [ T 1 , T 2 ] = i d p i = − e q i − q i +1 + e q i − 1 − q i 2[ T 1 + iT 2 , T 1 − iT 2 ] ⇒ ˙ H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclic Nahm eqns. ≡ Affine Toda eqns. Sutcliffe Ansatz C’td ▶ p i , q i real ( ) [ e q 1 − q 2 + e q 2 − q 3 + . . . + e q n − q 1 ] H = 1 p 2 1 + . . . + p 2 − . n 2 Toda ⇒ Nahm Affine Toda eqns. ⊂ Cyclic Nahm eqns. H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclic Nahm eqns. ≡ Affine Toda eqns. Sutcliffe Ansatz C’td ▶ p i , q i real ( ) [ e q 1 − q 2 + e q 2 − q 3 + . . . + e q n − q 1 ] H = 1 p 2 1 + . . . + p 2 − . n 2 Toda ⇒ Nahm Affine Toda eqns. ⊂ Cyclic Nahm eqns. ▶ G ⊂ SO (3) acts on triples t = ( T 1 , T 2 , T 3 ) ∈ ℝ 3 ⊗ SL ( n , ℂ ) via natural action on ℝ 3 and conjugation on SL ( n , ℂ ) ▶ g ′ ∈ SO (3) and g = 휌 ( g ′ ) ∈ SL ( n , ℂ ). Invariance of curve ⇒ g ( T 1 + iT 2 ) g − 1 = 휔 ( T 1 + iT 2 ) , gT 3 g − 1 = T 3 , g ( T 1 − iT 2 ) g − 1 = 휔 − 1 ( T 1 − iT 2 ) . H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Cyclic Nahm eqns. ≡ Affine Toda eqns. ▶ SO (3) action on SL ( n , ℂ ) ∼ 2 n − 1 ⊕ 2 n − 3 ⊕ . . . ⊕ 5 ⊕ 3 H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Cyclic Nahm eqns. ≡ Affine Toda eqns. ▶ SO (3) action on SL ( n , ℂ ) ∼ 2 n − 1 ⊕ 2 n − 3 ⊕ . . . ⊕ 5 ⊕ 3 ▶ Kostant ⇒ 휌 ( SO (3)) principal three dimensional subgroup. [ 2 휋 ] ▶ g = 휌 ( g ′ ) = exp n H , H semi-simple, generator Cartan TDS ▶ g ≡ Diag ( 휔 n − 1 , . . . , 휔, 1), gE ij g − 1 = 휔 j − i E ij . H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles Cyclic Nahm eqns. ≡ Affine Toda eqns. ▶ SO (3) action on SL ( n , ℂ ) ∼ 2 n − 1 ⊕ 2 n − 3 ⊕ . . . ⊕ 5 ⊕ 3 ▶ Kostant ⇒ 휌 ( SO (3)) principal three dimensional subgroup. [ 2 휋 ] ▶ g = 휌 ( g ′ ) = exp n H , H semi-simple, generator Cartan TDS ▶ g ≡ Diag ( 휔 n − 1 , . . . , 휔, 1), gE ij g − 1 = 휔 j − i E ij . ▶ For a cyclically invariant monopole ∑ ∑ T 3 = − i q ) / 2 E 훼 , e ( 훼, ˜ T 1 + iT 2 = ˜ p j H j 2 훼 ∈ ˆ j Δ H.W. Braden The Geometry of Monopoles: New and Old IV