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

The Geometry of Monopoles: New and Old III 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 III

Recall: To construct a su (2) charge n monopole we need ▶ Curve 풞 ⊂ T ℙ 1 : 0 = P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) H.W. Braden The Geometry of Monopoles: New and Old III

Recall: To construct a su (2) charge n monopole we need ▶ Curve 풞 ⊂ T ℙ 1 : 0 = P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) a r ( 휁 ) = ( − 1) r 휁 2 r a r ( − 1 ▶ deg a r ( 휁 ) ≤ 2 r 휁 ) ¯ [∏ r ) 1 / 2 ] ∏ r ( 훼 l k =1 ( 휁 − 훼 r )( 휁 + 1 a r ( 휁 ) = 휒 r 훼 r ) l =1 훼 l 훼 r ∈ ℂ , 휒 ∈ ℝ a r ( 휁 ) given by 2 r + 1 (real) parameters H.W. Braden The Geometry of Monopoles: New and Old III

Recall: To construct a su (2) charge n monopole we need ▶ Curve 풞 ⊂ T ℙ 1 : 0 = P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) a r ( 휁 ) = ( − 1) r 휁 2 r a r ( − 1 ▶ deg a r ( 휁 ) ≤ 2 r 휁 ) ¯ [∏ r ) 1 / 2 ] ∏ r ( 훼 l k =1 ( 휁 − 훼 r )( 휁 + 1 a r ( 휁 ) = 휒 r 훼 r ) l =1 훼 l 훼 r ∈ ℂ , 휒 ∈ ℝ a r ( 휁 ) given by 2 r + 1 (real) parameters ▶ Ercolani-Sinha Constraints: (∮ ) T ∮ 1 = 1 2 n + 1 1. U = 픟 1 훾 ∞ , . . . , 픟 g 훾 ∞ 2 휏 m . 2 휋횤 2. Ω = 훽 0 휂 n − 2 + 훽 1 ( 휁 ) 휂 n − 3 + . . . + 훽 n − 2 ( 휁 ) ∮ d 휁, Ω = − 2 훽 0 ∂ 풫 픢픰 ∂휂 픢픰 = n ⋅ 픞 + m ⋅ 픟 impose g transcendental constraints on 풞 ∑ n (2 j + 1) − g = ( n + 3)( n − 1) − ( n − 1) 2 = 4( n − 1) j =2 ▶ Flows and Theta Divisor: s U + C ∕∈ Θ H.W. Braden The Geometry of Monopoles: New and Old III

Recall: To construct a su (2) charge n monopole we need ▶ Curve 풞 ⊂ T ℙ 1 : 0 = P ( 휂, 휁 ) = 휂 n + a 1 ( 휁 ) 휂 n − 1 + . . . + a n ( 휁 ) a r ( 휁 ) = ( − 1) r 휁 2 r a r ( − 1 ▶ deg a r ( 휁 ) ≤ 2 r 휁 ) ¯ [∏ r ) 1 / 2 ] ∏ r ( 훼 l k =1 ( 휁 − 훼 r )( 휁 + 1 a r ( 휁 ) = 휒 r 훼 r ) l =1 훼 l 훼 r ∈ ℂ , 휒 ∈ ℝ a r ( 휁 ) given by 2 r + 1 (real) parameters ▶ Ercolani-Sinha Constraints: (∮ ) T ∮ 1 = 1 2 n + 1 1. U = 픟 1 훾 ∞ , . . . , 픟 g 훾 ∞ 2 휏 m . 2 휋횤 2. Ω = 훽 0 휂 n − 2 + 훽 1 ( 휁 ) 휂 n − 3 + . . . + 훽 n − 2 ( 휁 ) ∮ d 휁, Ω = − 2 훽 0 ∂ 풫 픢픰 ∂휂 픢픰 = n ⋅ 픞 + m ⋅ 픟 impose g transcendental constraints on 풞 ∑ n (2 j + 1) − g = ( n + 3)( n − 1) − ( n − 1) 2 = 4( n − 1) j =2 ▶ Flows and Theta Divisor: s U + C ∕∈ Θ ▶ Symmetry aids calculation of 휏 , U , K Q H.W. Braden The Geometry of Monopoles: New and Old III

Possible Charge 3 Curve A trigonal curve and its homology 6 ∏ ▶ w 3 = ( z − 휆 i ) ℛ : ( z , w ) → ( z , 휌 w ) , 휌 = exp { 2 i 휋/ 3 } i =1 H.W. Braden The Geometry of Monopoles: New and Old III

Possible Charge 3 Curve A trigonal curve and its homology 6 ∏ ▶ w 3 = ( z − 휆 i ) ℛ : ( z , w ) → ( z , 휌 w ) , 휌 = exp { 2 i 휋/ 3 } i =1 ▶ ℛ ( 픟 i ) = 픞 픦 , i = 1 , 2 , 3 , ℛ ( 픟 4 ) = − 픞 4 Aut ( 풞 ) = C 3 a 4 λ 2 λ 3 a 1 λ 2 a 2 λ 1 λ 4 b 1 a 3 λ 5 a 1 λ 6 λ 1 0 H.W. Braden The Geometry of Monopoles: New and Old III

Possible Charge 3 Curve Differentials and periods (Wellstein, 1899; Matsumoto, 2000; BE 2006) d u 4 = z 2 d z d u 1 = d z d u 2 = d z d u 3 = z d z w , w 2 , w 2 , w 2 ⎛ ⎞ ∮ ⎝ ⎠ 풜 = ( 풜 ki ) = d u i = ( x , b , c , d ) 픞 k i , k =1 ,..., 4 H = diag (1 , 1 , 1 , − 1) , Λ = diag ( 휌, 휌 2 , 휌 2 , 휌 2 ) ℬ = H 풜 Λ , ⎛ ⎞ ∮ ∮ ∮ ∮ ∑ ⎜ ⎟ ⎠ = 0 ⇔ 0 = x T H b = x T H c = x T H d d u k d u l − d u k d u l ⎝ i 픞 i 픟 i 픟 i 픞 i ( ) H − (1 − 휌 ) xx T 휏 픟 = 풜ℬ − 1 = 휌 x T H x x T H x < 0 Im 휏 픟 is positive definite if and only if ¯ H.W. Braden The Geometry of Monopoles: New and Old III

Possible Charge 3 Curve Solving the Ercolani-Sinha Constraints Theorem (∮ ) T ∮ 1 = 1 2 n + 1 U = 픟 1 훾 ∞ , . . . , 픟 g 훾 ∞ 2 휏 m ⇐ ⇒ 2 휋횤 1 6 휒 3 x = 휉 ( H n + 휌 2 m ) , 휉 = [ n T H n − m . n + m T H m ] where 휉 is real x 1 x 2 x 3 x 4 = = = = 휉, n 1 + 휌 2 m 1 n 2 + 휌 2 m 2 n 3 + 휌 2 m 3 − n 4 + 휌 2 m 4 x T H x ⇒ ¯ x T H x < 0 ⇐ = [ n T H n − m . n + m T H m ] x i / x j ∈ ℚ [ 휌 ] ¯ ∣ 휉 ∣ 2 3 ∑ ( n 2 i − n i m i + m 2 i ) − n 2 4 − m 2 = 4 − m 4 n 4 < 0 . i =1 H.W. Braden The Geometry of Monopoles: New and Old III

Symmetric 3-monopoles 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 , ▶ 풞 : b ∈ ℝ H.W. Braden The Geometry of Monopoles: New and Old III

Symmetric 3-monopoles 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 , ▶ 풞 : b ∈ ℝ ▶ √ √ b 2 + 4 − b + 3 ( 훼, 휌 2 훽, 휌훼, 훽, 휌 2 훼, 휌훽 ) , 훼 = > 0 , 훼훽 = − 1 2 H.W. Braden The Geometry of Monopoles: New and Old III

Symmetric 3-monopoles 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 , ▶ 풞 : b ∈ ℝ ▶ √ √ b 2 + 4 − b + 3 ( 훼, 휌 2 훽, 휌훼, 훽, 휌 2 훼, 휌훽 ) , 훼 = > 0 , 훼훽 = − 1 2 ▶ Aut ( 풞 ) → C 3 × S 3 ( 휁, 휂 ) �→ ( 휌휁, 휂 ) , ( 휁, 휂 ) �→ ( − 1 /휁, − 휂/휁 2 ) √ ∫ 훼 ( 1 ) w = − 2 휋 3 훼 3 , 1 d z 3; 1; − 훼 6 ℐ 1 ( 훼 ) = 2 F 1 9 0 √ 훽 ( 1 ) ∫ d z w = 2 휋 3 3 , 1 3; 1; − 훼 − 6 풥 1 ( 훼 ) = 2 F 1 9 훼 0 H.W. Braden The Geometry of Monopoles: New and Old III

Symmetric 3-monopoles 휂 3 + 휒 ( 휁 6 + b 휁 3 − 1) = 0 , ▶ 풞 : b ∈ ℝ ▶ √ √ b 2 + 4 − b + 3 ( 훼, 휌 2 훽, 휌훼, 훽, 휌 2 훼, 휌훽 ) , 훼 = > 0 , 훼훽 = − 1 2 ▶ Aut ( 풞 ) → C 3 × S 3 ( 휁, 휂 ) �→ ( 휌휁, 휂 ) , ( 휁, 휂 ) �→ ( − 1 /휁, − 휂/휁 2 ) √ ∫ 훼 ( 1 ) w = − 2 휋 3 훼 3 , 1 d z 3; 1; − 훼 6 ℐ 1 ( 훼 ) = 2 F 1 9 0 √ 훽 ( 1 ) ∫ d z w = 2 휋 3 3 , 1 3; 1; − 훼 − 6 풥 1 ( 훼 ) = 2 F 1 9 훼 0 ▶ x 1 = − (2 풥 1 + ℐ 1 ) 휌 − 2 ℐ 1 − 풥 1 , x 2 = 휌 x 1 , = 휌 2 x 1 , = 3( 풥 1 − ℐ 1 ) 휌 + 3 풥 1 , x 3 x 4 H.W. Braden The Geometry of Monopoles: New and Old III

Symmetric 3-monopoles Solving the Ercolani-Sinha Constraints To each pair of relatively prime integers ( n , m ) = 1 for which ( m + n )( m − 2 n ) < 0 we obtain a solution to the Ercolani-Sinha constraints for our curve with n = ( n , m − n , − m , 2 n − m ) , m = ( m , − n , n − m , − 3 n ) as follows. First we solve for t , where 2 F 1 ( 1 3 , 2 3 ; 1 , t ) 2 n − m m + n = 3 ; 1 , 1 − t ) . 2 F 1 ( 1 3 , 2 Then √ b 2 + 4 1 − 2 t t = − b + √ b = √ , , b 2 + 4 2 t (1 − t ) and with 훼 6 = t / (1 − t ) we obtain 휒 from 3 = − ( n 1 + m 1 ) 2 휋 훼 2 F 1 (1 3 , 2 1 휒 √ 3; 1 , t ) . 1 3 3 (1 + 훼 6 ) 3 H.W. Braden The Geometry of Monopoles: New and Old III

Ramanujan (1914) ∞ ∞ ∑ (1 + 6 m )( 1 2 ) m ( 1 2 ) m ( 1 ∑ (2 + 15 m )( 1 2 ) m ( 1 3 ) m ( 2 4 2 ) m 27 3 ) m 휋 = ; 4 휋 = ( m !) 3 ( 27 ) m ( m !) 3 4 m 2 m =0 m =0 √ ∞ ∑ (44 + 33 m )( 1 2 ) m ( 1 3 ) m ( 2 3 ) m 15 3 = ( m !) 3 ( 125 ) m 2 휋 4 m =0 H.W. Braden The Geometry of Monopoles: New and Old III

Ramanujan (1914) ∞ ∞ ∑ (1 + 6 m )( 1 2 ) m ( 1 2 ) m ( 1 ∑ (2 + 15 m )( 1 2 ) m ( 1 3 ) m ( 2 4 2 ) m 27 3 ) m 휋 = ; 4 휋 = ( m !) 3 ( 27 ) m ( m !) 3 4 m 2 m =0 m =0 √ ∞ ∑ (44 + 33 m )( 1 2 ) m ( 1 3 ) m ( 2 3 ) m 15 3 = ( m !) 3 ( 125 ) m 2 휋 4 m =0 휏 = i K ′ K ( k ) = 휋 2 2 F 1 (1 2 , 1 K ′ ( k ) = K ( k ′ ) , k ′ 2 = 1 − k 2 2; 1; k 2 ) , K , If k 1 = 1 − k ′ 1 + k ′ then 휏 1 = 2 휏 . Modular equation of degree n : 2 F 1 ( 1 2 , 1 2 F 1 ( 1 2 , 1 2 ; 1; 1 − 훼 ) 2 ; 1; 1 − 훽 ) n = 2 F 1 ( 1 2 , 1 2 F 1 ( 1 2 , 1 2 ; 1; 훼 ) 2 ; 1; 훽 ) ( 훼훽 ) 1 / 4 + ((1 − 훼 )(1 − 훽 )) 1 / 4 = 1 = ⇒ n = 3 H.W. Braden The Geometry of Monopoles: New and Old III

Modular equation of degree n and signature r ( r = 2 , 3 , 4 , 6) 2 F 1 ( 1 r , r − 1 2 F 1 ( 1 r , r − 1 r ; 1; 1 − 훼 ) r ; 1; 1 − 훽 ) n = 2 F 1 ( 1 r , r − 1 2 F 1 ( 1 r , r − 1 r ; 1; 훼 ) r ; 1; 훽 ) H.W. Braden The Geometry of Monopoles: New and Old III

Modular equation of degree n and signature r ( r = 2 , 3 , 4 , 6) 2 F 1 ( 1 r , r − 1 2 F 1 ( 1 r , r − 1 r ; 1; 1 − 훼 ) r ; 1; 1 − 훽 ) n = 2 F 1 ( 1 r , r − 1 2 F 1 ( 1 r , r − 1 r ; 1; 훼 ) r ; 1; 훽 ) ⇒ ( 훼훽 ) 1 / 3 + ((1 − 훼 )(1 − 훽 )) 1 / 3 = 1 n = 2 , r = 3 = √ ⇒ 훽 = 1 2 + 5 3 ⇒ 훽 1 / 3 + (1 − 훽 ) 1 / 3 = 2 1 / 3 = 훼 = 1 / 2 = 18 n =5 , r =3 ⇒ ( 훼훽 ) 1 / 3 +((1 − 훼 )(1 − 훽 )) 1 / 3 +3( 훼훽 (1 − 훼 )(1 − 훽 )) 1 / 6 =1 H.W. Braden The Geometry of Monopoles: New and Old III