Monopoles, Periods and Problems H.W. Braden Bath 2010 Monopole - PowerPoint PPT Presentation

Monopoles, Periods and Problems H.W. Braden Bath 2010 Monopole Results in collaboration with V.Z. Enolskii, A.D’Avanzo. Spectral curve programs with T.Northover. H.W. Braden Monopoles, Periods and Problems

Overview Zero Curvature / Lax Equations Spectral Curve 풞 ⊂ 풮 − → ↑ ↓ Reconstruction ← − Baker-Akhiezer Function ▶ BPS Monopoles ▶ Sigma Model reductions in AdS/CFT ▶ KP, KdV solitons ▶ Harmonic Maps ▶ SW Theory/Integrable Systems H.W. Braden Monopoles, Periods and Problems

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 Monopoles, Periods and Problems

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 Monopoles, Periods and Problems

Setting Spectral Curve ▶ [ d ds + M , A ] = 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 ) , deg a r ( 휁 ) ≤ 2 r ▶ 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 ▶ genus given by Riemann Hurwitz formula g monopole = ( n − 1) 2 H.W. Braden Monopoles, Periods and Problems

Setting Hitchin data H1 Reality conditions a r ( 휁 ) = ( − 1) r 휁 2 r a r ( − 1 휁 ) H2 L 휆 denote the holomorphic line bundle on T ℙ 1 defined by the transition function g 01 = exp ( − 휆휂/휁 ) L 휆 ( m ) ≡ L 휆 ⊗ 휋 ∗ 풪 ( m ) be similarly defined in terms of the transition function g 01 = 휁 m exp ( − 휆휂/휁 ). L 2 is trivial on 풞 and L 1 ( n − 1) is real. L 2 is trivial = ⇒ ∃ nowhere-vanishing holomorphic section. H3 H 0 ( 풞 , L 휆 ( n − 2)) = 0 for 휆 ∈ (0 , 2) H.W. Braden Monopoles, Periods and Problems

Spectral Curves Extrinsic Properties: Real Structure 풞 often comes with an antiholomorphic involution or real structure 휂/ ¯ 휁 2 , − 1 / ¯ ▶ Reverse orientation of lines ( 휂, 휁 ) → ( − ¯ 휁 ) a r ( 휁 ) = ( − 1) r 휁 2 r a r ( − 1 휁 ) = ⇒ ¯ [∏ r ) 1 / 2 ] ∏ r ( 훼 r , l 1 a r ( 휁 ) = 휒 r k =1 ( 휁 − 훼 r , k )( 휁 + 훼 r , k ) l =1 훼 r , l 훼 r , k ∈ ℂ , 휒 r ∈ ℝ , a r ( 휁 ) given by 2 r + 1 (real) parameters ▶ 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 Monopoles, Periods and Problems

Spectral Curves Extrinsic Properties: Rotations ▶ SO (3) induces an action on T ℙ 1 via PSU (2) ( p ) q ∣ p ∣ 2 + ∣ q ∣ 2 = 1 , ∈ PSU (2) , − ¯ q p ¯ 휁 → ¯ p 휁 − ¯ q 휂 q 휁 + p , 휂 → ( q 휁 + p ) 2 ▶ corresponds to a rotation by 휃 around n ∈ S 2 n 1 sin ( 휃/ 2) = Im q , n 2 sin ( 휃/ 2) = − Re q , n 3 sin ( 휃/ 2) = Im p , cos ( 휃/ 2) = − Re p . ▶ Invariant curves yield symmetric monopoles. H.W. Braden Monopoles, Periods and Problems

Spectral Curves Extrinsic Properties: Example of Cyclically Symmetric Monopoles p = 휔 1 / 2 q = 0 ▶ 휔 = exp(2 휋 i / n ), ¯ ( 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 Monopoles, Periods and Problems

Flows and Solutions The Ercolani-Sinha Constraints ▶ Meromorphic differentials describe flows { } ▶ L 2 trivial = − 2 휂 ⇒ f 0 ( 휂, 휁 ) = exp f 1 ( 휂, 휁 ) 휁 ∮ ( ) − 2 휂 dlog f 0 = d + dlog f 1 , exp dlog f 0 = 1 ∀ 휆 ∈ H 1 ( 풞 , ℤ ) 휁 휆 ▶ { 픞 i , 픟 i } g i =1 basis for H 1 ( 풞 , ℤ ): 픞 i ∩ 픟 j = − 픟 j ∩ 픞 i = 훿 ij ∮ g ∑ ▶ 훾 ∞ ( P ) = 1 2 dlog f 0 ( P ) + 횤휋 m j v j ( P ) , v j = 훿 jk 픞 k j =1 ∮ g ∑ 2 휋횤 U = 훾 ∞ = 횤휋 n k + 횤휋 m l 휏 lk , 2 U ∈ Λ l =1 픟 k ▶ H3 H 0 ( 풞 , 풪 ( L s ( n − 2))) = 0 ⇒ H 0 ( 풞 , 풪 ( L s )) = 0 , s ∈ (0 , 2) . → 풪 ( L s ( n − 2)) × a section of 휋 ∗ 풪 ( n − 2) ∣ 풞 ) ( 풪 ( L s ) ֒ L s trivial ⇐ ⇒ s U ∈ Λ , 2 U is a primitive vector in Λ H.W. Braden Monopoles, Periods and Problems

Flows and Solutions Differentials: Period constraints ▶ Ercolani-Sinha Constraints: The following are equivalent: 1. L 2 is trivial on 풞 . (∮ ) T ∮ 1 = 1 2 n + 1 2. 2 U ∈ Λ ⇐ ⇒ U = 픟 1 훾 ∞ , . . . , 픟 g 훾 ∞ 2 휏 m . 2 휋횤 3. There exists a 1-cycle 픠 = n ⋅ 픞 + m ⋅ 픟 such that for every holomorphic differential Ω = 훽 0 휂 n − 2 + 훽 1 ( 휁 ) 휂 n − 3 + . . . + 훽 n − 2 ( 휁 ) ∮ d 휁, Ω = − 2 훽 0 ∂ 풫 c ∂휂 ▶ ES constraints impose g transcendental constraints on curve n ∑ (2 j + 1) − g = ( n + 3)( n − 1) − ( n − 1) 2 = 4( n − 1) j =2 ⇒ 휃 ( 휆 U − ˜ ▶ H 0 ( 풞 , L 휆 ( n − 2)) ∕ = 0 ⇐ K ∣ 휏 ) = 0 where K = K + 흓 (( n − 2) ∑ n ˜ k =1 ∞ k ), K vector of Riemann constants H.W. Braden Monopoles, Periods and Problems

Cyclic Monopoles and Toda New Results ▶ 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 Monopoles, Periods and Problems

Cyclic Monopoles and Toda 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 Monopoles, Periods and Problems

Cyclic Monopoles and Toda 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 Monopoles, Periods and Problems

Cyclic Monopoles and Toda Cyclic Nahm eqns. ≡ Affine Toda eqns: New Result I ▶ 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 ▶ Sutcliffe follows if ˜ q i and ˜ p i may be chosen real. ˜ q i ∈ ℝ with SU ( n ) conjug. + overall SO (3) rotation. p i ∈ ℝ from T i ( s ) = − T † ˜ i ( s ) which also fixes T 1 − iT 2 . ▶ Any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. H.W. Braden Monopoles, Periods and Problems