Monopoles, Periods and Problems

Transcription

1 Bath 2010 Monopole Results in collaboration with V.Z. Enolskii, A.D Avanzo. Spectral curve programs with T.Northover.

2 Overview Equations Zero Curvature/Lax Spectral Curve C S Reconstruction Baker-Akhiezer Function BPS Monopoles Sigma Model reductions in AdS/CFT KP, KdV solitons Harmonic Maps SW Theory/Integrable Systems

3 Setting BPS Monopoles Reduction of F = F B i = 1 2 L = 1 2 Tr F ijf ij + Tr D i Φ D i Φ. 3 ε ijk F jk = D i Φ j,k=1 A monopole of charge n 1 2 Tr Φ(r)2 r 1 n 2r +O(r 2 ), r = x x x 2 3 Monopoles Nahm Data Hitchin Data

4 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: N1 Nahm s equation dt i ds = ε ijk [T j, T k ]. 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 i (s), T i(s) = Ti t (2 s). A(ζ) = T 1 + it 2 2iT 3 ζ + (T 1 it 2 )ζ 2 M(ζ) = it 3 + (T 1 it 2 )ζ dt i Nahm s eqn. ds = 1 3 ε ijk [T j, T k ] [ d 2 ds j,k=1 + M, A] = 0.

5 Setting Spectral Curve [ d ds + M, A] = 0, C : 0 = det(η1 n + A(ζ)) := P(η, ζ) P(η, ζ) = η n + a 1 (ζ)η n a n (ζ), deg a r (ζ) 2r Where does C lie? C S C monopole T P 1 := S (η, ζ) η d dζ T P1 Minitwistor description C σ model P 2 := S S = T Σ Hitchin Systems on a Riemann surface Σ S = K3 S a Poisson surface separation of variables Hilb [N] (S) X the total space of an appropriate line bundle L over S noncompact CY genus given by Riemann Hurwitz formula g monopole = (n 1) 2

6 Setting Hitchin data H1 Reality conditions a r (ζ) = ( 1) r ζ 2r a r ( 1 ζ ) H2 L λ denote the holomorphic line bundle on T P 1 defined by the transition function g 01 = exp( λη/ζ) L λ (m) L λ π O(m) be similarly defined in terms of the transition function g 01 = ζ m exp ( λη/ζ). L 2 is trivial on C and L 1 (n 1) is real. L 2 is trivial = nowhere-vanishing holomorphic section. H3 H 0 (C, L λ (n 2)) = 0 for λ (0, 2)

7 Spectral Curves Extrinsic Properties: Real Structure C often comes with an antiholomorphic involution or real structure Reverse orientation of lines (η, ζ) ( η/ ζ 2, 1/ ζ) a r (ζ) = ( 1) r ζ 2r a r ( ) = [ 1 ζ r ( ) ] αr,l 1/2 r a r (ζ) = χ r l=1 α r,l k=1 (ζ α r,k)(ζ + 1 α r,k ) α r,k C, χ r R, a r (ζ) given by 2r + 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.

8 Spectral Curves Extrinsic Properties: Rotations SO(3) induces an action on T P 1 via PSU(2) ( ) p q PSU(2), p 2 + q 2 = 1, q p ζ p ζ q q ζ + p, η η (q ζ + p) 2 corresponds to a rotation by θ around n S 2 n 1 sin (θ/2) = Im q, n 2 sin (θ/2) = Re q, n 3 sin (θ/2) = Im p, cos (θ/2) = Re p. Invariant curves yield symmetric monopoles.

9 Spectral Curves Extrinsic Properties: Example of Cyclically Symmetric Monopoles ω = exp(2πi/n), p = ω 1/2 q = 0 η i ζ j invariant for i + j 0 mod n (η, ζ) (ωη, ωζ) η n + a 1 η n 1 ζ + a 2 η n 2 ζ a n ζ n + βζ 2n + γ = 0 Impose reality conditions and centre a 1 = 0 η n + a 2 η n 2 ζ a n ζ n + βζ 2n + ( 1) n β = 0, a i R By an overall rotation we may choose β real x = η/ζ, ν = ζ n β, x n + a 2 x n a n + ν + ( 1)n β 2 Affine Toda Spectral Curve y = ν ( 1)n β 2 ν ν = 0 y 2 = (x n + a 2 x n a n ) 2 4( 1) n β 2

10 Flows and Solutions The Ercolani-Sinha Constraints Meromorphic differentials describe { } flows L 2 trivial = f 0 (η, ζ) = exp 2 η ζ f 1 (η, ζ) ( ) dlog f 0 = d 2 η ζ + dlogf 1, exp dlog f 0 = 1 λ H 1 (C,Z) {a i, b i } g i=1 basis for H 1(C, Z): a i b j = b j a i = δ ij γ (P) = 1 g 2 dlog f 0(P) + ıπ m j v j (P), v j = δ jk 2πıU = b k γ = ıπn k + ıπ j=1 λ g m l τ lk, l=1 a k 2U Λ H3 H 0 (C, O(L s (n 2))) = 0 H 0 (C, O(L s )) = 0, s (0, 2). (O(L s ) O(L s (n 2)) a section of π O(n 2) C ) L s trivial s U Λ, 2U is a primitive vector in Λ

11 Flows and Solutions Differentials: Period constraints Ercolani-Sinha Constraints: The following are equivalent: 1. L 2 is trivial on C. ( 2. 2U Λ U = 1 2πı b γ 1,..., ) T b γ g = 1 2 n τm. 3. There exists a 1-cycle c = n a + m b such that for every holomorphic differential Ω = β 0η n 2 + β 1 (ζ)η n β n 2 (ζ) dζ, Ω = 2β P 0 η c ES constraints impose g transcendental constraints on curve n (2j + 1) g = (n + 3)(n 1) (n 1) 2 = 4(n 1) j=2 H 0 (C, L λ (n 2)) = 0 θ(λu K τ) = 0 where K = K + φ ((n 2) n k=1 k), K vector of Riemann constants

12 Cyclic Monopoles and Toda New Results C monopole is an unbranched n : 1 cover C 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.

13 Cyclic Monopoles and Toda Sutcliffe Ansatz T 1 + it 2 = (T 1 it 2 ) T 0 e (q 1 q 2 )/ e (q 2 q 3 )/ = e (q n 1 q n)/2 e (qn q 1)/ 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 d ds T 3 = [T 1, T 2 ] = i 2 [T 1 + it 2, T 1 it 2 ] ṗ i = e q i q i+1 + e q i 1 q i

14 Cyclic Monopoles and Toda Sutcliffe Ansatz C td p i, q i real H = 1 2 Toda Nahm ( p pn 2 ) [ e q 1 q 2 + e q 2 q e qn q ] 1. 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 = ρ(g ) SL(n, C). Invariance of curve g(t 1 + it 2 )g 1 = ω(t 1 + it 2 ), gt 3 g 1 = T 3, g(t 1 it 2 )g 1 = ω 1 (T 1 it 2 ).

15 Cyclic Monopoles and Toda Cyclic Nahm eqns. Affine Toda eqns: New Result I SL(n, C) 2n 1 2n Kostant ρ(so(3)) principal three dimensional subgroup. g = ρ(g ) = exp [ 2π n H], H semi-simple, generator Cartan TDS g Diag(ω n 1,..., ω, 1), ge ij g 1 = ω j i E ij. For a cyclically invariant monopole T 1 + it 2 = α ˆΔ e (α, q)/2 E α, T 3 = i p j H j 2 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. j

16 Cyclic Monopoles and Toda Flows and Solutions: New Results II Theorem The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations. π : C monopole C Toda U = π u Jac(C monopole ) = π Jac(C Toda ) + Prym K Θsingular Jac(C monopole ), 2 K Λ, K = π e 1 Fay-Accola θ[ K](π z; τ monopole ) = c n θ[e i ](z; τ Toda ) i=1 θ-functions are still far from being a spectator sport. (L.V. Ahlfors)

17 Curves Fundamental Ingredients Homology basis {a i, b 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) ( ) ( ) A Period Matrix τ = BA 1 where = B a i du j b i du j normalized holomorphic differentials ω i, a i ω j = δ ij b i ω j = τ ij curves with lots of symmetries: evaluate τ via character theory w 2 = z 2g+2 1 (D 2g+2 ), w 2 = z(z 2g+1 1) (C 2g+1 ) 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

18 Curves Example: Klein s Curve and Problems C: x 3 y + y 3 z + z 3 x = 0 Aut(C) = PSL(2, 7) order 168. τ RL = 1+3i i i i i i i i i 7 8 τ = 1 e e 1, e = 1+i e Symplectic Equivalence of Period Matrices τ, τ ( ) A B M = Sp(2g, Z) M T JM = J C D ( τ 1 ) ( ) 1 M = 0 τ Action of Aut(C) on H 1 (C, Z)

19 The spectral curve of genus 4 w 3 + αwz 2 + βz 6 + γz 3 β = 0 a b b b σ k (a i ) = a i+k τ ˆCmonopole = b c d d σ k (b i ) = b i+k b d c d σ k (a 0 ) = a 0 b d d c σ k (b 0 ) b [ 1, 2 ] [ 1, 3 ] [ 2, 3 ] sheet 1 sheet 2 sheet 3

20 The spectral curve of genus 2 y 2 = (x 3 + αx 2ıβ + γ)(x 3 + αx + 2ıβ + γ) ( a ) τ = 3 b b c + 2d Figure: Projection of the previous basis

21 The Humbert Variety τ the period matrix of a genus 2 curve C. τ H Δ if there exist q i Z q 1 + q 2 τ 11 + q 3 τ 12 + q 4 τ 22 + q 5 (τ 2 12 τ 11 τ 22 ) = 0 q 2 3 4(q 1 q 5 + q 2 q 4 ) = Δ C covers elliptic curves E ± Δ = h 2 1, h N. Bierman-Humbert: τ H h 2 ) S Sp(4, Z), such that S τ = τ = ( 1 τ11 h 1 h τ 22 h 1 θ(z 1, z 2 τ) = θ 3 (z 1 + k ) h τ 1,1 k=0 θ [ k h 0 θ(z 1, z 2 τ) = θ 3 (z 1 τ 11 ) θ 3 (2z 2 4 τ 22 ) + θ 3 (z 1 + 1/2 τ 11 ) θ 2 (2z 2 4 τ 22 ) ] (hz2 h 2 τ ) 2,2

22 The Symmetric Monopole η 3 + χ(ζ 6 + bζ 3 1) = 0, b R Theorem To each pair of relatively prime integers (n, m) = 1 for which (m + n)(m 2n) < 0 we obtain a solution to the Ercolani-Sinha constraints for the symmetric curve as follows. First we solve for t, where 2n m m + n = 2 F 1 ( 1 3, 2 3 ; 1, t) 2F 1 ( 1 3, 2 3 ; 1, 1 t). Then b = 1 2t, t = b + b t(1 t) 2. With α 6 = t/(1 t) b then χ 1 2π 3 = (n + m) 3 α 2F 3 (1 + α 6 ) 1 1 ( 1 3 3, 2 ; 1, t). 3

23 The Symmetric Monopole η 3 + χ(ζ 6 + bζ 3 1) = 0, b R satisfies H1 and H2 n, m (n, m) = 1 (m + n)(m 2n) < 0 b = b(m, n) = p(m, n) = 3θ 2 3 θ 2 3 3(p(m, n) 6 45p(m, n) p(m, n) 2 27) ( 0 ( 0 9p(m, n)(p(m, n) 4 10p(m, n) 2 + 9) T (m,n) 2 T (m,n) 6 Expression for χ = χ(m, n) can be given. ) ), T (m, n) = 2ı 3 n + m 2n m

24 The Symmetric Monopole and H3 C monopole : η 3 + χ(ζ 6 + bζ 3 1) = 0, C Toda : y 2 = (x 3 + b) H3 H 0 (C monopole, L λ (n 2)) = 0 for λ (0, 2) θ(λu K; τ monopole ) = 0 for λ (0, 2) θ[e i ](λu; τ Toda ) = 0 for λ (0, 2) Bierman-Humbert+Weierstrass-Poincáre+Martens θ(λu K; τ monopole ) = 0 for λ [0, 2] at least one of the functions (k = 1, 0, 1 mod 3) h k (y) := θ 3 θ 2 ( i 3 y + k T 3 T ) + ( 1) k θ ( 2 y + k θ 3 3 T ) 3 also vanishes. y := y(λ) = λ (n + m)ρ/3, ρ = exp(2πı/3) T = 2i 3 n + m 2n m

25 An Elliptic function Conjecture and the Tetrahedral Monopole h k (y) := θ ( 3 i 3 y + k T θ 2 3 y := y(λ) = λ (n + m)ρ/3, ) T + ( 1) k θ ( 2 y + k θ 3 3 T ) 3 T = 2i 3 n + m 2n m Conjecture h 1 (y)h 0 (y)h 1 (y) vanishes 2( n 1) times on the interval λ (0, 2) θ ( τ 3 ( 3 τ) θ τ 2 3 τ) = θ ( τ ) ( 3 θ τ ) 3 Theorem Only (m, n) = (1, 1) and (0, 1) have no zeros within the range. Theorem The only curves η 3 + χ(ζ 6 + bζ 3 1) = 0 that Γ( 1 6 )Γ( 1 3 ) yield BPS monopoles have b = ±5 2, χ 1 3 = π 1 2 These correspond to tetrahedrally symmetric monopoles.