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

휂3 + 휒(휁6 + b휁3 − 1) = 0 have b = ±5 √ 2, 휒

1 3 = − 1

6 Γ( 1

6 )Γ( 1 3 )

2

1 6 휋 1 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

휂3 + 휒(휁6 + b휁3 − 1) = 0 have b = ±5 √ 2, 휒

1 3 = − 1

6 Γ( 1

6 )Γ( 1 3 )

2

1 6 휋 1 2 .

These correspond to tetrahedrally symmetric monopoles.

▶ SO(3) induces an action on Tℙ1 via PSU(2)

Invariant curves yield symmetric monopoles. ( p q −¯ q ¯ p ) ∈ PSU(2), ∣p∣2 + ∣q∣2 = 1, 휁 → ¯ p 휁 − ¯ q q 휁 + p, 휂 → 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

휂3 + 휒(휁6 + b휁3 − 1) = 0 have b = ±5 √ 2, 휒

1 3 = − 1

6 Γ( 1

6 )Γ( 1 3 )

2

1 6 휋 1 2 .

These correspond to tetrahedrally symmetric monopoles.

▶ SO(3) induces an action on Tℙ1 via PSU(2)

Invariant curves yield symmetric monopoles. ( p q −¯ q ¯ p ) ∈ PSU(2), ∣p∣2 + ∣q∣2 = 1, 휁 → ¯ p 휁 − ¯ q q 휁 + p, 휂 → 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

휂3 + 휒(휁6 + b휁3 − 1) = 0 have b = ±5 √ 2, 휒

1 3 = − 1

6 Γ( 1

6 )Γ( 1 3 )

2

1 6 휋 1 2 .

These correspond to tetrahedrally symmetric monopoles.

▶ SO(3) induces an action on Tℙ1 via PSU(2)

Invariant curves yield symmetric monopoles. ( p q −¯ q ¯ p ) ∈ PSU(2), ∣p∣2 + ∣q∣2 = 1, 휁 → ¯ p 휁 − ¯ q q 휁 + p, 휂 → 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(2휋i/n), p = 휔1/2 q = 0

(휂, 휁) → (휔휂, 휔휁) 휂i휁j invariant for i + j ≡ 0 mod n 휂n + a1휂n−1휁 + a2휂n−2휁2 + . . . + an휁n + 훽휁2n + 훾 = 0

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Spectral Curves

▶ 휔 = exp(2휋i/n), p = 휔1/2 q = 0

(휂, 휁) → (휔휂, 휔휁) 휂i휁j invariant for i + j ≡ 0 mod n 휂n + a1휂n−1휁 + a2휂n−2휁2 + . . . + an휁n + 훽휁2n + 훾 = 0

▶ Impose reality conditions and centre a1 = 0

휂n + a2휂n−2휁2 + . . . + an휁n + 훽휁2n + (−1)n ¯ 훽 = 0, ai ∈ 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(2휋i/n), p = 휔1/2 q = 0

(휂, 휁) → (휔휂, 휔휁) 휂i휁j invariant for i + j ≡ 0 mod n 휂n + a1휂n−1휁 + a2휂n−2휁2 + . . . + an휁n + 훽휁2n + 훾 = 0

▶ Impose reality conditions and centre a1 = 0

휂n + a2휂n−2휁2 + . . . + an휁n + 훽휁2n + (−1)n ¯ 훽 = 0, ai ∈ R By an overall rotation we may choose 훽 real

▶ x = 휂/휁, 휈 = 휁n훽,

xn + a2xn−2 + . . . + an + 휈 + (−1)n∣훽∣2 휈 = 0

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Spectral Curves

▶ 휔 = exp(2휋i/n), p = 휔1/2 q = 0

(휂, 휁) → (휔휂, 휔휁) 휂i휁j invariant for i + j ≡ 0 mod n 휂n + a1휂n−1휁 + a2휂n−2휁2 + . . . + an휁n + 훽휁2n + 훾 = 0

▶ Impose reality conditions and centre a1 = 0

휂n + a2휂n−2휁2 + . . . + an휁n + 훽휁2n + (−1)n ¯ 훽 = 0, ai ∈ R By an overall rotation we may choose 훽 real

▶ x = 휂/휁, 휈 = 휁n훽,

xn + a2xn−2 + . . . + an + 휈 + (−1)n∣훽∣2 휈 = 0

▶ Affine Toda Spectral Curve y = 휈 − (−1)n∣훽∣2 휈

y2 = (xn + a2xn−2 + . . . + an)2 − 4(−1)n∣훽∣2

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Overview

▶ 풞monopole is an unbranched n : 1 cover 풞Toda

gmonopole = (n − 1)2, gToda = (n − 1)

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Overview

▶ 풞monopole is an unbranched n : 1 cover 풞Toda

gmonopole = (n − 1)2, gToda = (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

▶ 풞monopole is an unbranched n : 1 cover 풞Toda

gmonopole = (n − 1)2, gToda = (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

▶ 풞monopole is an unbranched n : 1 cover 풞Toda

gmonopole = (n − 1)2, gToda = (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

▶ 풞monopole is an unbranched n : 1 cover 풞Toda

gmonopole = (n − 1)2, gToda = (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

T1 + iT2 = (T1 − iT2)T = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ e(q1−q2)/2 . . . e(q2−q3)/2 . . . . . . ... . . . . . . e(qn−1−qn)/2 e(qn−q1)/2 . . . ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T3 = − i 2 Diag(p1, p2, . . . , pn) d ds (T1 + iT2) = i[T3, T1 + iT2] ⇒ pi − pi+1 = ˙ qi − ˙ qi+1 d ds T3 = [T1, T2] = i 2[T1 + iT2, T1 − iT2] ⇒ ˙ pi = −eqi−qi+1 + eqi−1−qi

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclic Nahm eqns. ≡ Affine Toda eqns.

Sutcliffe Ansatz C’td

▶ pi, qi real

H = 1 2 ( p2

1 + . . . + p2 n

) − [ eq1−q2 + eq2−q3 + . . . + eqn−q1] . 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

▶ pi, qi real

H = 1 2 ( p2

1 + . . . + p2 n

) − [ eq1−q2 + eq2−q3 + . . . + eqn−q1] . Toda ⇒ Nahm Affine Toda eqns. ⊂ Cyclic Nahm eqns.

▶ G ⊂ SO(3) acts on triples t = (T1, T2, T3) ∈ ℝ3 ⊗ SL(n, ℂ)

via natural action on ℝ3 and conjugation on SL(n, ℂ)

▶ g′ ∈ SO(3) and g = 휌(g′) ∈ SL(n, ℂ). Invariance of curve ⇒

g(T1 + iT2)g−1 = 휔(T1 + iT2), gT3g−1 = T3, g(T1 − iT2)g−1 = 휔−1(T1 − iT2).

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) ∼ 2n − 1 ⊕ 2n − 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) ∼ 2n − 1 ⊕ 2n − 3 ⊕ . . . ⊕ 5 ⊕ 3 ▶ Kostant ⇒ 휌(SO(3)) principal three dimensional subgroup. ▶ g = 휌(g′) = exp

[ 2휋

n H

] , H semi-simple, generator Cartan TDS

▶ g ≡ Diag(휔n−1, . . . , 휔, 1), gEijg−1 = 휔j−i Eij.

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) ∼ 2n − 1 ⊕ 2n − 3 ⊕ . . . ⊕ 5 ⊕ 3 ▶ Kostant ⇒ 휌(SO(3)) principal three dimensional subgroup. ▶ g = 휌(g′) = exp

[ 2휋

n H

] , H semi-simple, generator Cartan TDS

▶ g ≡ Diag(휔n−1, . . . , 휔, 1), gEijg−1 = 휔j−i Eij. ▶ For a cyclically invariant monopole

T1 + iT2 = ∑

훼∈ ˆ Δ

e(훼,˜

q)/2 E훼,

T3 = − i 2 ∑

j

˜ pj Hj

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) ∼ 2n − 1 ⊕ 2n − 3 ⊕ . . . ⊕ 5 ⊕ 3 ▶ Kostant ⇒ 휌(SO(3)) principal three dimensional subgroup. ▶ g = 휌(g′) = exp

[ 2휋

n H

] , H semi-simple, generator Cartan TDS

▶ g ≡ Diag(휔n−1, . . . , 휔, 1), gEijg−1 = 휔j−i Eij. ▶ For a cyclically invariant monopole

T1 + iT2 = ∑

훼∈ ˆ Δ

e(훼,˜

q)/2 E훼,

T3 = − i 2 ∑

j

˜ pj Hj

▶ Sutcliffe follows if ˜

qi and ˜ pi may be chosen real. ˜ qi ∈ ℝ with SU(n) conjug. + overall SO(3) rotation. ˜ pi ∈ ℝ from Ti(s) = −T †

i (s) which also fixes T1 − iT2.

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) ∼ 2n − 1 ⊕ 2n − 3 ⊕ . . . ⊕ 5 ⊕ 3 ▶ Kostant ⇒ 휌(SO(3)) principal three dimensional subgroup. ▶ g = 휌(g′) = exp

[ 2휋

n H

] , H semi-simple, generator Cartan TDS

▶ g ≡ Diag(휔n−1, . . . , 휔, 1), gEijg−1 = 휔j−i Eij. ▶ For a cyclically invariant monopole

T1 + iT2 = ∑

훼∈ ˆ Δ

e(훼,˜

q)/2 E훼,

T3 = − i 2 ∑

j

˜ pj Hj

▶ Sutcliffe follows if ˜

qi and ˜ pi may be chosen real. ˜ qi ∈ ℝ with SU(n) conjug. + overall SO(3) rotation. ˜ pi ∈ ℝ from Ti(s) = −T †

i (s) which also fixes T1 − iT2. ▶ Theorem Any cyclically symmetric monopole is gauge

equivalent to Nahm data given by Sutcliffe’s ansatz, and so

• btained from the affine Toda equations.

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Flows and Solutions

Theorem

The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations.

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Flows and Solutions

Theorem

The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations.

▶ 휋 : ˆ

풞monopole → 풞Toda := ˆ 풞/Cn n : 1 unbranched cover

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Flows and Solutions

Theorem

The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations.

▶ 휋 : ˆ

풞monopole → 풞Toda := ˆ 풞/Cn n : 1 unbranched cover

▶ 휆U + C = 휋∗(휆u + e1),

u, e1 ∈ Jac(풞Toda)

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Flows and Solutions

Theorem

The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations.

▶ 휋 : ˆ

풞monopole → 풞Toda := ˆ 풞/Cn n : 1 unbranched cover

▶ 휆U + C = 휋∗(휆u + e1),

u, e1 ∈ Jac(풞Toda)

▶ C ∈ Θsingular ⊂ Jac(풞monopole),

2C ∈ Λ, C = 휋∗e1

H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Flows and Solutions

Theorem

The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations.

▶ 휋 : ˆ

풞monopole → 풞Toda := ˆ 풞/Cn n : 1 unbranched cover

▶ 휆U + C = 휋∗(휆u + e1),

u, e1 ∈ Jac(풞Toda)

▶ C ∈ Θsingular ⊂ Jac(풞monopole),

2C ∈ Λ, C = 휋∗e1

▶ Fay-Accola

휃[C](휋∗z; ˆ 휏monopole) = c

n

∏

i=1

휃[ei](z; 휏Toda)

”휃-functions are still far from being a spectator sport.”(L.V. Ahlfors) H.W. Braden The Geometry of Monopoles: New and Old IV

Cyclically Symmetric Monopoles

Flows and Solutions

Theorem

The Ercolani-Sinha vector is invariant under the group of symmetries of the spectral curve arising from rotations.

▶ 휋 : ˆ

풞monopole → 풞Toda := ˆ 풞/Cn n : 1 unbranched cover

▶ 휆U + C = 휋∗(휆u + e1),

u, e1 ∈ Jac(풞Toda)

▶ C ∈ Θsingular ⊂ Jac(풞monopole),

2C ∈ Λ, C = 휋∗e1

▶ Fay-Accola

휃[C](휋∗z; ˆ 휏monopole) = c

n

∏

i=1

휃[ei](z; 휏Toda)

”휃-functions are still far from being a spectator sport.”(L.V. Ahlfors)

▶

휃(3z1, z2, z2, z2; ˆ 휏) 휃(z1, z2; 휏)휃(z1 + 1/3, z2; 휏)휃(z1 − 1/3, z2; 휏) = c

H.W. Braden The Geometry of Monopoles: New and Old IV

The C3 cyclically symmetric spectral curve of genus 4

ˆ 풞 : w3 + 훼wz2 + 훽z6 + 훾z3 − 훽 = 0 (훼 = 0 Symmetric Monple) C3 : (z, w) → (휌z, 휌w), 휌 = exp(2휋i/3) 휏 ˆ

풞monopole =

⎛ ⎜ ⎜ ⎝ a b b b b c d d b d c d b d d c ⎞ ⎟ ⎟ ⎠ 휎k

∗(픞i) = 픞i+k

휎k

∗(픟i) = 픟i+k

휎k

∗(픞0) = 픞0

휎k

∗(픟0) ∼ 픟0

12 2 3 4 5 6 7 1 8 9 10 11 12 7 2 3 4 5 6 1 8 9 10 11

[ 1 , 3 ] [ 1 , 2 ] [ 2 , 3 ]

sheet 1 sheet 2 sheet 3

H.W. Braden The Geometry of Monopoles: New and Old IV

The C3 Cyclically Symmetric Monopole

The spectral curve of genus 2

풞 = ˆ 풞/C3 : y2 = (x3 + 훼x + 훾)2 + 4훽2 휏 = ( a

3

b b c + 2d )

2 3 4 5 6 1 2 3 4 5 6 1

Figure: Projection of the previous basis

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

ES conditions

▶ 했 := 휋(픢픰)

Y 2 = (X 3 + a X + g)2 + 4 ES conditions ≡ ∮

했

dX Y = 0

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

ES conditions

▶ 했 := 휋(픢픰)

Y 2 = (X 3 + a X + g)2 + 4 ES conditions ≡ ∮

했

dX Y = 0

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

ES conditions

▶ 했 := 휋(픢픰)

Y 2 = (X 3 + a X + g)2 + 4 ES conditions ≡ ∮

했

dX Y = 0

▶ With a = 훼/훽2/3, g = 훾/훽 and 훽 defined by

6훽1/3 = ∮

했

XdX Y we may recover the monopole spectral curve.

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

Properties

Figure: A log-log plot of the asymptotic behaviour of 훼 versus 훾

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

The Richelot Transform

▶

∮

했

dX Y Y 2 = (X 3 + a X + g)2 + 4

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

The Richelot Transform

▶

∮

했

dX Y Y 2 = (X 3 + a X + g)2 + 4

▶ a, b ∈ ℝ+

a0 = a, b0 = b, an+1 = an + bn 2 , bn+1 = √ anbn lim

n→∞ an = lim n→∞ bn = M(a, b)

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

The Richelot Transform

▶

∮

했

dX Y Y 2 = (X 3 + a X + g)2 + 4

▶ a, b ∈ ℝ+

a0 = a, b0 = b, an+1 = an + bn 2 , bn+1 = √ anbn lim

n→∞ an = lim n→∞ bn = M(a, b)

AGM Gauss ∫ 휋/2 d휙 √ a2 cos2 휙 + b2 sin2 휙 = 휋 2M(a, b)

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

The Richelot Transform

▶

∮

했

dX Y Y 2 = (X 3 + a X + g)2 + 4

▶ a, b ∈ ℝ+

a0 = a, b0 = b, an+1 = an + bn 2 , bn+1 = √ anbn lim

n→∞ an = lim n→∞ bn = M(a, b)

AGM Gauss ∫ 휋/2 d휙 √ a2 cos2 휙 + b2 sin2 휙 = 휋 2M(a, b) ℰn → ℰn+1, ℰn : y2

n = xn(xn − a2 n)(xn − b2 n)

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

The Richelot Transform

▶

∮

했

dX Y Y 2 = (X 3 + a X + g)2 + 4

▶

a a’ b’ b c’ c v w w’ u u’ v’

Figure 1. Roots of 푃, 푄, 푅 and 푈, 푉, 푊

P(x) = (x − a)(x − a′) Q(x) = (x − b)(x − b′) R(x) = (x − c)(x − c′) U(x) = (x − u)(x − u′) V (x) = (x − v)(x − v′) W (x) = (x − w)(x − w′)

▶ Degree 2 correspondence 풵 ⊂ 풞 × 풞′

풞 : y2 + P(x)Q(x)R(x) = 0, 풞′ : Δy′2 + U(x′)V (x′)W (x′) = 0 풵 : { P(x)U(x′) + Q(x)V (x′) = 0, yy′ = P(x)U(x′)(x − x′).

H.W. Braden The Geometry of Monopoles: New and Old IV

C3 Cyclically Symmetric Monopoles

The Richelot Transform

▶

∮

했

dX Y Y 2 = (X 3 + a X + g)2 + 4

▶

a a’ b’ b c’ c v w w’ u u’ v’

Figure 1. Roots of 푃, 푄, 푅 and 푈, 푉, 푊

P(x) = (x − a)(x − a′) Q(x) = (x − b)(x − b′) R(x) = (x − c)(x − c′) U(x) = (x − u)(x − u′) V (x) = (x − v)(x − v′) W (x) = (x − w)(x − w′)

▶

∫ a′

a

dx √ −P(x)Q(x)R(x) = 2 √ Δ ∫ w

v

dx √ −U(x)V (x)W (x)

H.W. Braden The Geometry of Monopoles: New and Old IV

Summary

There remain two types of outstanding difficulty when implementing the algebro-geometric solution of integrable systems.

H.W. Braden The Geometry of Monopoles: New and Old IV

Summary

There remain two types of outstanding difficulty when implementing the algebro-geometric solution of integrable systems.

▶ Transcendental constraints: 풞 constrained by requiring periods

• f a given meromorphic differential to be specified.

ℒ2 trivial U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m.

H.W. Braden The Geometry of Monopoles: New and Old IV

Summary

There remain two types of outstanding difficulty when implementing the algebro-geometric solution of integrable systems.

▶ Transcendental constraints: 풞 constrained by requiring periods

• f a given meromorphic differential to be specified.

ℒ2 trivial U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m. ▶ Flows and Theta Divisor:

H0(풞, ℒ) = 0 휃 (tU + C∣휏) = 0

H.W. Braden The Geometry of Monopoles: New and Old IV