Symmetry, Curves and Monopoles H.W. Braden EMPG Edinburgh, November - - PowerPoint PPT Presentation

▶
symmetry curves and monopoles
SMART_READER_LITE
LIVE PREVIEW

Symmetry, Curves and Monopoles H.W. Braden EMPG Edinburgh, November - - PowerPoint PPT Presentation

Feb 10, 2023 347 likes •882 views

Symmetry, Curves and Monopoles H.W. Braden EMPG Edinburgh, November 2011 Curve results with T.P. Northover. Monopole Results in collaboration with V.Z. Enolski, A.DAvanzo. H.W. Braden Symmetry, Curves and Monopoles Overview Zero Curvature

slide-1
SLIDE 1

Symmetry, Curves and Monopoles

H.W. Braden EMPG Edinburgh, November 2011 Curve results with T.P. Northover. Monopole Results in collaboration with V.Z. Enolski, A.D’Avanzo.

H.W. Braden Symmetry, Curves and Monopoles

slide-2
SLIDE 2

Overview

Equations Zero Curvature/Lax − → Spectral Curve 풞 ⊂ 풮 ↑ ↓ Reconstruction ← − Baker-Akhiezer Function

H.W. Braden Symmetry, Curves and Monopoles

slide-3
SLIDE 3

Overview

Equations Zero Curvature/Lax − → Spectral Curve 풞 ⊂ 풮 ↑ ↓ Reconstruction ← − Baker-Akhiezer Function tU + C ∈ Jac(풞)

H.W. Braden Symmetry, Curves and Monopoles

slide-4
SLIDE 4

Overview

Equations Zero Curvature/Lax − → Spectral Curve 풞 ⊂ 풮 ↑ ↓ Reconstruction ← − Baker-Akhiezer Function tU + C ∈ Jac(풞)

▶ BPS Monopoles ▶ Sigma Model reductions in AdS/CFT ▶ KP, KdV solitons ▶ Harmonic Maps ▶ SW Theory/Integrable Systems

H.W. Braden Symmetry, Curves and Monopoles

slide-5
SLIDE 5

Overview

Equations Zero Curvature/Lax − → Spectral Curve 풞 ⊂ 풮 ↑ ↓ Reconstruction ← − Baker-Akhiezer Function tU + C ∈ Jac(풞) Difficulties:

▶ Transcendental constraints. ℒ2 trivial

H.W. Braden Symmetry, Curves and Monopoles

slide-6
SLIDE 6

Overview

Equations Zero Curvature/Lax − → Spectral Curve 풞 ⊂ 풮 ↑ ↓ Reconstruction ← − Baker-Akhiezer Function tU + C ∈ Jac(풞) Difficulties:

▶ Transcendental constraints. ℒ2 trivial ▶ Flows and Theta Divisor.

H0(풞, ℒ) = 0

H.W. Braden Symmetry, Curves and Monopoles

slide-7
SLIDE 7

Overview

Equations Zero Curvature/Lax − → Spectral Curve 풞 ⊂ 풮 ↑ ↓ Reconstruction ← − Baker-Akhiezer Function tU + C ∈ Jac(풞) Difficulties:

▶ Transcendental constraints. ℒ2 trivial ▶ Flows and Theta Divisor.

H0(풞, ℒ) = 0 휃 (tU + C∣휏)

H.W. Braden Symmetry, Curves and Monopoles

slide-8
SLIDE 8

Setting

BPS Monopoles

▶ Reduction of F = ∗F

L = −1 2Tr FijF ij + Tr DiΦ DiΦ.

▶ Bi = 1

2

3

∑

j,k=1

휖ijkF jk = DiΦ

▶ A monopole of charge n

√ −1 2Tr Φ(r)2

  • r→∞

∼ 1− n 2r +O(r−2), r = √ x2

1 + x2 2 + x2 3 ▶ Monopoles ↔ Nahm Data ↔ Hitchin Data

H.W. Braden Symmetry, Curves and Monopoles

slide-9
SLIDE 9

Setting

BPS Monopoles: Nahm Data for charge n SU(2) monopoles

Three n × n matrices Ti(s) with s ∈ [0, 2] satisfying the following: N1 Nahm’s equation dTi ds = 1 2

3

∑

j,k=1

휖ijk[Tj, Tk]. N2 Ti(s) is regular for s ∈ (0, 2) and has simple poles at s = 0, 2. Residues form su(2) irreducible n-dimensional representation. N3 Ti(s) = −T †

i (s),

Ti(s) = T t

i (2 − s).

H.W. Braden Symmetry, Curves and Monopoles

slide-10
SLIDE 10

Setting

BPS Monopoles: Nahm Data for charge n SU(2) monopoles

Three n × n matrices Ti(s) with s ∈ [0, 2] satisfying the following: N1 Nahm’s equation dTi ds = 1 2

3

∑

j,k=1

휖ijk[Tj, Tk]. N2 Ti(s) is regular for s ∈ (0, 2) and has simple poles at s = 0, 2. Residues form su(2) irreducible n-dimensional representation. N3 Ti(s) = −T †

i (s),

Ti(s) = T t

i (2 − s).

A(휁) = T1 + iT2 − 2iT3휁 + (T1 − iT2)휁2 M(휁) = −iT3 + (T1 − iT2)휁 Nahm’s eqn. dTi ds = 1 2

3

∑

j,k=1

휖ijk[Tj, Tk] ⇐ ⇒ [ d ds + M, A] = 0.

H.W. Braden Symmetry, Curves and Monopoles

slide-11
SLIDE 11

Spectral Curves

풞 ⊂ 풮

▶ [ d

ds + M(휁), A(휁)] = 0, 풞 : 0 = det(휂1n + A(휁)) := P(휂, 휁) P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁)

H.W. Braden Symmetry, Curves and Monopoles

slide-12
SLIDE 12

Spectral Curves

풞 ⊂ 풮

▶ [ d

ds + M(휁), A(휁)] = 0, 풞 : 0 = det(휂1n + A(휁)) := P(휂, 휁) P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁)

▶ Where does 풞 lie?

풞 ⊂ 풮

H.W. Braden Symmetry, Curves and Monopoles

slide-13
SLIDE 13

Spectral Curves

풞 ⊂ 풮

▶ [ d

ds + M(휁), A(휁)] = 0, 풞 : 0 = det(휂1n + A(휁)) := P(휂, 휁) P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁)

▶ Where does 풞 lie?

풞 ⊂ 풮

▶ 풞monopole ⊂ Tℙ1 := 풮

(휂, 휁) → 휂 d d휁 ∈ Tℙ1 Minitwistor description

H.W. Braden Symmetry, Curves and Monopoles

slide-14
SLIDE 14

Spectral Curves

풞 ⊂ 풮

▶ [ d

ds + M(휁), A(휁)] = 0, 풞 : 0 = det(휂1n + A(휁)) := P(휂, 휁) P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁)

▶ Where does 풞 lie?

풞 ⊂ 풮

▶ 풞monopole ⊂ Tℙ1 := 풮

(휂, 휁) → 휂 d d휁 ∈ Tℙ1 Minitwistor description

▶ 풞휎−model ⊂ ℙ2 := 풮 ▶ 풮 = T ∗Σ Hitchin Systems on a Riemann surface Σ ▶ 풮 = K3 ▶ 풮 a Poisson surface ▶ separation of variables ↔ Hilb[N](풮) ▶ X the total space of an appropriate line bundle ℒ over 풮 ↔

noncompact CY

H.W. Braden Symmetry, Curves and Monopoles

slide-15
SLIDE 15

Spectral Curves

풞 ⊂ 풮

▶ [ d

ds + M(휁), A(휁)] = 0, 풞 : 0 = det(휂1n + A(휁)) := P(휂, 휁) P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁)

▶ Where does 풞 lie?

풞 ⊂ 풮

▶ 풞monopole ⊂ Tℙ1 := 풮

(휂, 휁) → 휂 d d휁 ∈ Tℙ1 Minitwistor description

▶ 풞휎−model ⊂ ℙ2 := 풮 ▶ 풮 = T ∗Σ Hitchin Systems on a Riemann surface Σ ▶ 풮 = K3 ▶ 풮 a Poisson surface ▶ separation of variables ↔ Hilb[N](풮) ▶ X the total space of an appropriate line bundle ℒ over 풮 ↔

noncompact CY

▶ Symmetry: 풞 ⊂ ℙa,b,c

[X, Y , Z] ∼ [휆aX, 휆bY , 휆cZ], 휆 ∈ ℂ∗

H.W. Braden Symmetry, Curves and Monopoles

slide-16
SLIDE 16

Spectral Curves: data

▶ Homology basis {훾i}2g i=1 = {픞i, 픟i}g i=1

▶ algorithm for branched covers of ℙ1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries

▶ Holomorphic differentials dui (i = 1, . . . , g)

H.W. Braden Symmetry, Curves and Monopoles

slide-17
SLIDE 17

Spectral Curves: data

▶ Homology basis {훾i}2g i=1 = {픞i, 픟i}g i=1

▶ algorithm for branched covers of ℙ1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries

▶ Holomorphic differentials dui (i = 1, . . . , g) ▶ Period Matrix 휏 = ℬ풜−1 where

Π := (풜 ℬ ) = (∮

픞i duj

∮

픟i duj

)

▶ 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

H.W. Braden Symmetry, Curves and Monopoles

slide-18
SLIDE 18

Spectral Curves: data

▶ Homology basis {훾i}2g i=1 = {픞i, 픟i}g i=1

▶ algorithm for branched covers of ℙ1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries

▶ Holomorphic differentials dui (i = 1, . . . , g) ▶ Period Matrix 휏 = ℬ풜−1 where

Π := (풜 ℬ ) = (∮

픞i duj

∮

픟i duj

)

▶ 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

▶ normalized holomorphic differentials 휔i,

∮

픞i 휔j = 훿ij

∮

픟i 휔j = 휏ij ▶ 풞 often has an antiholomorphic involution/real structure

H.W. Braden Symmetry, Curves and Monopoles

slide-19
SLIDE 19

Spectral Curves: data

▶ Homology basis {훾i}2g i=1 = {픞i, 픟i}g i=1

▶ algorithm for branched covers of ℙ1 (Tretkoff & Tretkoff) ▶ poor if curve has symmetries

▶ Holomorphic differentials dui (i = 1, . . . , g) ▶ Period Matrix 휏 = ℬ풜−1 where

Π := (풜 ℬ ) = (∮

픞i duj

∮

픟i duj

)

▶ 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

▶ normalized holomorphic differentials 휔i,

∮

픞i 휔j = 훿ij

∮

픟i 휔j = 휏ij ▶ 풞 often has an antiholomorphic involution/real structure

▶ 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 Symmetry, Curves and Monopoles

slide-20
SLIDE 20

Spectral Curves: Flows

휃 (tU + C∣휏)

H.W. Braden Symmetry, Curves and Monopoles

slide-21
SLIDE 21

Spectral Curves: Flows

휃 (tU + C∣휏)

▶ Meromorphic differentials describe flows ⇐

⇒ U

H.W. Braden Symmetry, Curves and Monopoles

slide-22
SLIDE 22

Spectral Curves: Flows

휃 (tU + C∣휏)

▶ Meromorphic differentials describe flows ⇐

⇒ U

▶ 휃(e ∣ 휏) = 0 ⇐

⇒ e ∈ Θ ⊂ Jac 풞

▶ e ≡ 휙Q

(g−1 ∑

i=1

Pi ) + KQ, 휙Q(P) := ∫ P

Q

휔 multe 휃 = i (g−1 ∑

i=1

Pi ) = dim H1(풞, ℒ∑g−1

i=1 Pi) = dim H0(풞, ℒ∑g−1 i=1 Pi) H.W. Braden Symmetry, Curves and Monopoles

slide-23
SLIDE 23

Spectral Curves: Flows

휃 (tU + C∣휏)

▶ Meromorphic differentials describe flows ⇐

⇒ U

▶ 휃(e ∣ 휏) = 0 ⇐

⇒ e ∈ Θ ⊂ Jac 풞

▶ e ≡ 휙Q

(g−1 ∑

i=1

Pi ) + KQ, 휙Q(P) := ∫ P

Q

휔 multe 휃 = i (g−1 ∑

i=1

Pi ) = dim H1(풞, ℒ∑g−1

i=1 Pi) = dim H0(풞, ℒ∑g−1 i=1 Pi)

▶ −KQ = 휙∗ (Δ − (g − 1)Q) = 휙Q (Δ),

deg Δ = g − 1, 2Δ ≡ 풦풞

H.W. Braden Symmetry, Curves and Monopoles

slide-24
SLIDE 24

Spectral Curves: Flows

휃 (tU + C∣휏)

▶ Meromorphic differentials describe flows ⇐

⇒ U

▶ 휃(e ∣ 휏) = 0 ⇐

⇒ e ∈ Θ ⊂ Jac 풞

▶ e ≡ 휙Q

(g−1 ∑

i=1

Pi ) + KQ, 휙Q(P) := ∫ P

Q

휔 multe 휃 = i (g−1 ∑

i=1

Pi ) = dim H1(풞, ℒ∑g−1

i=1 Pi) = dim H0(풞, ℒ∑g−1 i=1 Pi)

▶ −KQ = 휙∗ (Δ − (g − 1)Q) = 휙Q (Δ),

deg Δ = g − 1, 2Δ ≡ 풦풞

▶ 풞 often constrained by fixing periods of a given meromorphic

differential

▶ BPS Monopoles ▶ Sigma Model reductions in AdS/CFT ▶ Harmonic Maps H.W. Braden Symmetry, Curves and Monopoles

slide-25
SLIDE 25

Symmetry

Why? Can be used to simplify the period matrix and integrals.

H.W. Braden Symmetry, Curves and Monopoles

slide-26
SLIDE 26

Symmetry

Why? Can be used to simplify the period matrix and integrals. 휎 ∈ Aut(풞) 휎∗휔j = 휔kLk

j , 휎∗

(픞i 픟i ) = M (픞i 픟i ) := (A B C D ) (픞i 픟i ) , M ∈ Sp(2g, ℤ)

H.W. Braden Symmetry, Curves and Monopoles

slide-27
SLIDE 27

Symmetry

Why? Can be used to simplify the period matrix and integrals. 휎 ∈ Aut(풞) 휎∗휔j = 휔kLk

j , 휎∗

(픞i 픟i ) = M (픞i 픟i ) := (A B C D ) (픞i 픟i ) , M ∈ Sp(2g, ℤ) ∮

휎∗훾

휔 = ∮

훾

휎∗휔 ⇐ ⇒ (A B C D ) (풜 ℬ ) = (풜 ℬ ) L ⇐ ⇒ MΠ = ΠL Restricts 휏: 휏B휏 + 휏A − D휏 − C = 0 Curves with lots of symmetries: evaluate 휏 via character theory

H.W. Braden Symmetry, Curves and Monopoles

slide-28
SLIDE 28

Symmetry

Why? Can be used to simplify the period matrix and integrals. 휎 ∈ Aut(풞) 휎∗휔j = 휔kLk

j , 휎∗

(픞i 픟i ) = M (픞i 픟i ) := (A B C D ) (픞i 픟i ) , M ∈ Sp(2g, ℤ) ∮

휎∗훾

휔 = ∮

훾

휎∗휔 ⇐ ⇒ (A B C D ) (풜 ℬ ) = (풜 ℬ ) L ⇐ ⇒ MΠ = ΠL Restricts 휏: 휏B휏 + 휏A − D휏 − C = 0 Curves with lots of symmetries: evaluate 휏 via character theory

▶ How can one specify homology cycles?

H.W. Braden Symmetry, Curves and Monopoles

slide-29
SLIDE 29

Symmetry

H.W. Braden Symmetry, Curves and Monopoles

slide-30
SLIDE 30

Symmetry

Why? Can be used to simplify the period matrix and integrals. 휎 ∈ Aut(풞) 휎∗휔j = 휔kLk

j , 휎∗

(픞i 픟i ) = M (픞i 픟i ) := (A B C D ) (픞i 픟i ) , M ∈ Sp(2g, ℤ) ∮

휎∗훾

휔 = ∮

훾

휎∗휔 ⇐ ⇒ (A B C D ) (풜 ℬ ) = (풜 ℬ ) L ⇐ ⇒ MΠ = ΠL Restricts 휏: 휏B휏 + 휏A − D휏 − C = 0 Curves with lots of symmetries: evaluate 휏 via character theory

▶ How can one specify homology cycles? ▶ How to determine M, 휎∗(훾) = M.훾? extcurves ▶ How to determine a good basis {훾i}?

H.W. Braden Symmetry, Curves and Monopoles

slide-31
SLIDE 31

Calculation

Example: Klein’s Curve and Problems

▶ 풞: X 3Y + Y 3Z + Z 3X = 0 ▶ Aut(풞) = PSL(2, 7) order 168.

H.W. Braden Symmetry, Curves and Monopoles

slide-32
SLIDE 32

Calculation

Example: Klein’s Curve and Problems

▶ 풞: X 3Y + Y 3Z + Z 3X = 0 ▶ Aut(풞) = PSL(2, 7) order 168. ▶ 휏RL =

⎛ ⎜ ⎝

−1+3i √ 7 8 −1−i √ 7 4 −3+i √ 7 8 −1−i √ 7 4 1+i √ 7 2 −1−i √ 7 4 −3+i √ 7 8 −1−i √ 7 4 7+3i √ 7 8

⎞ ⎟ ⎠

H.W. Braden Symmetry, Curves and Monopoles

slide-33
SLIDE 33

Calculation

Example: Klein’s Curve and Problems

▶ 풞: X 3Y + Y 3Z + Z 3X = 0 ▶ Aut(풞) = PSL(2, 7) order 168. ▶ 풞: w7 = (z − 1)(z − 휌)2(z − 휌2)4, 휌 = exp(2휋i/3)

S h e e t 1 S h e e t 2 S h e e t 3 S h e e t 4 S h e e t 5 S h e e t 6 S h e e t 7

Figure: Homology basis in (z, w) coordinates

H.W. Braden Symmetry, Curves and Monopoles

slide-34
SLIDE 34

Calculation

Example: Klein’s Curve and Problems

▶ 풞: X 3Y + Y 3Z + Z 3X = 0 ▶ Aut(풞) = PSL(2, 7) order 168. ▶ 풞: w7 = (z − 1)(z − 휌)2(z − 휌2)4, 휌 = exp(2휋i/3) ▶ 휏 = 1

2 ⎛ ⎝ e 1 1 1 e 1 1 1 e ⎞ ⎠, e = −1+i

√ 7 2

KQ = i √ 7 (3, −1, 5) Q = (z, w) = (휌, 0)

H.W. Braden Symmetry, Curves and Monopoles

slide-35
SLIDE 35

Calculation

Example: Klein’s Curve and Problems

▶ 풞: X 3Y + Y 3Z + Z 3X = 0 ▶ Aut(풞) = PSL(2, 7) order 168. ▶ 풞: w7 = (z − 1)(z − 휌)2(z − 휌2)4, 휌 = exp(2휋i/3) ▶ 휏 = 1

2 ⎛ ⎝ e 1 1 1 e 1 1 1 e ⎞ ⎠, e = −1+i

√ 7 2

KQ = i √ 7 (3, −1, 5) Q = (z, w) = (휌, 0)

▶ This depends on finding a good adapted basis simplifying the

action of Aut(풞) on H1(풞, ℤ)

▶ Symplectic Equivalence of Period Matrices 휏, 휏 ′

M = (A B C D ) ∈ Sp(2g, ℤ) ⇔ MTJM = J ( 휏 ′ −1 ) M (1 휏 ) = 0

H.W. Braden Symmetry, Curves and Monopoles

slide-36
SLIDE 36

Calculation: The spectral curve of genus 4

ˆ 풞 : w3 + 훼wz2 + 훽z6 + 훾z3 − 훽 = 0 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 Symmetry, Curves and Monopoles

slide-37
SLIDE 37

Calculation

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 Symmetry, Curves and Monopoles

slide-38
SLIDE 38

BPS Monopoles

Hitchin data

H1 풞 ⊂ Tℙ1 Reality conditions ar(휁) = (−1)r휁2rar(−1 휁 ) H2 ℒ2 is trivial on 풞 and ℒ1(n − 1) is real.

H.W. Braden Symmetry, Curves and Monopoles

slide-39
SLIDE 39

BPS Monopoles

Hitchin data

H1 풞 ⊂ Tℙ1 Reality conditions ar(휁) = (−1)r휁2rar(−1 휁 ) H2 ℒ2 is trivial on 풞 and ℒ1(n − 1) is real. ⇐ ⇒ Ercolani-Sinha Constraints: 2U ∈ Λ ⇐ ⇒ U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m

H.W. Braden Symmetry, Curves and Monopoles

slide-40
SLIDE 40

BPS Monopoles

Hitchin data

H1 풞 ⊂ Tℙ1 Reality conditions ar(휁) = (−1)r휁2rar(−1 휁 ) H2 ℒ2 is trivial on 풞 and ℒ1(n − 1) is real. ⇐ ⇒ Ercolani-Sinha Constraints: 2U ∈ Λ ⇐ ⇒ U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m

⇐ ⇒ ∃ 1-cycle 픢픰 = n ⋅ 픞 + m ⋅ 픟 s.t. (n, m) (풜 ℬ ) = −2(0, . . . , 0, 1), dug = 휂n−2

∂풫 ∂휂

d휁, First transcendental constraint. Number Theory+Ramanujan

H.W. Braden Symmetry, Curves and Monopoles

slide-41
SLIDE 41

BPS Monopoles

Hitchin data

H1 풞 ⊂ Tℙ1 Reality conditions ar(휁) = (−1)r휁2rar(−1 휁 ) H2 ℒ2 is trivial on 풞 and ℒ1(n − 1) is real. ⇐ ⇒ Ercolani-Sinha Constraints: 2U ∈ Λ ⇐ ⇒ U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m

⇐ ⇒ ∃ 1-cycle 픢픰 = n ⋅ 픞 + m ⋅ 픟 s.t. (n, m) (풜 ℬ ) = −2(0, . . . , 0, 1), dug = 휂n−2

∂풫 ∂휂

d휁, First transcendental constraint. Number Theory+Ramanujan H3 H0(풞, ℒs(n − 2)) = 0 for s ∈ (0, 2)⇐ ⇒ 휃(sU + C ∣ 휏) ∕= 0 C = KQ + 휙Q ( (n − 2)

n

∑

k=1

∞k ) Second transcendental constraint.

H.W. Braden Symmetry, Curves and Monopoles

slide-42
SLIDE 42

Cyclically Symmetric Monopoles

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

( 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 Symmetry, Curves and Monopoles

slide-43
SLIDE 43

Cyclically Symmetric Monopoles

▶ SO(3) induces an action on Tℙ1 via PSU(2) ▶ Invariant curves yield symmetric monopoles.

H.W. Braden Symmetry, Curves and Monopoles

slide-44
SLIDE 44

Cyclically Symmetric Monopoles

▶ SO(3) induces an action on Tℙ1 via PSU(2) ▶ Invariant curves yield symmetric monopoles. ▶ 휔 = exp(2휋i/n), (휂, 휁) → (휔휂, 휔휁)

Cn symmetric (centred) charge-n monopole curve of form ˆ 풞 : 휂n+a2휂n−2휁2+. . .+an휁n+훽휁2n+(−1)n훽 = 0, ai, 훽 ∈ R

▶ ˆ

풞 a n : 1 unbranched cover Affine Toda Spectral Curve 풞 := ˆ 풞/Cn y2 = (xn + a2xn−2 + . . . + an)2 − 4(−1)n훽2 gmonopole = (n − 1)2, gToda = (n − 1)

H.W. Braden Symmetry, Curves and Monopoles

slide-45
SLIDE 45

Cyclically Symmetric Monopoles

▶ SO(3) induces an action on Tℙ1 via PSU(2) ▶ Invariant curves yield symmetric monopoles. ▶ 휔 = exp(2휋i/n), (휂, 휁) → (휔휂, 휔휁)

Cn symmetric (centred) charge-n monopole curve of form ˆ 풞 : 휂n+a2휂n−2휁2+. . .+an휁n+훽휁2n+(−1)n훽 = 0, ai, 훽 ∈ R

▶ ˆ

풞 a n : 1 unbranched cover Affine Toda Spectral Curve 풞 := ˆ 풞/Cn y2 = (xn + a2xn−2 + . . . + an)2 − 4(−1)n훽2 gmonopole = (n − 1)2, gToda = (n − 1)

▶ Sutcliffe’s Ansatz: Cyclic Nahm eqns. ⊃ Affine Toda eqns.

H.W. Braden Symmetry, Curves and Monopoles

slide-46
SLIDE 46

Cyclically Symmetric Monopoles

▶ SO(3) induces an action on Tℙ1 via PSU(2) ▶ Invariant curves yield symmetric monopoles. ▶ 휔 = exp(2휋i/n), (휂, 휁) → (휔휂, 휔휁)

Cn symmetric (centred) charge-n monopole curve of form ˆ 풞 : 휂n+a2휂n−2휁2+. . .+an휁n+훽휁2n+(−1)n훽 = 0, ai, 훽 ∈ R

▶ ˆ

풞 a n : 1 unbranched cover Affine Toda Spectral Curve 풞 := ˆ 풞/Cn y2 = (xn + a2xn−2 + . . . + an)2 − 4(−1)n훽2 gmonopole = (n − 1)2, gToda = (n − 1)

▶ Sutcliffe’s Ansatz: Cyclic Nahm eqns. ⊃ Affine Toda eqns. ▶ Cyclic Nahm eqns. ≡ Affine Toda eqns.

Theorem

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 Symmetry, Curves and Monopoles

slide-47
SLIDE 47

Cyclically Symmetric Monopoles

▶ Cyclic monopoles ≡ (particular) Affine Toda solns.

H.W. Braden Symmetry, Curves and Monopoles

slide-48
SLIDE 48

Cyclically Symmetric Monopoles

▶ Cyclic monopoles ≡ (particular) Affine Toda solns. ▶ 휋 : ˆ

풞 → 풞 := ˆ 풞/Cn 휆U + C = 휋∗(휆u + c), u, c ∈ Jac(풞Toda)

H.W. Braden Symmetry, Curves and Monopoles

slide-49
SLIDE 49

Cyclically Symmetric Monopoles

▶ Cyclic monopoles ≡ (particular) Affine Toda solns. ▶ 휋 : ˆ

풞 → 풞 := ˆ 풞/Cn 휆U + C = 휋∗(휆u + c), u, c ∈ Jac(풞Toda)

▶ Fay-Accola

휃[C](휋∗z; 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 Symmetry, Curves and Monopoles

slide-50
SLIDE 50

C3 Cyclically Symmetric Monopoles

▶ 했 := 휋(픢픰)

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

했

dX Y = 0

H.W. Braden Symmetry, Curves and Monopoles

slide-51
SLIDE 51

C3 Cyclically Symmetric Monopoles

▶ 했 := 휋(픢픰)

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

했

dX Y = 0

H.W. Braden Symmetry, Curves and Monopoles

slide-52
SLIDE 52

C3 Cyclically Symmetric Monopoles

▶ 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 Symmetry, Curves and Monopoles