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 ⊂ Tℙ1 : 0 = P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(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 ⊂ Tℙ1 : 0 = P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁) ▶ deg ar(휁) ≤ 2r

ar(휁) = (−1)r휁2rar(−1 ¯ 휁 ) ar(휁) = 휒r [∏r

l=1

(

훼l 훼l

)1/2] ∏r

k=1(휁 − 훼r)(휁 + 1 훼r )

훼r ∈ ℂ, 휒 ∈ ℝ ar(휁) given by 2r + 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 ⊂ Tℙ1 : 0 = P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁) ▶ deg ar(휁) ≤ 2r

ar(휁) = (−1)r휁2rar(−1 ¯ 휁 ) ar(휁) = 휒r [∏r

l=1

(

훼l 훼l

)1/2] ∏r

k=1(휁 − 훼r)(휁 + 1 훼r )

훼r ∈ ℂ, 휒 ∈ ℝ ar(휁) given by 2r + 1 (real) parameters

▶ Ercolani-Sinha Constraints:

• 1. U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m.

• 2. Ω = 훽0휂n−2 + 훽1(휁)휂n−3 + . . . + 훽n−2(휁)

∂풫 ∂휂

d휁, ∮

픢픰

Ω = −2훽0 픢픰 = n ⋅ 픞 + m ⋅ 픟 impose g transcendental constraints on 풞

n

∑

j=2

(2j + 1) − g = (n + 3)(n − 1) − (n − 1)2 = 4(n − 1)

▶ Flows and Theta Divisor:

sU + 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 ⊂ Tℙ1 : 0 = P(휂, 휁) = 휂n + a1(휁)휂n−1 + . . . + an(휁) ▶ deg ar(휁) ≤ 2r

ar(휁) = (−1)r휁2rar(−1 ¯ 휁 ) ar(휁) = 휒r [∏r

l=1

(

훼l 훼l

)1/2] ∏r

k=1(휁 − 훼r)(휁 + 1 훼r )

훼r ∈ ℂ, 휒 ∈ ℝ ar(휁) given by 2r + 1 (real) parameters

▶ Ercolani-Sinha Constraints:

• 1. U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m.

• 2. Ω = 훽0휂n−2 + 훽1(휁)휂n−3 + . . . + 훽n−2(휁)

∂풫 ∂휂

d휁, ∮

픢픰

Ω = −2훽0 픢픰 = n ⋅ 픞 + m ⋅ 픟 impose g transcendental constraints on 풞

n

∑

j=2

(2j + 1) − g = (n + 3)(n − 1) − (n − 1)2 = 4(n − 1)

▶ Flows and Theta Divisor:

sU + C ∕∈ Θ

▶ Symmetry aids calculation of 휏, U, KQ

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

Possible Charge 3 Curve

A trigonal curve and its homology

▶ w3 = 6

∏

i=1

(z − 휆i) ℛ : (z, w) → (z, 휌w), 휌 = exp{2i휋/3}

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

Possible Charge 3 Curve

A trigonal curve and its homology

▶ w3 = 6

∏

i=1

(z − 휆i) ℛ : (z, w) → (z, 휌w), 휌 = exp{2i휋/3}

▶ ℛ(픟i) = 픞픦,

i = 1, 2, 3, ℛ(픟4) = −픞4 Aut(풞) = C3

a1 a2 a3 a4 λ1 λ4 λ6 λ2 λ3 λ5

a1 b1 λ1 λ2

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

Possible Charge 3 Curve

Differentials and periods (Wellstein, 1899; Matsumoto, 2000; BE 2006)

du1 = dz w , du2 = dz w2 , du3 = zdz w2 , du4 = z2dz w2 풜 = (풜ki) = ⎛ ⎝ ∮

픞k

dui ⎞ ⎠

i,k=1,...,4

= (x, b, c, d) ℬ = H풜Λ, H = diag(1, 1, 1, −1), Λ = diag(휌, 휌2, 휌2, 휌2) ∑

i

⎛ ⎜ ⎝ ∮

픞i

duk ∮

픟i

dul − ∮

픟i

duk ∮

픞i

dul ⎞ ⎟ ⎠ = 0 ⇔ 0 = xTHb = xTHc = xTHd 휏픟 = 풜ℬ−1 = 휌 ( H − (1 − 휌) xxT xTHx ) Im 휏픟 is positive definite if and only if ¯ xTHx < 0

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

Possible Charge 3 Curve

Solving the Ercolani-Sinha Constraints

Theorem

U =

1 2휋횤

(∮

픟1 훾∞, . . . ,

∮

픟g 훾∞

)T = 1

2n + 1 2휏m ⇐

⇒ x = 휉(Hn + 휌2m), 휉 = 6휒

1 3

[nTHn − m.n + mTHm] where 휉 is real x1 n1 + 휌2m1 = x2 n2 + 휌2m2 = x3 n3 + 휌2m3 = x4 −n4 + 휌2m4 = 휉, xi/xj ∈ ℚ[휌] ¯ xTHx < 0 ⇐ ⇒ ¯ xTHx ∣휉∣2 = [nTHn − m.n + mTHm] =

3

∑

i=1

(n2

i − nimi + m2 i ) − n2 4 − m2 4 − m4n4 < 0.

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

Symmetric 3-monopoles

▶ 풞 :

휂3 + 휒(휁6 + b휁3 − 1) = 0, b ∈ ℝ

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

Symmetric 3-monopoles

▶ 풞 :

휂3 + 휒(휁6 + b휁3 − 1) = 0, b ∈ ℝ

▶

(훼, 휌2훽, 휌훼, 훽, 휌2훼, 휌훽), 훼 =

3

√ −b + √ b2 + 4 2 > 0, 훼훽 = −1

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

Symmetric 3-monopoles

▶ 풞 :

휂3 + 휒(휁6 + b휁3 − 1) = 0, b ∈ ℝ

▶

(훼, 휌2훽, 휌훼, 훽, 휌2훼, 휌훽), 훼 =

3

√ −b + √ b2 + 4 2 > 0, 훼훽 = −1

▶ Aut(풞) → C3 × S3 (휁, 휂) → (휌휁, 휂), (휁, 휂) → (−1/휁, −휂/휁2)

ℐ1(훼) =

훼

∫ dz w = −2휋 √ 3훼 9

2F1

(1 3, 1 3; 1; −훼6 ) 풥1(훼) =

훽

∫ dz w = 2휋 √ 3 9훼

2F1

(1 3, 1 3; 1; −훼−6 )

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

Symmetric 3-monopoles

▶ 풞 :

휂3 + 휒(휁6 + b휁3 − 1) = 0, b ∈ ℝ

▶

(훼, 휌2훽, 휌훼, 훽, 휌2훼, 휌훽), 훼 =

3

√ −b + √ b2 + 4 2 > 0, 훼훽 = −1

▶ Aut(풞) → C3 × S3 (휁, 휂) → (휌휁, 휂), (휁, 휂) → (−1/휁, −휂/휁2)

ℐ1(훼) =

훼

∫ dz w = −2휋 √ 3훼 9

2F1

(1 3, 1 3; 1; −훼6 ) 풥1(훼) =

훽

∫ dz w = 2휋 √ 3 9훼

2F1

(1 3, 1 3; 1; −훼−6 )

▶

x1 = −(2풥1 + ℐ1)휌 − 2ℐ1 − 풥1, x2 = 휌 x1, x3 = 휌2x1, x4 = 3(풥1 − ℐ1)휌 + 3풥1,

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 − 2n) < 0 we obtain a solution to the Ercolani-Sinha constraints for our curve with n = (n, m − n, −m, 2n − m), m = (m, −n, n − m, −3n) as

• follows. First we solve for t, where

2n − m m + n =

2F1( 1 3, 2 3; 1, t) 2F1( 1 3, 2 3; 1, 1 − t).

Then b = 1 − 2t √ t(1 − t) , t = −b + √ b2 + 4 2 √ b2 + 4 , and with 훼6 = t/(1 − t) we obtain 휒 from 휒

1 3 = −(n1 + m1) 2휋

3 √ 3 훼 (1 + 훼6)

1 3

2F1(1

3, 2 3; 1, t).

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

Ramanujan (1914)

4 휋 =

∞

∑

m=0

(1 + 6m)( 1

2)m( 1 2)m( 1 2)m

(m!)34m ; 27 4휋 =

∞

∑

m=0

(2 + 15m)( 1

2)m( 1 3)m( 2 3)m

(m!)3 ( 27

2

)m 15 √ 3 2휋 =

∞

∑

m=0

(44 + 33m)( 1

2)m( 1 3)m( 2 3)m

(m!)3 ( 125

4

)m

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

Ramanujan (1914)

4 휋 =

∞

∑

m=0

(1 + 6m)( 1

2)m( 1 2)m( 1 2)m

(m!)34m ; 27 4휋 =

∞

∑

m=0

(2 + 15m)( 1

2)m( 1 3)m( 2 3)m

(m!)3 ( 27

2

)m 15 √ 3 2휋 =

∞

∑

m=0

(44 + 33m)( 1

2)m( 1 3)m( 2 3)m

(m!)3 ( 125

4

)m 휏 = i K ′ K , K(k) = 휋 2 2F1(1 2, 1 2; 1; k2), K ′(k) = K(k′), k′2 = 1−k2 If k1 = 1 − k′ 1 + k′ then 휏1 = 2휏. Modular equation of degree n: n

2F1( 1 2, 1 2; 1; 1 − 훼) 2F1( 1 2, 1 2; 1; 훼)

=

2F1( 1 2, 1 2; 1; 1 − 훽) 2F1( 1 2, 1 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)

n

2F1( 1 r , r−1 r ; 1; 1 − 훼) 2F1( 1 r , r−1 r ; 1; 훼)

=

2F1( 1 r , r−1 r ; 1; 1 − 훽) 2F1( 1 r , r−1 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)

n

2F1( 1 r , r−1 r ; 1; 1 − 훼) 2F1( 1 r , r−1 r ; 1; 훼)

=

2F1( 1 r , r−1 r ; 1; 1 − 훽) 2F1( 1 r , r−1 r ; 1; 훽)

n = 2, r = 3 = ⇒ (훼훽)1/3 + ((1 − 훼)(1 − 훽))1/3 = 1 훼 = 1/2 = ⇒ 훽1/3 + (1 − 훽)1/3 = 21/3 = ⇒ 훽 = 1 2 + 5 √ 3 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

Modular equation of degree n and signature r (r = 2, 3, 4, 6)

n

2F1( 1 r , r−1 r ; 1; 1 − 훼) 2F1( 1 r , r−1 r ; 1; 훼)

=

2F1( 1 r , r−1 r ; 1; 1 − 훽) 2F1( 1 r , r−1 r ; 1; 훽)

n = 2, r = 3 = ⇒ (훼훽)1/3 + ((1 − 훼)(1 − 훽))1/3 = 1 훼 = 1/2 = ⇒ 훽1/3 + (1 − 훽)1/3 = 21/3 = ⇒ 훽 = 1 2 + 5 √ 3 18 n m

2n−m m+n

t b 2 1 1

1 2

1 2

1 2 + 5 √ 3 18

−5 √ 2 1 1

1 2 1 2 − 5 √ 3 18

5 √ 2 4 −1 3 (63 + 171

3

√ 2 − 18

3

√ 4)/250

1 3(44 + 38

3

√ 2 + 26

3

√ 4) 5 −2 4

1 2 + 153 √ 3−99 √ 2 250

9 √ 458 + 187 √ 6

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

Symmetric 3-monopoles

The H3 constraint

Flows and Theta Divisor: 휃(sU + C) ∕= 0 C = KQ + 휙Q ( (n − 2)

n

∑

k=1

∞k ) , Q a branchpoint C = KQ

1,000 x 0.5 2,000 −2,000 3,000 0.0 −1,000 1.0 2.0 1.5 −3,000 1,000 −500 −1,000 1.5 0.5 500 2.0 1.0 0.0 1,500 x

Conjecture: No solutions to H3 apart from (n, m) ∈ {(1, 0), (1, 1)}

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

Symmetric 3-monopoles

The H3 constraint and Symmetry

풞 = (x, y)

• 휋∗

풞∗ ℰ+ = (z+, w+)

• ℰ− = (z−, w−)

휋− 휋+

• 휋1

ℰ1 = (z1, w1)

• 휋2

ℰ2 = (z2, w2)

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

Symmetric 3-monopoles

The H3 constraint and Symmetry

풞 = (x, y)

• 휋∗

풞∗ ℰ+ = (z+, w+)

• ℰ− = (z−, w−)

휋− 휋+

• 휋1

ℰ1 = (z1, w1)

• 휋2

ℰ2 = (z2, w2)

▶ 풞 covers the g풞∗ = 2 hyperelliptic curve

풞∗ = {(휇, 휈)∣휈2 = (휇3 + b)2 + 4}

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

Symmetric 3-monopoles

The H3 constraint and Symmetry

풞 = (x, y)

• 휋∗

풞∗ ℰ+ = (z+, w+)

• ℰ− = (z−, w−)

휋− 휋+

• 휋1

ℰ1 = (z1, w1)

• 휋2

ℰ2 = (z2, w2)

▶ 풞 covers the g풞∗ = 2 hyperelliptic curve

풞∗ = {(휇, 휈)∣휈2 = (휇3 + b)2 + 4}

▶ 풞∗ covers two-sheetedly elliptic curves ℰ± with Jacobi moduli

k±

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

Symmetric 3-monopoles

The H3 constraint and Symmetry

풞 = (x, y)

• 휋∗

풞∗ ℰ+ = (z+, w+)

• ℰ− = (z−, w−)

휋− 휋+

• 휋1

ℰ1 = (z1, w1)

• 휋2

ℰ2 = (z2, w2)

▶ 풞 covers the g풞∗ = 2 hyperelliptic curve

풞∗ = {(휇, 휈)∣휈2 = (휇3 + b)2 + 4}

▶ 풞∗ covers two-sheetedly elliptic curves ℰ± with Jacobi moduli

k±

▶ Parameterize (2횤 − b)

1 3

(b2 + 4)

1 6

= 1 + 2휌 + p 1 + 2휌 − p with p = 3휗2

3(0∣3휏)

휗2

3(0∣휏)

k+ = 휗2

2(0∣휏)

휗2

3(0∣휏),

k− = 휗2

2(0∣3휏)

휗2

3(0∣3휏)

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

The Symmetric Monopole

Rewriting H2

휂3 + 휒(휁6 + b휁3 − 1) = 0, b ∈ ℝ satisfies H1 and H2 ⇔ ∃ n, m (n, m) = 1, (m + n)(m − 2n) < 0 b = b(m, n) = − √ 3(p(m, n)6 − 45p(m, n)4 + 135p(m, n)2 − 27) 9p(m, n)(p(m, n)4 − 10p(m, n)2 + 9) p(m, n) = 3휗2

3

( 0∣ 풯 (m,n)

2

) 휗2

3

( 0∣ 풯 (m,n)

6

) , 풯 (m, n) = 2횤 √ 3 n + m 2n − m Expression for 휒 = 휒(m, n) can be given. What about H3? Reduces to questions of 휃 for 풞∗.

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

The Humbert Variety

휏 the period matrix of a genus 2 curve 풞∗.

▶ 휏 ∈ ℋΔ if there exist qi ∈ ℤ

q1 + q2휏11 + q3휏12 + q4휏22 + q5(휏 2

12 − 휏11휏22) = 0

q2

3 − 4(q1q5 + q2q4) = Δ

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

The Humbert Variety

휏 the period matrix of a genus 2 curve 풞∗.

▶ 휏 ∈ ℋΔ if there exist qi ∈ ℤ

q1 + q2휏11 + q3휏12 + q4휏22 + q5(휏 2

12 − 휏11휏22) = 0

q2

3 − 4(q1q5 + q2q4) = Δ ▶ 풞∗ covers elliptic curves ℰ± ⇔ Δ = h2 ≥ 1, h ∈ ℕ.

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

The Humbert Variety

휏 the period matrix of a genus 2 curve 풞∗.

▶ 휏 ∈ ℋΔ if there exist qi ∈ ℤ

q1 + q2휏11 + q3휏12 + q4휏22 + q5(휏 2

12 − 휏11휏22) = 0

q2

3 − 4(q1q5 + q2q4) = Δ ▶ 풞∗ covers elliptic curves ℰ± ⇔ Δ = h2 ≥ 1, h ∈ ℕ. ▶ Bierman-Humbert: 휏 ∈ ℋh2 ⇒ ∃ 픖 ∈ Sp(4, ℤ), such that

픖 ∘ 휏 = ˜ 휏 = ( ˜ 휏11

1 h 1 h

˜ 휏22 )

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

The Humbert Variety

휏 the period matrix of a genus 2 curve 풞∗.

▶ 휏 ∈ ℋΔ if there exist qi ∈ ℤ

q1 + q2휏11 + q3휏12 + q4휏22 + q5(휏 2

12 − 휏11휏22) = 0

q2

3 − 4(q1q5 + q2q4) = Δ ▶ 풞∗ covers elliptic curves ℰ± ⇔ Δ = h2 ≥ 1, h ∈ ℕ. ▶ Bierman-Humbert: 휏 ∈ ℋh2 ⇒ ∃ 픖 ∈ Sp(4, ℤ), such that

픖 ∘ 휏 = ˜ 휏 = ( ˜ 휏11

1 h 1 h

˜ 휏22 )

▶ 휃(z1, z2 ∣ ˜

휏) =

h−1

∑

k=0

휗3 ( z1 + k h ∣ ˜ 휏1,1 ) 휃 [ k

h

] ( hz2 ∣ h2˜ 휏2,2 )

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

The Humbert Variety

휏 the period matrix of a genus 2 curve 풞∗.

▶ 휏 ∈ ℋΔ if there exist qi ∈ ℤ

q1 + q2휏11 + q3휏12 + q4휏22 + q5(휏 2

12 − 휏11휏22) = 0

q2

3 − 4(q1q5 + q2q4) = Δ ▶ 풞∗ covers elliptic curves ℰ± ⇔ Δ = h2 ≥ 1, h ∈ ℕ. ▶ Bierman-Humbert: 휏 ∈ ℋh2 ⇒ ∃ 픖 ∈ Sp(4, ℤ), such that

픖 ∘ 휏 = ˜ 휏 = ( ˜ 휏11

1 h 1 h

˜ 휏22 )

▶ 휃(z1, z2 ∣ ˜

휏) =

h−1

∑

k=0

휗3 ( z1 + k h ∣ ˜ 휏1,1 ) 휃 [ k

h

] ( hz2 ∣ h2˜ 휏2,2 )

▶ 휃(z1, z2 ∣ ˜

휏) = 휗3 (z1 ∣ ˜ 휏11) 휗3 (2z2 ∣ 4˜ 휏22) + 휗3 (z1 + 1/2 ∣ ˜ 휏11) 휗2 (2z2 ∣ 4˜ 휏22)

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

The Symmetric Monopole

H3 and an elliptic function conjecture

▶ Bierman-Humbert+Weierstrass-Poinc´

are+Martens 휃(휆U − ˜ K; 휏monopole) = 0 for 휆 ∈ [0, 2] ⇔ at least one of the functions (k = −1, 0, 1 mod 3) hk(y) := 휗3 휗2 ( i √ 3 y + k 풯 3

• 풯

) + (−1)k 휗2 휗3 ( y + k 3 ∣ 풯 3 ) also vanishes. y := y(휆) = 휆 (n + m)휌/3, 풯 = 2i √ 3 n + m 2n − m 휌 = exp(2휋횤/3)

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

The Symmetric Monopole

H3 and an elliptic function conjecture

▶ Bierman-Humbert+Weierstrass-Poinc´

are+Martens 휃(휆U − ˜ K; 휏monopole) = 0 for 휆 ∈ [0, 2] ⇔ at least one of the functions (k = −1, 0, 1 mod 3) hk(y) := 휗3 휗2 ( i √ 3 y + k 풯 3

• 풯

) + (−1)k 휗2 휗3 ( y + k 3 ∣ 풯 3 ) also vanishes. y := y(휆) = 휆 (n + m)휌/3, 풯 = 2i √ 3 n + m 2n − m 휌 = exp(2휋횤/3)

▶ Conjecture h−1(y)h0(y)h1(y) vanishes 2(∣n∣ − 1) times on the

interval 휆 ∈ (0, 2)

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

The Symmetric Monopole

H3 and an elliptic function conjecture

▶ Bierman-Humbert+Weierstrass-Poinc´

are+Martens 휃(휆U − ˜ K; 휏monopole) = 0 for 휆 ∈ [0, 2] ⇔ at least one of the functions (k = −1, 0, 1 mod 3) hk(y) := 휗3 휗2 ( i √ 3 y + k 풯 3

• 풯

) + (−1)k 휗2 휗3 ( y + k 3 ∣ 풯 3 ) also vanishes. y := y(휆) = 휆 (n + m)휌/3, 풯 = 2i √ 3 n + m 2n − m 휌 = exp(2휋횤/3)

▶ Conjecture h−1(y)h0(y)h1(y) vanishes 2(∣n∣ − 1) times on the

interval 휆 ∈ (0, 2)

▶ Monopole requires 휗3

( 휏

3 ∣ 휏

) 휗2 ( 휏

3 ∣ 휏

) = 휗2 ( 1

3 ∣ 휏 3

) 휗3 ( 1

3 ∣ 휏 3

) An identity (new?)

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

The Symmetric Monopole

H3 and an elliptic function conjecture

▶ Bierman-Humbert+Weierstrass-Poinc´

are+Martens 휃(휆U − ˜ K; 휏monopole) = 0 for 휆 ∈ [0, 2] ⇔ at least one of the functions (k = −1, 0, 1 mod 3) hk(y) := 휗3 휗2 ( i √ 3 y + k 풯 3

• 풯

) + (−1)k 휗2 휗3 ( y + k 3 ∣ 풯 3 ) also vanishes. y := y(휆) = 휆 (n + m)휌/3, 풯 = 2i √ 3 n + m 2n − m 휌 = exp(2휋횤/3)

▶ Conjecture h−1(y)h0(y)h1(y) vanishes 2(∣n∣ − 1) times on the

interval 휆 ∈ (0, 2)

▶ Monopole requires 휗3

( 휏

3 ∣ 휏

) 휗2 ( 휏

3 ∣ 휏

) = 휗2 ( 1

3 ∣ 휏 3

) 휗3 ( 1

3 ∣ 휏 3

) An identity (new?)

▶ Numerically hk(y) = 0 ⇔ hk(휌y) = 0 ⇔ hk(휌2y) = 0 True?

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

Summary: The Tetrahedral Monopole

▶ Theorem Only (n, m) = (1, 1) and (1, 0) have no zeros within

the range. Conjecture: unsolved

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

Summary: The Tetrahedral Monopole

▶ Theorem Only (n, m) = (1, 1) and (1, 0) have no zeros within

the range. Conjecture: unsolved

▶ Theorem The only curves 휂3 + 휒(휁6 + b휁3 − 1) = 0 that

yield BPS monopoles 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. The monopole is expressible in terms of elliptic functions; this was studied by Hitchin, Manton, Murray.

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

Summary: The Tetrahedral Monopole

▶ Theorem Only (n, m) = (1, 1) and (1, 0) have no zeros within

the range. Conjecture: unsolved

▶ Theorem The only curves 휂3 + 휒(휁6 + b휁3 − 1) = 0 that

yield BPS monopoles 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. The monopole is expressible in terms of elliptic functions; this was studied by Hitchin, Manton, Murray.

▶ We know the Ercolani-Sinha vector for these curves.

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

Summary: The Tetrahedral Monopole

▶ Theorem Only (n, m) = (1, 1) and (1, 0) have no zeros within

the range. Conjecture: unsolved

▶ Theorem The only curves 휂3 + 휒(휁6 + b휁3 − 1) = 0 that

yield BPS monopoles 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. The monopole is expressible in terms of elliptic functions; this was studied by Hitchin, Manton, Murray.

▶ We know the Ercolani-Sinha vector for these curves. ▶ When does sU + C ∕∈ Θ?

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