martin speight- a curious limit of the faddeev-skyrme model
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8/3/2019 Martin Speight- A curious limit of the Faddeev-Skyrme model
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A curious limit of the Faddeev-Skyrme model
Martin Speight
University of Leeds
May 15, 2010
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The problem
ϕ : M → N , (M ,g ) Riemannian, (N ,ω) symplectic
E (ϕ) =1
2ϕ∗ω2 =
1
2
Z M ϕ∗ω∧ ∗ϕ∗ω.
Existence of critical points? Stability? (Index of Hessian)cf Harmonic map problem: (N ,h ) Riemannian
E D (ϕ) =1
2dϕ2
Joint work with Martin Svensson (Odense). Also systematically
studied by Slobodeanu and some special cases by De Carli and
Ferreira (M Lorentzian)
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Motivation: Faddeev-Skyrme model
M =R3, N = S 2 ⊂ R
3, V : N → [0,∞)
E FS =Z
M
1
2∑
i
∂ϕ∂x i
2 +α2∑i < j
ϕ ·∂ϕ∂x i
×∂ϕ∂x j
2
+ V (ϕ)
=1
2dϕ2 +
α
2ϕ∗ω2 +
Z M
V ϕ
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Motivation: Faddeev-Skyrme model
M =R3, N = S 2 ⊂ R
3, V : N → [0,∞)
E FS =Z
M
1
2∑
i
∂ϕ∂x i
2 +α2∑i < j
ϕ ·∂ϕ∂x i
× ∂ϕ∂x j
2
+ V (ϕ)
=1
2dϕ2 +
α
2ϕ∗ω2 +
Z M
V ϕ
α> 0 parameter. M = R3, usually take V = 0. Then can set
α = 1 WLOG. Interesting (knot solitons) but heavily numerical.
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Motivation: Faddeev-Skyrme model
M =R3, N = S 2 ⊂ R
3, V : N → [0,∞)
E FS =Z
M
1
2∑
i
∂ϕ∂x i
2 + α2∑i < j
ϕ ·∂ϕ∂x i
× ∂ϕ∂x j
2
+ V (ϕ)
=1
2dϕ2 +
α
2ϕ∗ω2 +
Z M
V ϕ
α> 0 parameter. M = R3, usually take V = 0. Then can set
α = 1 WLOG. Interesting (knot solitons) but heavily numerical.
Other M (e.g. M = S 3) α free parameter.
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Motivation: Faddeev-Skyrme model
M =R3, N = S 2 ⊂ R
3, V : N → [0,∞)
E FS =Z
M
1
2∑
i
∂ϕ∂x i
2 + α2∑i < j
ϕ ·∂ϕ∂x i
× ∂ϕ∂x j
2
+ V (ϕ)
=1
2dϕ2 +
α
2ϕ∗ω2 +
Z M
V ϕ
α> 0 parameter. M = R3, usually take V = 0. Then can set
α = 1 WLOG. Interesting (knot solitons) but heavily numerical.
Other M (e.g. M = S 3) α free parameter.
Extreme limits: α → 0, harmonic map problemα → ∞, our problem
M i i F dd Sk d l
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Motivation: Faddeev-Skyrme model
M =R3, N = S 2 ⊂ R
3, V : N → [0,∞)
E FS =Z
M
1
2∑
i
∂ϕ∂x i
2 + α2∑i < j
ϕ ·∂ϕ∂x i
× ∂ϕ∂x j
2
+ V (ϕ)
=1
2dϕ2 +
α
2ϕ∗ω2 +
Z M
V ϕ
α> 0 parameter. M = R3, usually take V = 0. Then can set
α = 1 WLOG. Interesting (knot solitons) but heavily numerical.
Other M (e.g. M = S 3) α free parameter.
Extreme limits: α → 0, harmonic map problemα → ∞, our problem
Conformally invariant if dim M = 4, infinite dimensional symmetry
group (symplectic diffeos of (N ,ω))
Minimizer in generator of π4(S
2
)? (”Pure FS instanton”)
Fi d d i i
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First and second variation
Smooth variation ϕt of ϕ0 = ϕ : M → N .
Fi t d d i ti
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First and second variation
Smooth variation ϕt of ϕ0 = ϕ : M → N .
X =∂ϕ
∂t
t =0 ∈ Γ(ϕ−1
TN )
dE (ϕt )
dt
t =0
=Z
M ω(X ,dϕ(δϕ∗ω))
Fi t d d i ti
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First and second variation
Smooth variation ϕt of ϕ0 = ϕ : M → N .
X =∂ϕ
∂t
t =0 ∈ Γ(ϕ−1
TN )
dE (ϕt )
dt
t =0
=Z
M ω(X ,dϕ(δϕ∗ω))
ϕ critical ⇔ dϕδϕ∗ω = 0
First and second variation
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First and second variation
Smooth variation ϕt of ϕ0 = ϕ : M → N .
X =∂ϕ
∂t
t =0 ∈ Γ(ϕ−1
TN )
dE (ϕt )
dt
t =0
=Z
M ω(X ,dϕ(δϕ∗ω))
ϕ critical ⇔ dϕδϕ∗ω = 0
If ϕ is critical and ω = h (J ·, ·) = Kahler form on N
d 2E (ϕt )
dt 2
t =0
= X ,L ϕX L2
where
L ϕX = −J (∇ϕδϕ∗ωX +dϕ(δdϕ∗ιX ω))
(Ins)stability: spectrum of L ϕ
Coclosed maps
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Coclosed maps
dϕδ(ϕ∗ω) = 0
ϕ : M → N is coclosed if δϕ∗ω = 0
Clearly coclosed ⇒ critical
Coclosed maps
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Coclosed maps
dϕδ(ϕ∗ω) = 0
ϕ : M → N is coclosed if δϕ∗ω = 0
Clearly coclosed ⇒ critical
Prop: If ϕ is coclosed it minimizes E in its homotopy class.
Coclosed maps
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Coclosed maps
dϕδ(ϕ∗ω) = 0
ϕ : M → N is coclosed if δϕ∗ω = 0
Clearly coclosed ⇒ critical
Prop: If ϕ is coclosed it minimizes E in its homotopy class.
Proof: Let ϕ be coclosed and define functional
F (ψ ) = ϕ∗ω,ψ ∗ω.
Then F is a homotopy invariant of ψ (homotopy lemma). Hence,
if ψ ∼ ϕ,
E (ϕ) = 12
F (ϕ) = 12
F (ψ ) ≤ 12
ϕ∗ωψ ∗ω.
Hence ϕ∗ω ≤ ψ ∗ω.
Coclosed maps
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Coclosed maps
dϕδ(ϕ∗ω) = 0
ϕ : M → N is coclosed if δϕ∗ω = 0
Clearly coclosed ⇒ critical
Prop: If ϕ is coclosed it minimizes E in its homotopy class.
Proof: Let ϕ be coclosed and define functional
F (ψ ) = ϕ∗ω,ψ ∗ω.
Then F is a homotopy invariant of ψ (homotopy lemma). Hence,
if ψ ∼ ϕ,
E (ϕ) = 12
F (ϕ) = 12
F (ψ ) ≤ 12
ϕ∗ωψ ∗ω.
Hence ϕ∗ω ≤ ψ ∗ω.
E.g. Id : N → N , p 1 : N × P → N are global minimimizers
Coclosed maps
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Coclosed maps
dϕδ(ϕ∗ω) = 0
ϕ : M → N is coclosed if δϕ∗ω = 0
Clearly coclosed ⇒ critical
Prop: If ϕ is coclosed it minimizes E in its homotopy class.
Proof: Let ϕ be coclosed and define functional
F (ψ ) = ϕ∗ω,ψ ∗ω.
Then F is a homotopy invariant of ψ (homotopy lemma). Hence,
if ψ ∼ ϕ,
E (ϕ) = 12
F (ϕ) = 12
F (ψ ) ≤ 12
ϕ∗ωψ ∗ω.
Hence ϕ∗ω ≤ ψ ∗ω.
E.g. Id : N → N , p 1 : N × P → N are global minimimizers
Cor: If ϕ is critical and a.e. immersive, then ϕ minimizes E in its
htpy class
Two dimensional domains
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Two dimensional domains
Let M = Σ, a two-dimensional compact oriented Riemannian
mfd, N = symplectic mfd
Two dimensional domains
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Two dimensional domains
Let M = Σ, a two-dimensional compact oriented Riemannian
mfd, N = symplectic mfd
Thm: ϕ : Σ → N is critical ⇔ ϕ is coclosed
Two dimensional domains
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Two dimensional domains
Let M = Σ, a two-dimensional compact oriented Riemannian
mfd, N = symplectic mfd
Thm: ϕ : Σ → N is critical ⇔ ϕ is coclosed
Proof: ϕ∗ω = f ∗ 1 for some f ∈ C ∞(Σ). Then δϕ∗ω = ∗df , so
δϕ∗ω = J ∇f . ϕ critical ⇒
dϕ(J ∇f ) = 0 ⇒ ϕ∗ω(J ∇f ,∇f ) = 0 ⇒ −f |∇f |2 = 0.
Hence f = constant.
Two dimensional domains
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Two dimensional domains
Let M = Σ, a two-dimensional compact oriented Riemannian
mfd, N = symplectic mfd
Thm: ϕ : Σ → N is critical ⇔ ϕ is coclosed
Proof: ϕ∗ω = f ∗ 1 for some f ∈ C ∞(Σ). Then δϕ∗ω = ∗df , so
δϕ∗ω = J ∇f . ϕ critical ⇒
dϕ(J ∇f ) = 0 ⇒ ϕ∗ω(J ∇f ,∇f ) = 0 ⇒ −f |∇f |2 = 0.
Hence f = constant.
Hence, on Σ, only critical points are global minimizers
Two dimensional domains
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o d e s o a do a s
Let M = Σ, a two-dimensional compact oriented Riemannian
mfd, N = symplectic mfd
Thm: ϕ : Σ → N is critical ⇔ ϕ is coclosed
Proof: ϕ∗ω = f ∗ 1 for some f ∈ C ∞(Σ). Then δϕ∗ω = ∗df , so
δϕ∗ω = J ∇f . ϕ critical ⇒
dϕ(J ∇f ) = 0 ⇒ ϕ∗ω(J ∇f ,∇f ) = 0 ⇒ −f |∇f |2 = 0.
Hence f = constant.
Hence, on Σ, only critical points are global minimizers
Cor: Let ϕ : Σ → S 2 be critical. Then E (ϕ) = 0 or Σ = S 2 and
ϕ = Id up to symmetry.
Two dimensional domains
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Let M = Σ, a two-dimensional compact oriented Riemannian
mfd, N = symplectic mfd
Thm: ϕ : Σ → N is critical ⇔ ϕ is coclosed
Proof: ϕ∗ω = f ∗ 1 for some f ∈ C ∞(Σ). Then δϕ∗ω = ∗df , so
δϕ∗ω = J ∇f . ϕ critical ⇒
dϕ(J ∇f ) = 0 ⇒ ϕ∗ω(J ∇f ,∇f ) = 0 ⇒ −f |∇f |2 = 0.
Hence f = constant.
Hence, on Σ, only critical points are global minimizers
Cor: Let ϕ : Σ → S 2 be critical. Then E (ϕ) = 0 or Σ = S 2 and
ϕ = Id up to symmetry.Proof: f is constant. If f = 0, E (ϕ) = 0. Otherwise, ϕ has no
critical points, so is a covering map. But π1(S 2) = 0, so Σ = S 2
and ϕ is an (area preserving) diffeomorphism.
Two dimensional domains
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Let M = Σ, a two-dimensional compact oriented Riemannian
mfd, N = symplectic mfd
Thm: ϕ : Σ → N is critical ⇔ ϕ is coclosed
Proof: ϕ∗ω = f ∗ 1 for some f ∈ C ∞(Σ). Then δϕ∗ω = ∗df , so
δϕ∗ω = J ∇f . ϕ critical ⇒
dϕ(J ∇f ) = 0 ⇒ ϕ∗ω(J ∇f ,∇f ) = 0 ⇒ −f |∇f |2 = 0.
Hence f = constant.
Hence, on Σ, only critical points are global minimizers
Cor: Let ϕ : Σ → S 2 be critical. Then E (ϕ) = 0 or Σ = S 2 and
ϕ = Id up to symmetry.Proof: f is constant. If f = 0, E (ϕ) = 0. Otherwise, ϕ has no
critical points, so is a covering map. But π1(S 2) = 0, so Σ = S 2
and ϕ is an (area preserving) diffeomorphism.
Physically, rather surprising
Critical submersions
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Submersion ϕ : M → N , T x M = V x ⊕H x = kerdϕx ⊕ (kerdϕx )⊥
Critical submersions
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Submersion ϕ : M → N , T x M = V x ⊕H x = kerdϕx ⊕ (kerdϕx )⊥
F-structure F ∈ Γ(End (TM )) s.t. F 3 +F = 0,
F X =
0 X ∈ V x
JX X ∈H x
where we’ve used isomorphism dϕx :H x → T ϕ(x )N
Critical submersions
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Submersion ϕ : M → N , T x M = V x ⊕H x = kerdϕx ⊕ (kerdϕx )⊥
F-structure F ∈ Γ(End (TM )) s.t. F 3 +F = 0,
F X =
0 X ∈ V x
JX X ∈H x
where we’ve used isomorphism dϕx :H x → T ϕ(x )N
If ϕ is Riemannian (meaning dϕx :H x → T ϕ(x )N is an isometry)
then
δϕ∗ω = −divF
Critical submersions
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Submersion ϕ : M → N , T x M = V x ⊕H x = kerdϕx ⊕ (kerdϕx )⊥
F-structure F ∈ Γ(End (TM )) s.t. F 3 +F = 0,
F X =
0 X ∈ V x
JX X ∈H x
where we’ve used isomorphism dϕx :H x → T ϕ(x )N
If ϕ is Riemannian (meaning dϕx :H x → T ϕ(x )N is an isometry)
then
δϕ∗ω = −divF
So a Riemannian submersion (M ,g ) → (N ,h ,J ) is critical iff
divF is vertical
Critical submersions
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Submersion ϕ : M → N , T x M = V x ⊕H x = kerdϕx ⊕ (kerdϕx )⊥
F-structure F ∈ Γ(End (TM )) s.t. F 3 +F = 0,
F X =
0 X ∈ V x
JX X ∈H x
where we’ve used isomorphism dϕx :H x → T ϕ(x )N
If ϕ is Riemannian (meaning dϕx :H x → T ϕ(x )N is an isometry)
then
δϕ∗ω = −divF
So a Riemannian submersion (M ,g ) → (N ,h ,J ) is critical iff
divF is vertical
(In fact, iff harmonic, hence iff a harmonic morphism)
Critical submersions
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Example: H < K < G , G compact simple Lie group, G /K
compact Hermitian symmetric space
Critical submersions
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Example: H < K < G , G compact simple Lie group, G /K
compact Hermitian symmetric space
ϕ : G /H → G /K has T x M ≡ q = h⊥, T ϕ(x )N ≡ p = k⊥
H x ≡ p, V x ≡ q∩ k
Critical submersions
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Example: H < K < G , G compact simple Lie group, G /K
compact Hermitian symmetric space
ϕ : G /H → G /K has T x M ≡ q = h⊥, T ϕ(x )N ≡ p = k⊥
H x ≡ p, V x ≡ q∩ k
F = adJ for some J ∈ z(k)
Critical submersions
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Example: H < K < G , G compact simple Lie group, G /K
compact Hermitian symmetric space
ϕ : G /H → G /K has T x M ≡ q = h⊥, T ϕ(x )N ≡ p = k⊥
H x ≡ p, V x ≡ q∩ k
F = adJ for some J ∈ z(k)
Y ,divF
= 1
2 ∑i [X
i ,Y ], [J ,X
i ]
Critical submersions
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Example: H < K < G , G compact simple Lie group, G /K
compact Hermitian symmetric space
ϕ : G /H → G /K has T x M ≡ q = h⊥, T ϕ(x )N ≡ p = k⊥
H x ≡ p, V x ≡ q∩ k
F = adJ for some J ∈ z(k)Y ,divF = 1
2 ∑i [X
i ,Y ], [J ,X
i ]
Hence divF vertical
Critical submersions
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Example: H < K < G , G compact simple Lie group, G /K
compact Hermitian symmetric space
ϕ : G /H → G /K has T x M ≡ q = h⊥, T ϕ(x )N ≡ p = k⊥
H x ≡ p, V x ≡ q∩ k
F = adJ for some J ∈ z(k)Y ,divF = 1
2 ∑i [X
i ,Y ], [J ,X
i ]
Hence divF vertical
Example: G = SU (n ), K = S (U (k ) × U (n − k )), H = S (U (1)n )Flag(Cn ) → Gr k (C
n ) has divF = 0
Coclosed, hence global minimizer
Critical submersions
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Example: H < K < G , G compact simple Lie group, G /K
compact Hermitian symmetric space
ϕ : G /H → G /K has T x M ≡ q = h⊥, T ϕ(x )N ≡ p = k⊥
H x ≡ p, V x ≡ q∩ k
F = adJ for some J ∈ z(k)Y ,divF = 1
2∑
i [X i ,Y ], [J ,X i ]
Hence divF vertical
Example: G = SU (n ), K = S (U (k ) × U (n − k )), H = S (U (1)n )Flag(Cn ) → Gr k (C
n ) has divF = 0
Coclosed, hence global minimizer
Example: G = SU (2), K = S (U (1) × U (1)), H = e
Hopf : S 3 → S 2 has divF = 0 (of course)
But it’s still a global minimizer!
Energy bound M 3 → S 2
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A map ϕ : M 3 → S 2 is algebraically inessential if [ϕ∗ω] = 0.
Such maps are classified up to homotopy by
Q (ϕ) = 116π2
Z M
A ∧dA where dA = ϕ∗ω
Energy bound M 3 → S 2
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A map ϕ : M 3 → S 2 is algebraically inessential if [ϕ∗ω] = 0.
Such maps are classified up to homotopy by
Q (ϕ) =1
16π2
Z M
A ∧dA where dA = ϕ∗ω
Thm: Let ϕ : M 3 → S 2 be algebraically inessential. Then
E (ϕ) ≥ 8π2 λ1Q (ϕ)
where λ1 > 0 is the lowest eigenvalue of the Laplacian coexact
one-forms on M .
Energy bound M 3 → S 2
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A map ϕ : M 3 → S 2 is algebraically inessential if [ϕ∗ω] = 0.
Such maps are classified up to homotopy by
Q (ϕ) =1
16π2
Z M
A ∧dA where dA = ϕ∗ω
Thm: Let ϕ : M 3 → S 2 be algebraically inessential. Then
E (ϕ) ≥ 8π2 λ1Q (ϕ)
where λ1 > 0 is the lowest eigenvalue of the Laplacian coexact
one-forms on M .
Proof: Choose A s.t. dA = ϕ∗ω. WLOG can assume A is
coeaxact (Hodge decomposition). Hence
E (ϕ) = 1
2dA,dA = 1
2A,∆A ≥ λ1
2A2.
Energy bound M 3 → S 2
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A map ϕ : M 3 → S 2 is algebraically inessential if [ϕ∗ω] = 0.
Such maps are classified up to homotopy by
Q (ϕ) =1
16π2
Z M
A ∧dA where dA = ϕ∗ω
Thm: Let ϕ : M 3 → S 2 be algebraically inessential. Then
E (ϕ) ≥ 8π2 λ1Q (ϕ)
where λ1 > 0 is the lowest eigenvalue of the Laplacian coexact
one-forms on M .
Proof: Choose A s.t. dA = ϕ∗ω. WLOG can assume A is
coeaxact (Hodge decomposition). Hence
E (ϕ) = 1
2dA,dA = 1
2A,∆A ≥ λ1
2A2.
But
Q (ϕ) =1
16π2
A,−∗dA ≤1
16π2
AdA ≤1
16π2
2
λ1
E (ϕ) 2E (ϕ).
Energy bound M 3 → S 2
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Fact: Hopf : S 3 → S 2 attains this bound, hence minimizes E in
its homotopy class.
Energy bound M 3 → S 2
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Fact: Hopf : S 3 → S 2 attains this bound, hence minimizes E in
its homotopy class.
Cor: Hopf : S 3 → S 2 is a stable critical point of the full FS energyE D +αE for all α ≥ 1. (Conjectured by Ward, proved
independently by Isobe.)
Baby Skyrmions ϕ : R2 → S 2
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E (ϕ) =1
2αdϕ2 +
1
2ϕ∗ω2 +
Z R2
V ϕ
ϕ(∞) = ϕ0 ∈ V −1(0), so fields labelled by integer charge
n = (4π)−1R ϕ∗ω
Baby Skyrmions ϕ : R2 → S 2
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E (ϕ) =1
2αdϕ2 +
1
2ϕ∗ω2 +
Z R2
V ϕ
ϕ(∞) = ϕ0 ∈ V −1(0), so fields labelled by integer charge
n = (4π)−1R ϕ∗ω
Limit α → 0: Again, has infinite dimensional symmetry group
E (ϕ p ) = E (ϕ)
for any area-preserving diffeo R2 → R2
Baby Skyrmions ϕ : R2 → S 2
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E (ϕ) =1
2αdϕ2 +
1
2ϕ∗ω2 +
Z R2
V ϕ
ϕ(∞) = ϕ0 ∈ V −1(0), so fields labelled by integer charge
n = (4π)−1R ϕ∗ω
Limit α → 0: Again, has infinite dimensional symmetry group
E (ϕ p ) = E (ϕ)
for any area-preserving diffeo R2 → R2
Assume V (ϕ) = 12
U (ϕ)2 where U : S 2 → [0,∞) is smooth.
Bogomol’nyi type bound:
0 ≤1
2ϕ∗ω− ∗U ϕ2 = E (ϕ) −
Z R2ϕ∗(U ω)
E (ϕ) ≥ U Z R2ϕ∗ω equality iff ϕ∗ω = ∗U ϕ
Baby Skyrmions ϕ : R2 → S 2
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Example U (ϕ) = 1 −ϕ3, for which U = 1. Bog eqn has charge
1 solutionϕG = (w (r ) cosθ,w (r ) sinθ,z (r )),
z (r ) = 1 − 2e −r 2/2, w (r ) =
1 − z (r )2
and “plastic” deformations
Baby Skyrmions ϕ : R2 → S 2
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Example U (ϕ) = 1 −ϕ3, for which U = 1. Bog eqn has charge
1 solutionϕG = (w (r ) cosθ,w (r ) sinθ,z (r )),
z (r ) = 1 − 2e −r 2/2, w (r ) =
1 − z (r )2
and “plastic” deformations
Higher n solns? Let X = p ∈R2 : rank dϕp < 2 = ϕ−1(0,0,1).
Then ϕ is an area-preserving surjection
(R2\X ,dx ∧ dy ) → (S 2
×,Ω)
where Ω = ω/U . Hence X has no bounded connected
component. Weird...
Baby Skyrmions ϕ : R2 → S 2
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...but possible! Area-preserving diffeo
p : (0,∞) ×R→R2 (x ,y ) → (log x ,xy )
Charge 1 solution of Bog eqn
ϕ(x ,y ) =
(ϕG p )(x ,y ) x > 0
(0,0,1) x ≤ 0
Baby Skyrmions ϕ : R2 → S 2
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...but possible! Area-preserving diffeo
p : (0,∞) ×R→R2 (x ,y ) → (log x ,xy )
Charge 1 solution of Bog eqn
ϕ(x ,y ) =
(ϕG p )(x ,y ) x > 0
(0,0,1) x ≤ 0
It’s C 2. Translate it to the right, compose it with area preserving
diffeo
q : R× (−π
2,π
2) → R
2 (x ,y ) → (x cos2 y , tan y )
and extend. Get C 2 charge 1 solution which is constant outside
strip (0,∞) × (−π2, π
2). Can patch n of these together, for any n .
Baby Skyrmions ϕ : R2 → S 2
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...but possible! Area-preserving diffeo
p : (0,∞) ×R→R2 (x ,y ) → (log x ,xy )
Charge 1 solution of Bog eqn
ϕ(x ,y ) =
(ϕG p )(x ,y ) x > 0
(0,0,1) x ≤ 0
It’s C 2. Translate it to the right, compose it with area preserving
diffeo
q : R× (−π
2,π
2) → R
2 (x ,y ) → (x cos2 y , tan y )
and extend. Get C 2 charge 1 solution which is constant outside
strip (0,∞) × (−π2, π
2). Can patch n of these together, for any n .
“Moduli space” of minimizers, even for n = 1, is very complicated.
Baby Skyrmions ϕ : R2 → S 2
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...but possible! Area-preserving diffeo
p : (0,∞) ×R→R2 (x ,y ) → (log x ,xy )
Charge 1 solution of Bog eqn
ϕ(x ,y ) =
(ϕG p )(x ,y ) x > 0
(0,0,1) x ≤ 0
It’s C 2. Translate it to the right, compose it with area preserving
diffeo
q : R× (−π
2,π
2) → R
2 (x ,y ) → (x cos2 y , tan y )
and extend. Get C 2 charge 1 solution which is constant outside
strip (0,∞) × (−π2, π
2). Can patch n of these together, for any n .
“Moduli space” of minimizers, even for n = 1, is very complicated.
Cf model on S
2
with V = 0: very different.
Concluding remarks
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Variational problem for E = 12
ϕ∗ω2 is mathematically
interesting, and leads to insight into the full FS model on compact
domains.
Case M = Σ2 finished
Critical point in generator of π4(S 2)? What about
π3(S 2)\−1,0,1?
Are there any unstable critical points?
Case M =R2 with potential is very strange. What about M =R3?