the geometry of soliton moduli spaces · 2009. 12. 14. · low energy dynamics reduces to geodesic...
TRANSCRIPT
![Page 1: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/1.jpg)
The Geometry of Soliton Moduli Spaces
Martin SpeightUniversity of Leeds, UK
December 14, 2009
![Page 2: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/2.jpg)
Topological solitons
Smooth, spatially localized, lump-like solutions of relativisticnonlinear wave equations
Stable for topological reasons
Like strings. . .hypothetical particles, resolve many theoretical puzzles in HEP
. . . only better!exist in real world: magnetic flux tubes in superconductors,magnetic bubbles, optical pulses, crystal dislocations
![Page 3: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/3.jpg)
Topological solitons
Smooth, spatially localized, lump-like solutions of relativisticnonlinear wave equations
Stable for topological reasons
Like strings. . .hypothetical particles, resolve many theoretical puzzles in HEP
. . . only better!exist in real world: magnetic flux tubes in superconductors,magnetic bubbles, optical pulses, crystal dislocations
![Page 4: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/4.jpg)
Topological solitons
Smooth, spatially localized, lump-like solutions of relativisticnonlinear wave equations
Stable for topological reasons
Like strings. . .hypothetical particles, resolve many theoretical puzzles in HEP
. . . only better!exist in real world: magnetic flux tubes in superconductors,magnetic bubbles, optical pulses, crystal dislocations
![Page 5: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/5.jpg)
Topological solitons
Smooth, spatially localized, lump-like solutions of relativisticnonlinear wave equations
Stable for topological reasons
Like strings. . .hypothetical particles, resolve many theoretical puzzles in HEP
. . . only better!exist in real world: magnetic flux tubes in superconductors,magnetic bubbles, optical pulses, crystal dislocations
![Page 6: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/6.jpg)
Soliton moduli spaces
Interesting special case: static solitons exert no net force on eachother
Moduli space of static n-soliton solutions Mn, dimMn = n dimM1
Low energy dynamics reduces to geodesic motion in Mn!
Soliton dynamics←→ Riemannian geometry
![Page 7: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/7.jpg)
Soliton moduli spaces
Interesting special case: static solitons exert no net force on eachother
Moduli space of static n-soliton solutions Mn, dimMn = n dimM1
Low energy dynamics reduces to geodesic motion in Mn!
Soliton dynamics←→ Riemannian geometry
![Page 8: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/8.jpg)
Soliton moduli spaces
Interesting special case: static solitons exert no net force on eachother
Moduli space of static n-soliton solutions Mn, dimMn = n dimM1
Low energy dynamics reduces to geodesic motion in Mn!
Soliton dynamics←→ Riemannian geometry
![Page 9: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/9.jpg)
Soliton moduli spaces
Interesting special case: static solitons exert no net force on eachother
Moduli space of static n-soliton solutions Mn, dimMn = n dimM1
Low energy dynamics reduces to geodesic motion in Mn!
Soliton dynamics←→ Riemannian geometry
![Page 10: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/10.jpg)
Plan
Planar antiferromagnets→ CP1 model
The Bogomol’nyi argument, Mn
The metric on Mn, soliton scattering
Other solitons
Open problems
![Page 11: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/11.jpg)
Antiferromagnets
Square spin lattice: S : Z×Z→ S2
Neighbouring spin like to anti-align
Lattice energy: H := ∑i,j
[2+Sij · (Si,j+1 +Si+1,j)]
Dynamics:dSij
dτ=−Sij ×
∂H∂Sij
First order, spin couples to nearest neighbours.
Continuum limit?
![Page 12: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/12.jpg)
Antiferromagnets
Square spin lattice: S : Z×Z→ S2
Neighbouring spin like to anti-align
Lattice energy: H := ∑i,j
[2+Sij · (Si,j+1 +Si+1,j)]
Dynamics:dSij
dτ=−Sij ×
∂H∂Sij
First order, spin couples to nearest neighbours.
Continuum limit?
![Page 13: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/13.jpg)
Antiferromagnets
Square spin lattice: S : Z×Z→ S2
Neighbouring spin like to anti-align
Lattice energy: H := ∑i,j
[2+Sij · (Si,j+1 +Si+1,j)]
Dynamics:dSij
dτ=−Sij ×
∂H∂Sij
First order, spin couples to nearest neighbours.
Continuum limit?
![Page 14: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/14.jpg)
Antiferromagnets
Square spin lattice: S : Z×Z→ S2
Neighbouring spin like to anti-align
Lattice energy: H := ∑i,j
[2+Sij · (Si,j+1 +Si+1,j)]
Dynamics:dSij
dτ=−Sij ×
∂H∂Sij
First order, spin couples to nearest neighbours.
Continuum limit?
![Page 15: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/15.jpg)
Antiferromagnets
Square spin lattice: S : Z×Z→ S2
Neighbouring spin like to anti-align
Lattice energy: H := ∑i,j
[2+Sij · (Si,j+1 +Si+1,j)]
Dynamics:dSij
dτ=−Sij ×
∂H∂Sij
First order, spin couples to nearest neighbours.
Continuum limit?
![Page 16: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/16.jpg)
Antiferromagnets
![Page 17: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/17.jpg)
Antiferromagnets
![Page 18: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/18.jpg)
Antiferromagnets
αβ
![Page 19: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/19.jpg)
Antiferromagnets
αβ
A
B
−2,1
−2,1
![Page 20: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/20.jpg)
Antiferromagnets
dAαβ
dτ= −(Bα,β−1 +Bαβ +Bα−1,β +Bα−1,β−1)
dBαβ
dτ= −(Aα+1,β +Aα+1,β+1 +Aα,β+1 +Aα,β)
δ
x = αδ, y = βδ, t = 2τδ
Assumption:
Aα,β
Bα,β
}δ→0−→
{A(x ,y)B(x ,y)
Replace Aα+1,β by A+δAx + 12 δ2Axx + · · · etc
Work to order δ2
![Page 21: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/21.jpg)
Antiferromagnets
dAαβ
dτ= −(Bα,β−1 +Bαβ +Bα−1,β +Bα−1,β−1)
dBαβ
dτ= −(Aα+1,β +Aα+1,β+1 +Aα,β+1 +Aα,β)
δ
x = αδ, y = βδ, t = 2τδ
Assumption:
Aα,β
Bα,β
}δ→0−→
{A(x ,y)B(x ,y)
Replace Aα+1,β by A+δAx + 12 δ2Axx + · · · etc
Work to order δ2
![Page 22: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/22.jpg)
Antiferromagnets
dAαβ
dτ= −(Bα,β−1 +Bαβ +Bα−1,β +Bα−1,β−1)
dBαβ
dτ= −(Aα+1,β +Aα+1,β+1 +Aα,β+1 +Aα,β)
δ
x = αδ, y = βδ, t = 2τδ
Assumption:
Aα,β
Bα,β
}δ→0−→
{A(x ,y)B(x ,y)
Replace Aα+1,β by A+δAx + 12 δ2Axx + · · · etc
Work to order δ2
![Page 23: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/23.jpg)
Antiferromagnets
2δAt = −A× [4B−2δ(Bx +By)+δ2(Bxx +Byy +Bxy)]
2δBt = −B× [4A+2δ(Ax +Ay)+δ2(Axx +Ayy +Axy)]
New fields: m =12(A+B) ϕ =
12(A−B)
|m|= O(δ) , |ϕ|= 1+O(δ2)
mt = −(∂x +∂y)[m×ϕ]+δ
4[2ϕ× (ϕxx +ϕyy +ϕxy) (1)
δϕt = 4m×ϕ−δϕ× (ϕx +ϕy)+O(δ2) (2)
Solve (2): m =δ
4
[ϕ×ϕt −ϕx −ϕy
]+O(δ2)
Subst in (1): ϕ×ϕtt = ϕ× (ϕxx +ϕyy)+O(δ)
![Page 24: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/24.jpg)
Antiferromagnets
2δAt = −A× [4B−2δ(Bx +By)+δ2(Bxx +Byy +Bxy)]
2δBt = −B× [4A+2δ(Ax +Ay)+δ2(Axx +Ayy +Axy)]
New fields: m =12(A+B) ϕ =
12(A−B)
|m|= O(δ) , |ϕ|= 1+O(δ2)
mt = −(∂x +∂y)[m×ϕ]+δ
4[2ϕ× (ϕxx +ϕyy +ϕxy) (1)
δϕt = 4m×ϕ−δϕ× (ϕx +ϕy)+O(δ2) (2)
Solve (2): m =δ
4
[ϕ×ϕt −ϕx −ϕy
]+O(δ2)
Subst in (1): ϕ×ϕtt = ϕ× (ϕxx +ϕyy)+O(δ)
![Page 25: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/25.jpg)
Antiferromagnets
2δAt = −A× [4B−2δ(Bx +By)+δ2(Bxx +Byy +Bxy)]
2δBt = −B× [4A+2δ(Ax +Ay)+δ2(Axx +Ayy +Axy)]
New fields: m =12(A+B) ϕ =
12(A−B)
|m|= O(δ) , |ϕ|= 1+O(δ2)
mt = −(∂x +∂y)[m×ϕ]+δ
4[2ϕ× (ϕxx +ϕyy +ϕxy) (1)
δϕt = 4m×ϕ−δϕ× (ϕx +ϕy)+O(δ2) (2)
Solve (2): m =δ
4
[ϕ×ϕt −ϕx −ϕy
]+O(δ2)
Subst in (1): ϕ×ϕtt = ϕ× (ϕxx +ϕyy)+O(δ)
![Page 26: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/26.jpg)
Antiferromagnets
2δAt = −A× [4B−2δ(Bx +By)+δ2(Bxx +Byy +Bxy)]
2δBt = −B× [4A+2δ(Ax +Ay)+δ2(Axx +Ayy +Axy)]
New fields: m =12(A+B) ϕ =
12(A−B)
|m|= O(δ) , |ϕ|= 1+O(δ2)
mt = −(∂x +∂y)[m×ϕ]+δ
4[2ϕ× (ϕxx +ϕyy +ϕxy) (1)
δϕt = 4m×ϕ−δϕ× (ϕx +ϕy)+O(δ2) (2)
Solve (2): m =δ
4
[ϕ×ϕt −ϕx −ϕy
]+O(δ2)
Subst in (1): ϕ×ϕtt = ϕ× (ϕxx +ϕyy)+O(δ)
![Page 27: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/27.jpg)
Antiferromagnets
2δAt = −A× [4B−2δ(Bx +By)+δ2(Bxx +Byy +Bxy)]
2δBt = −B× [4A+2δ(Ax +Ay)+δ2(Axx +Ayy +Axy)]
New fields: m =12(A+B) ϕ =
12(A−B)
|m|= O(δ) , |ϕ|= 1+O(δ2)
mt = −(∂x +∂y)[m×ϕ]+δ
4[2ϕ× (ϕxx +ϕyy +ϕxy) (1)
δϕt = 4m×ϕ−δϕ× (ϕx +ϕy)+O(δ2) (2)
Solve (2): m =δ
4
[ϕ×ϕt −ϕx −ϕy
]+O(δ2)
Subst in (1): ϕ×ϕtt = ϕ× (ϕxx +ϕyy)+O(δ)
![Page 28: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/28.jpg)
Antiferromagnets
2δAt = −A× [4B−2δ(Bx +By)+δ2(Bxx +Byy +Bxy)]
2δBt = −B× [4A+2δ(Ax +Ay)+δ2(Axx +Ayy +Axy)]
New fields: m =12(A+B) ϕ =
12(A−B)
|m|= O(δ) , |ϕ|= 1+O(δ2)
mt = −(∂x +∂y)[m×ϕ]+δ
4[2ϕ× (ϕxx +ϕyy +ϕxy) (1)
δϕt = 4m×ϕ−δϕ× (ϕx +ϕy)+O(δ2) (2)
Solve (2): m =δ
4
[ϕ×ϕt −ϕx −ϕy
]+O(δ2)
Subst in (1): ϕ×ϕtt = ϕ× (ϕxx +ϕyy)+O(δ)
![Page 29: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/29.jpg)
Antiferromagnets
Leading order
ϕ×�ϕ = ϕ×(ϕtt −ϕxx −ϕyy
)= 0
�ϕ− (ϕ ·�ϕ)ϕ = 0 (∗)
Nonlinear wave equation! Lorentz invariant!
Variational formulation: action of field ϕ : R×Σ→ S2
S[ϕ] =12
ZR×Σ
(|ϕt |2−|ϕx |2−|ϕy |2
)dt dx dy
ϕ solves (∗) iff ϕ a critical point of S.
Physicists call this the CP1 model
![Page 30: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/30.jpg)
Antiferromagnets
Leading order
ϕ×�ϕ = ϕ×(ϕtt −ϕxx −ϕyy
)= 0
�ϕ− (ϕ ·�ϕ)ϕ = 0 (∗)
Nonlinear wave equation! Lorentz invariant!
Variational formulation: action of field ϕ : R×Σ→ S2
S[ϕ] =12
ZR×Σ
(|ϕt |2−|ϕx |2−|ϕy |2
)dt dx dy
ϕ solves (∗) iff ϕ a critical point of S.
Physicists call this the CP1 model
![Page 31: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/31.jpg)
Antiferromagnets
Leading order
ϕ×�ϕ = ϕ×(ϕtt −ϕxx −ϕyy
)= 0
�ϕ− (ϕ ·�ϕ)ϕ = 0 (∗)
Nonlinear wave equation! Lorentz invariant!
Variational formulation: action of field ϕ : R×Σ→ S2
S[ϕ] =12
ZR×Σ
(|ϕt |2−|ϕx |2−|ϕy |2
)dt dx dy
ϕ solves (∗) iff ϕ a critical point of S.
Physicists call this the CP1 model
![Page 32: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/32.jpg)
Antiferromagnets
Leading order
ϕ×�ϕ = ϕ×(ϕtt −ϕxx −ϕyy
)= 0
�ϕ− (ϕ ·�ϕ)ϕ = 0 (∗)
Nonlinear wave equation! Lorentz invariant!
Variational formulation: action of field ϕ : R×Σ→ S2
S[ϕ] =12
ZR×Σ
(|ϕt |2−|ϕx |2−|ϕy |2
)dt dx dy
ϕ solves (∗) iff ϕ a critical point of S.
Physicists call this the CP1 model
![Page 33: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/33.jpg)
The Bogomol’nyi argument (Belavin, Polyakov, Lichnerowicz)
ϕtt −∆ϕ− [ϕ · (ϕtt −∆ϕ)]ϕ = 0
Static solutions are critical points of potential energy
E =12
ZΣ|ϕx |2 + |ϕy |2
Assume ϕ(x)→ ϕ0 fixed as |x| → ∞. Then ϕ extends to a ctsmap Σ∪{∞} ∼= S2→ S2
Fields fall into disjoint homotopy classes labelled byn = degϕ ∈ Z. WLOG can assume n ≥ 0Topological lower energy bound:
0 ≤ 12
ZΣ|ϕx +ϕ×ϕy |2 = E−
ZΣ
ϕ · (ϕx ×ϕy) = E−4πn
E ≥ 4πn
E = 4πn ⇔ ϕx +ϕ×ϕy = 0 1st order PDE!
![Page 34: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/34.jpg)
The Bogomol’nyi argument (Belavin, Polyakov, Lichnerowicz)
ϕtt −∆ϕ− [ϕ · (ϕtt −∆ϕ)]ϕ = 0
Static solutions are critical points of potential energy
E =12
ZΣ|ϕx |2 + |ϕy |2
Assume ϕ(x)→ ϕ0 fixed as |x| → ∞. Then ϕ extends to a ctsmap Σ∪{∞} ∼= S2→ S2
Fields fall into disjoint homotopy classes labelled byn = degϕ ∈ Z. WLOG can assume n ≥ 0Topological lower energy bound:
0 ≤ 12
ZΣ|ϕx +ϕ×ϕy |2 = E−
ZΣ
ϕ · (ϕx ×ϕy) = E−4πn
E ≥ 4πn
E = 4πn ⇔ ϕx +ϕ×ϕy = 0 1st order PDE!
![Page 35: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/35.jpg)
The Bogomol’nyi argument (Belavin, Polyakov, Lichnerowicz)
ϕtt −∆ϕ− [ϕ · (ϕtt −∆ϕ)]ϕ = 0
Static solutions are critical points of potential energy
E =12
ZΣ|ϕx |2 + |ϕy |2
Assume ϕ(x)→ ϕ0 fixed as |x| → ∞. Then ϕ extends to a ctsmap Σ∪{∞} ∼= S2→ S2
Fields fall into disjoint homotopy classes labelled byn = degϕ ∈ Z. WLOG can assume n ≥ 0Topological lower energy bound:
0 ≤ 12
ZΣ|ϕx +ϕ×ϕy |2 = E−
ZΣ
ϕ · (ϕx ×ϕy) = E−4πn
E ≥ 4πn
E = 4πn ⇔ ϕx +ϕ×ϕy = 0 1st order PDE!
![Page 36: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/36.jpg)
The Bogomol’nyi argument (Belavin, Polyakov, Lichnerowicz)
ϕtt −∆ϕ− [ϕ · (ϕtt −∆ϕ)]ϕ = 0
Static solutions are critical points of potential energy
E =12
ZΣ|ϕx |2 + |ϕy |2
Assume ϕ(x)→ ϕ0 fixed as |x| → ∞. Then ϕ extends to a ctsmap Σ∪{∞} ∼= S2→ S2
Fields fall into disjoint homotopy classes labelled byn = degϕ ∈ Z. WLOG can assume n ≥ 0
Topological lower energy bound:
0 ≤ 12
ZΣ|ϕx +ϕ×ϕy |2 = E−
ZΣ
ϕ · (ϕx ×ϕy) = E−4πn
E ≥ 4πn
E = 4πn ⇔ ϕx +ϕ×ϕy = 0 1st order PDE!
![Page 37: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/37.jpg)
The Bogomol’nyi argument (Belavin, Polyakov, Lichnerowicz)
ϕtt −∆ϕ− [ϕ · (ϕtt −∆ϕ)]ϕ = 0
Static solutions are critical points of potential energy
E =12
ZΣ|ϕx |2 + |ϕy |2
Assume ϕ(x)→ ϕ0 fixed as |x| → ∞. Then ϕ extends to a ctsmap Σ∪{∞} ∼= S2→ S2
Fields fall into disjoint homotopy classes labelled byn = degϕ ∈ Z. WLOG can assume n ≥ 0Topological lower energy bound:
0 ≤ 12
ZΣ|ϕx +ϕ×ϕy |2 = E−
ZΣ
ϕ · (ϕx ×ϕy) = E−4πn
E ≥ 4πn
E = 4πn ⇔ ϕx +ϕ×ϕy = 0 1st order PDE!
![Page 38: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/38.jpg)
The Bogomol’nyi argument (Belavin, Polyakov, Lichnerowicz)
ϕtt −∆ϕ− [ϕ · (ϕtt −∆ϕ)]ϕ = 0
Static solutions are critical points of potential energy
E =12
ZΣ|ϕx |2 + |ϕy |2
Assume ϕ(x)→ ϕ0 fixed as |x| → ∞. Then ϕ extends to a ctsmap Σ∪{∞} ∼= S2→ S2
Fields fall into disjoint homotopy classes labelled byn = degϕ ∈ Z. WLOG can assume n ≥ 0Topological lower energy bound:
0 ≤ 12
ZΣ|ϕx +ϕ×ϕy |2 = E−
ZΣ
ϕ · (ϕx ×ϕy) = E−4πn
E ≥ 4πn
E = 4πn ⇔ ϕx +ϕ×ϕy = 0 1st order PDE!
![Page 39: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/39.jpg)
The Bogomol’nyi argument
ϕx +ϕ×ϕy = 0 [⇔ ϕy −ϕ×ϕx = 0]
In fact, it’s the Cauchy Riemann condition!
S2 is a complex manifold. Can turn each tangent space into acomplex vector space of dimension 1.Need to define (a+ ib)X for any X ∈ TϕS2,a,b ∈ R.Suffices to define iX , i.e. need linear map
J : TϕS2→ TϕS2 s.t. J2 =−1
Then (a+ ib)X := aX +bJX . JX =−ϕ×X does the job.
Σ = R2 ∼= C is also a complex manifold.JΣ∂x = ∂y , JΣ∂y =−∂x
Bogomol’nyi equation equivalent to dϕ◦ JΣ = J ◦dϕ
that is, ϕ : Σ→ S2 is holomorphic
![Page 40: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/40.jpg)
The Bogomol’nyi argument
ϕx +ϕ×ϕy = 0 [⇔ ϕy −ϕ×ϕx = 0]
In fact, it’s the Cauchy Riemann condition!
S2 is a complex manifold. Can turn each tangent space into acomplex vector space of dimension 1.
Need to define (a+ ib)X for any X ∈ TϕS2,a,b ∈ R.Suffices to define iX , i.e. need linear map
J : TϕS2→ TϕS2 s.t. J2 =−1
Then (a+ ib)X := aX +bJX . JX =−ϕ×X does the job.
Σ = R2 ∼= C is also a complex manifold.JΣ∂x = ∂y , JΣ∂y =−∂x
Bogomol’nyi equation equivalent to dϕ◦ JΣ = J ◦dϕ
that is, ϕ : Σ→ S2 is holomorphic
![Page 41: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/41.jpg)
The Bogomol’nyi argument
ϕx +ϕ×ϕy = 0 [⇔ ϕy −ϕ×ϕx = 0]
In fact, it’s the Cauchy Riemann condition!
S2 is a complex manifold. Can turn each tangent space into acomplex vector space of dimension 1.Need to define (a+ ib)X for any X ∈ TϕS2,a,b ∈ R.
Suffices to define iX , i.e. need linear map
J : TϕS2→ TϕS2 s.t. J2 =−1
Then (a+ ib)X := aX +bJX . JX =−ϕ×X does the job.
Σ = R2 ∼= C is also a complex manifold.JΣ∂x = ∂y , JΣ∂y =−∂x
Bogomol’nyi equation equivalent to dϕ◦ JΣ = J ◦dϕ
that is, ϕ : Σ→ S2 is holomorphic
![Page 42: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/42.jpg)
The Bogomol’nyi argument
ϕx +ϕ×ϕy = 0 [⇔ ϕy −ϕ×ϕx = 0]
In fact, it’s the Cauchy Riemann condition!
S2 is a complex manifold. Can turn each tangent space into acomplex vector space of dimension 1.Need to define (a+ ib)X for any X ∈ TϕS2,a,b ∈ R.Suffices to define iX , i.e. need linear map
J : TϕS2→ TϕS2 s.t. J2 =−1
Then (a+ ib)X := aX +bJX .
JX =−ϕ×X does the job.
Σ = R2 ∼= C is also a complex manifold.JΣ∂x = ∂y , JΣ∂y =−∂x
Bogomol’nyi equation equivalent to dϕ◦ JΣ = J ◦dϕ
that is, ϕ : Σ→ S2 is holomorphic
![Page 43: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/43.jpg)
The Bogomol’nyi argument
ϕx +ϕ×ϕy = 0 [⇔ ϕy −ϕ×ϕx = 0]
In fact, it’s the Cauchy Riemann condition!
S2 is a complex manifold. Can turn each tangent space into acomplex vector space of dimension 1.Need to define (a+ ib)X for any X ∈ TϕS2,a,b ∈ R.Suffices to define iX , i.e. need linear map
J : TϕS2→ TϕS2 s.t. J2 =−1
Then (a+ ib)X := aX +bJX . JX =−ϕ×X does the job.
Σ = R2 ∼= C is also a complex manifold.JΣ∂x = ∂y , JΣ∂y =−∂x
Bogomol’nyi equation equivalent to dϕ◦ JΣ = J ◦dϕ
that is, ϕ : Σ→ S2 is holomorphic
![Page 44: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/44.jpg)
The Bogomol’nyi argument
ϕx +ϕ×ϕy = 0 [⇔ ϕy −ϕ×ϕx = 0]
In fact, it’s the Cauchy Riemann condition!
S2 is a complex manifold. Can turn each tangent space into acomplex vector space of dimension 1.Need to define (a+ ib)X for any X ∈ TϕS2,a,b ∈ R.Suffices to define iX , i.e. need linear map
J : TϕS2→ TϕS2 s.t. J2 =−1
Then (a+ ib)X := aX +bJX . JX =−ϕ×X does the job.
Σ = R2 ∼= C is also a complex manifold.JΣ∂x = ∂y , JΣ∂y =−∂x
Bogomol’nyi equation equivalent to dϕ◦ JΣ = J ◦dϕ
that is, ϕ : Σ→ S2 is holomorphic
![Page 45: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/45.jpg)
The Bogomol’nyi argument
ϕx +ϕ×ϕy = 0 [⇔ ϕy −ϕ×ϕx = 0]
In fact, it’s the Cauchy Riemann condition!
S2 is a complex manifold. Can turn each tangent space into acomplex vector space of dimension 1.Need to define (a+ ib)X for any X ∈ TϕS2,a,b ∈ R.Suffices to define iX , i.e. need linear map
J : TϕS2→ TϕS2 s.t. J2 =−1
Then (a+ ib)X := aX +bJX . JX =−ϕ×X does the job.
Σ = R2 ∼= C is also a complex manifold.JΣ∂x = ∂y , JΣ∂y =−∂x
Bogomol’nyi equation equivalent to dϕ◦ JΣ = J ◦dϕ
that is, ϕ : Σ→ S2 is holomorphic
![Page 46: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/46.jpg)
Moduli space
R2
2
S
u(z)x + iy = z
(z)φ
u(z) =a0 +a1z + · · ·+anzn
b0 +b1z + · · ·+bnzn
Boundary condition: ϕ(∞) = (0,0,1)⇒ bn = 0.
u(z) =a0 +a1z + · · ·+an−1zn−1 + zn
b0 +b1z + · · ·+bn−1zn−1
Moduli space Mn = Rat∗n ⊂ C2n, open
![Page 47: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/47.jpg)
Moduli space
R2
2
S
u(z)x + iy = z
(z)φ
u(z) =a0 +a1z + · · ·+anzn
b0 +b1z + · · ·+bnzn
Boundary condition: ϕ(∞) = (0,0,1)⇒ bn = 0.
u(z) =a0 +a1z + · · ·+an−1zn−1 + zn
b0 +b1z + · · ·+bn−1zn−1
Moduli space Mn = Rat∗n ⊂ C2n, open
![Page 48: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/48.jpg)
Moduli space
R2
2
S
u(z)x + iy = z
(z)φ
u(z) =a0 +a1z + · · ·+anzn
b0 +b1z + · · ·+bnzn
Boundary condition: ϕ(∞) = (0,0,1)⇒ bn = 0.
u(z) =a0 +a1z + · · ·+an−1zn−1 + zn
b0 +b1z + · · ·+bn−1zn−1
Moduli space Mn = Rat∗n ⊂ C2n, open
![Page 49: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/49.jpg)
Moduli space
R2
2
S
u(z)x + iy = z
(z)φ
u(z) =a0 +a1z + · · ·+anzn
b0 +b1z + · · ·+bnzn
Boundary condition: ϕ(∞) = (0,0,1)⇒ bn = 0.
u(z) =a0 +a1z + · · ·+an−1zn−1 + zn
b0 +b1z + · · ·+bn−1zn−1
Moduli space Mn = Rat∗n ⊂ C2n, open
![Page 50: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/50.jpg)
Moduli space
M1 = C×C×
u(z) =a0 + z
b0
Position −a0, width |b0|, orientation arg(b0)
![Page 51: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/51.jpg)
Moduli space
M2, complicated manifold dimC M2 = 4.
Expect energy to localize around zeros of u. OK if well-separated
Lose identity when close, e.g. u = z2
![Page 52: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/52.jpg)
Moduli space
M2, complicated manifold dimC M2 = 4.
Expect energy to localize around zeros of u. OK if well-separated
Lose identity when close, e.g. u = z2
![Page 53: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/53.jpg)
Moduli space
M2, complicated manifold dimC M2 = 4.
Expect energy to localize around zeros of u. OK if well-separated
Lose identity when close, e.g. u = z2
![Page 54: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/54.jpg)
Geodesic approximation (Ward, after Manton)
E
Q
Mn
n
4π n
E =12
ZΣ|ϕx |2 + |ϕy |2
S =Z
dt
{12
(ZΣ|ϕt |2
)−E(ϕ)
}
Evolution conserves Etot = E(ϕ)+12
ZΣ|ϕt |2
Constrain ϕ(t) to Mn
S|TMn =Z
dt{12
γ(ϕ, ϕ)−4πn}
Geodesic motion on (Mn,γ) where γ = L2 metric.
![Page 55: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/55.jpg)
Geodesic approximation (Ward, after Manton)
E
Q
Mn
n
4π n
E =12
ZΣ|ϕx |2 + |ϕy |2
S =Z
dt
{12
(ZΣ|ϕt |2
)−E(ϕ)
}
Evolution conserves Etot = E(ϕ)+12
ZΣ|ϕt |2
Constrain ϕ(t) to Mn
S|TMn =Z
dt{12
γ(ϕ, ϕ)−4πn}
Geodesic motion on (Mn,γ) where γ = L2 metric.
![Page 56: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/56.jpg)
Geodesic approximation (Ward, after Manton)
E
Q
Mn
n
4π n
E =12
ZΣ|ϕx |2 + |ϕy |2
S =Z
dt
{12
(ZΣ|ϕt |2
)−E(ϕ)
}
Evolution conserves Etot = E(ϕ)+12
ZΣ|ϕt |2
Constrain ϕ(t) to Mn
S|TMn =Z
dt{12
γ(ϕ, ϕ)−4πn}
Geodesic motion on (Mn,γ) where γ = L2 metric.
![Page 57: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/57.jpg)
Geodesic approximation (Ward, after Manton)
E
Q
Mn
n
4π n
E =12
ZΣ|ϕx |2 + |ϕy |2
S =Z
dt
{12
(ZΣ|ϕt |2
)−E(ϕ)
}
Evolution conserves Etot = E(ϕ)+12
ZΣ|ϕt |2
Constrain ϕ(t) to Mn
S|TMn =Z
dt{12
γ(ϕ, ϕ)−4πn}
Geodesic motion on (Mn,γ) where γ = L2 metric.
![Page 58: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/58.jpg)
L2 metric
S2
Σ
φ
X ∈ TϕMn is a section of ϕ−1TS2
Concretely, a map Σ→ R3 s.t. X(x) ·ϕ(x) = 0 everywhere. Then
γ(X ,Y ) =ZΣ
X ·Y .
Soγ(ϕ, ϕ) =
ZΣ|ϕt |2
Geodesic motion is constant speed motion along “straightestpossible” curve
![Page 59: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/59.jpg)
L2 metric
S2
Σ
φ
X ∈ TϕMn is a section of ϕ−1TS2
Concretely, a map Σ→ R3 s.t. X(x) ·ϕ(x) = 0 everywhere. Then
γ(X ,Y ) =ZΣ
X ·Y .
Soγ(ϕ, ϕ) =
ZΣ|ϕt |2
Geodesic motion is constant speed motion along “straightestpossible” curve
![Page 60: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/60.jpg)
L2 metric
S2
Σ
φ
X ∈ TϕMn is a section of ϕ−1TS2
Concretely, a map Σ→ R3 s.t. X(x) ·ϕ(x) = 0 everywhere. Then
γ(X ,Y ) =ZΣ
X ·Y .
Soγ(ϕ, ϕ) =
ZΣ|ϕt |2
Geodesic motion is constant speed motion along “straightestpossible” curve
![Page 61: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/61.jpg)
L2 metric
S2
Σ
φ
X ∈ TϕMn is a section of ϕ−1TS2
Concretely, a map Σ→ R3 s.t. X(x) ·ϕ(x) = 0 everywhere. Then
γ(X ,Y ) =ZΣ
X ·Y .
Soγ(ϕ, ϕ) =
ZΣ|ϕt |2
Geodesic motion is constant speed motion along “straightestpossible” curve
![Page 62: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/62.jpg)
Two-lump scattering (Ward, Leese)
u(z) =z2 +a1z +a0
b1z +b0
In general, geodesic flow very complicated
Trick: reduce dimension by finding totally geodesic submanifolds
Discrete group of isometries of M2, Z2×Z2, generated by
u(z) 7→ u(−z), u(z) 7→ u(z)
Fixed point set M2 ⊂M2
u(z) = λ(z2 +µ), λ ∈ R\{0},µ ∈ R
Lumps of equal width ∼ λ−12 located where z2 =−µ
![Page 63: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/63.jpg)
Two-lump scattering (Ward, Leese)
u(z) =z2 +a1z +a0
b1z +b0
In general, geodesic flow very complicated
Trick: reduce dimension by finding totally geodesic submanifolds
Discrete group of isometries of M2, Z2×Z2, generated by
u(z) 7→ u(−z), u(z) 7→ u(z)
Fixed point set M2 ⊂M2
u(z) = λ(z2 +µ), λ ∈ R\{0},µ ∈ R
Lumps of equal width ∼ λ−12 located where z2 =−µ
![Page 64: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/64.jpg)
Two-lump scattering (Ward, Leese)
u(z) =z2 +a1z +a0
b1z +b0
In general, geodesic flow very complicated
Trick: reduce dimension by finding totally geodesic submanifolds
Discrete group of isometries of M2, Z2×Z2, generated by
u(z) 7→ u(−z), u(z) 7→ u(z)
Fixed point set M2 ⊂M2
u(z) = λ(z2 +µ), λ ∈ R\{0},µ ∈ R
Lumps of equal width ∼ λ−12 located where z2 =−µ
![Page 65: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/65.jpg)
Two-lump scattering (Ward, Leese)
u(z) =z2 +a1z +a0
b1z +b0
In general, geodesic flow very complicated
Trick: reduce dimension by finding totally geodesic submanifolds
Discrete group of isometries of M2, Z2×Z2, generated by
u(z) 7→ u(−z), u(z) 7→ u(z)
Fixed point set M2 ⊂M2
u(z) = λ(z2 +µ), λ ∈ R\{0},µ ∈ R
Lumps of equal width ∼ λ−12 located where z2 =−µ
![Page 66: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/66.jpg)
Two-lump scattering (Ward, Leese)
u(z) =z2 +a1z +a0
b1z +b0
In general, geodesic flow very complicated
Trick: reduce dimension by finding totally geodesic submanifolds
Discrete group of isometries of M2, Z2×Z2, generated by
u(z) 7→ u(−z), u(z) 7→ u(z)
Fixed point set M2 ⊂M2
u(z) = λ(z2 +µ), λ ∈ R\{0},µ ∈ R
Lumps of equal width ∼ λ−12 located where z2 =−µ
![Page 67: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/67.jpg)
Two-lump scattering
u(z) = λ(z2 +µ), λ ∈ R\{0},µ ∈ R
µ
λx
y
![Page 68: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/68.jpg)
Two-lump scattering
![Page 69: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/69.jpg)
Two-lump scattering
![Page 70: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/70.jpg)
Two-lump scattering
![Page 71: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/71.jpg)
Two-lump scattering
![Page 72: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/72.jpg)
Two-lump scattering
![Page 73: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/73.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1
Two natural structures on each tangent space, J and γ. Are theycompatible?Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is HermitianEven better, ∇J = 0 (J invariant under parallel transport)γ is KahlerAlternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )Kahler⇒ dω = 0Consequence: centre of mass motion decouples, Mn = C×Mred
n
γ = 4πnγEuc + γred
![Page 74: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/74.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1Two natural structures on each tangent space, J and γ. Are theycompatible?
Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is HermitianEven better, ∇J = 0 (J invariant under parallel transport)γ is KahlerAlternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )Kahler⇒ dω = 0Consequence: centre of mass motion decouples, Mn = C×Mred
n
γ = 4πnγEuc + γred
![Page 75: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/75.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1Two natural structures on each tangent space, J and γ. Are theycompatible?Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is Hermitian
Even better, ∇J = 0 (J invariant under parallel transport)γ is KahlerAlternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )Kahler⇒ dω = 0Consequence: centre of mass motion decouples, Mn = C×Mred
n
γ = 4πnγEuc + γred
![Page 76: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/76.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1Two natural structures on each tangent space, J and γ. Are theycompatible?Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is HermitianEven better, ∇J = 0 (J invariant under parallel transport)γ is Kahler
Alternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )Kahler⇒ dω = 0Consequence: centre of mass motion decouples, Mn = C×Mred
n
γ = 4πnγEuc + γred
![Page 77: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/77.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1Two natural structures on each tangent space, J and γ. Are theycompatible?Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is HermitianEven better, ∇J = 0 (J invariant under parallel transport)γ is KahlerAlternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )Kahler⇒ dω = 0Consequence: centre of mass motion decouples, Mn = C×Mred
n
γ = 4πnγEuc + γred
![Page 78: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/78.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1Two natural structures on each tangent space, J and γ. Are theycompatible?Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is HermitianEven better, ∇J = 0 (J invariant under parallel transport)γ is KahlerAlternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )
Kahler⇒ dω = 0Consequence: centre of mass motion decouples, Mn = C×Mred
n
γ = 4πnγEuc + γred
![Page 79: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/79.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1Two natural structures on each tangent space, J and γ. Are theycompatible?Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is HermitianEven better, ∇J = 0 (J invariant under parallel transport)γ is KahlerAlternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )Kahler⇒ dω = 0
Consequence: centre of mass motion decouples, Mn = C×Mredn
γ = 4πnγEuc + γred
![Page 80: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/80.jpg)
Kahler property (Ruback)
Mn ⊂ C2n is itself a complex manifold, has a J : TϕMn→ TϕMn,J2 =−1Two natural structures on each tangent space, J and γ. Are theycompatible?Yes: γ(JX ,JY ) = γ(X ,Y ) for all X ,Y ∈ TϕMn
γ is HermitianEven better, ∇J = 0 (J invariant under parallel transport)γ is KahlerAlternative characterization: define Kahler form
ω(X ,Y ) = γ(JX ,Y )
Hermitian⇒ ω(Y ,X) =−ω(X ,Y )Kahler⇒ dω = 0Consequence: centre of mass motion decouples, Mn = C×Mred
n
γ = 4πnγEuc + γred
![Page 81: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/81.jpg)
Important features
Integer topological charge n
Topological energy bound E ≥ E0nAttained by solutions of a first order nonlinear PDE system,“holomorphic”
Moduli space of energy minimizers Mn is a finite-dimensionalcomplex manifold, with a natural Riemannian metric
Metric is Kahler
Geodesics in Mn ↔ slow n-soliton trajectories
90◦ head-on scattering
![Page 82: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/82.jpg)
Important features
Integer topological charge n
Topological energy bound E ≥ E0nAttained by solutions of a first order nonlinear PDE system,“holomorphic”
Moduli space of energy minimizers Mn is a finite-dimensionalcomplex manifold, with a natural Riemannian metric
Metric is Kahler
Geodesics in Mn ↔ slow n-soliton trajectories
90◦ head-on scattering
![Page 83: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/83.jpg)
Important features
Integer topological charge n
Topological energy bound E ≥ E0nAttained by solutions of a first order nonlinear PDE system,“holomorphic”
Moduli space of energy minimizers Mn is a finite-dimensionalcomplex manifold, with a natural Riemannian metric
Metric is Kahler
Geodesics in Mn ↔ slow n-soliton trajectories
90◦ head-on scattering
![Page 84: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/84.jpg)
Important features
Integer topological charge n
Topological energy bound E ≥ E0nAttained by solutions of a first order nonlinear PDE system,“holomorphic”
Moduli space of energy minimizers Mn is a finite-dimensionalcomplex manifold, with a natural Riemannian metric
Metric is Kahler
Geodesics in Mn ↔ slow n-soliton trajectories
90◦ head-on scattering
![Page 85: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/85.jpg)
Important features
Integer topological charge n
Topological energy bound E ≥ E0nAttained by solutions of a first order nonlinear PDE system,“holomorphic”
Moduli space of energy minimizers Mn is a finite-dimensionalcomplex manifold, with a natural Riemannian metric
Metric is Kahler
Geodesics in Mn ↔ slow n-soliton trajectories
90◦ head-on scattering
![Page 86: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/86.jpg)
Important features
Integer topological charge n
Topological energy bound E ≥ E0nAttained by solutions of a first order nonlinear PDE system,“holomorphic”
Moduli space of energy minimizers Mn is a finite-dimensionalcomplex manifold, with a natural Riemannian metric
Metric is Kahler
Geodesics in Mn ↔ slow n-soliton trajectories
90◦ head-on scattering
![Page 87: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/87.jpg)
Other solitons
Obvious generalization:
ϕ : Σ→ N, E =12
ZΣ|dϕ|2
Σ,N Riemannian mfds (harmonic map problem)
Antiferromagnets: Σ = C, N = S2
Inhomogeneous antiferromagnets: Σ = Σ2, N = S2
Σ,N Kahler, keeps key features: n = [ϕ∗ω]Mn = holn(Σ,N), Kahler
![Page 88: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/88.jpg)
Other solitons
Obvious generalization:
ϕ : Σ→ N, E =12
ZΣ|dϕ|2
Σ,N Riemannian mfds (harmonic map problem)
Antiferromagnets: Σ = C, N = S2
Inhomogeneous antiferromagnets: Σ = Σ2, N = S2
Σ,N Kahler, keeps key features: n = [ϕ∗ω]Mn = holn(Σ,N), Kahler
![Page 89: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/89.jpg)
Other solitons
Obvious generalization:
ϕ : Σ→ N, E =12
ZΣ|dϕ|2
Σ,N Riemannian mfds (harmonic map problem)
Antiferromagnets: Σ = C, N = S2
Inhomogeneous antiferromagnets: Σ = Σ2, N = S2
Σ,N Kahler, keeps key features: n = [ϕ∗ω]Mn = holn(Σ,N), Kahler
![Page 90: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/90.jpg)
Other solitons
Obvious generalization:
ϕ : Σ→ N, E =12
ZΣ|dϕ|2
Σ,N Riemannian mfds (harmonic map problem)
Antiferromagnets: Σ = C, N = S2
Inhomogeneous antiferromagnets: Σ = Σ2, N = S2
Σ,N Kahler, keeps key features: n = [ϕ∗ω]Mn = holn(Σ,N), Kahler
![Page 91: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/91.jpg)
Other solitons: Gauge theory
ϕ a section of a complex vector bundle W over Σ
∇ = unitary connexion on W , curvature F
E =12
ZΣ|∇ϕ|2 + |F |2 +U(ϕ)
n = Chern class of W
E ≥ E0n, equality iff (ϕ,∇) satisfy “self-duality” conditionVortices: Σ2, W = line bundle
n = c1(Σ) =RΣ F
Mn = Sn(Σ), Kahler
Monopoles: Σ3 = R3, W = C2 bundlen = subtleMn = Rat∗n, hyperkahler (Kahler w.r.t. three different complexstructures I,J,K , satisfying quaternion algebra)
![Page 92: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/92.jpg)
Other solitons: Gauge theory
ϕ a section of a complex vector bundle W over Σ
∇ = unitary connexion on W , curvature F
E =12
ZΣ|∇ϕ|2 + |F |2 +U(ϕ)
n = Chern class of W
E ≥ E0n, equality iff (ϕ,∇) satisfy “self-duality” condition
Vortices: Σ2, W = line bundlen = c1(Σ) =
RΣ F
Mn = Sn(Σ), Kahler
Monopoles: Σ3 = R3, W = C2 bundlen = subtleMn = Rat∗n, hyperkahler (Kahler w.r.t. three different complexstructures I,J,K , satisfying quaternion algebra)
![Page 93: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/93.jpg)
Other solitons: Gauge theory
ϕ a section of a complex vector bundle W over Σ
∇ = unitary connexion on W , curvature F
E =12
ZΣ|∇ϕ|2 + |F |2 +U(ϕ)
n = Chern class of W
E ≥ E0n, equality iff (ϕ,∇) satisfy “self-duality” conditionVortices: Σ2, W = line bundle
n = c1(Σ) =RΣ F
Mn = Sn(Σ), Kahler
Monopoles: Σ3 = R3, W = C2 bundlen = subtleMn = Rat∗n, hyperkahler (Kahler w.r.t. three different complexstructures I,J,K , satisfying quaternion algebra)
![Page 94: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/94.jpg)
Other solitons: Gauge theory
ϕ a section of a complex vector bundle W over Σ
∇ = unitary connexion on W , curvature F
E =12
ZΣ|∇ϕ|2 + |F |2 +U(ϕ)
n = Chern class of W
E ≥ E0n, equality iff (ϕ,∇) satisfy “self-duality” conditionVortices: Σ2, W = line bundle
n = c1(Σ) =RΣ F
Mn = Sn(Σ), Kahler
Monopoles: Σ3 = R3, W = C2 bundlen = subtleMn = Rat∗n, hyperkahler (Kahler w.r.t. three different complexstructures I,J,K , satisfying quaternion algebra)
![Page 95: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/95.jpg)
Other solitons: Gauge theory
Instantons: Σ4 = S4,R4, W = C2 bundle, no ϕ,n = c2(W ) =
RΣ tr(F ∧F)
Mn = {self-dual connexions} (meaning ∗F = F ), also hyperkahler
Calorons: Σ4 = S1×R3,T 2×R2, . . .
Monopoles = translation invariant instantons
Vortices = SO(3) invariant instantons
![Page 96: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/96.jpg)
Other solitons: Gauge theory
Instantons: Σ4 = S4,R4, W = C2 bundle, no ϕ,n = c2(W ) =
RΣ tr(F ∧F)
Mn = {self-dual connexions} (meaning ∗F = F ), also hyperkahler
Calorons: Σ4 = S1×R3,T 2×R2, . . .
Monopoles = translation invariant instantons
Vortices = SO(3) invariant instantons
![Page 97: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/97.jpg)
Other solitons: Gauge theory
Instantons: Σ4 = S4,R4, W = C2 bundle, no ϕ,n = c2(W ) =
RΣ tr(F ∧F)
Mn = {self-dual connexions} (meaning ∗F = F ), also hyperkahler
Calorons: Σ4 = S1×R3,T 2×R2, . . .
Monopoles = translation invariant instantons
Vortices = SO(3) invariant instantons
![Page 98: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/98.jpg)
Open questions: Geometry
Volume, diameter of Mn?
Curvature properties?
Periodic geodesics?
Ergodicity?
Symplectic geometry of (Mn,ω)?
![Page 99: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/99.jpg)
Open questions: Quantization
ClassicalField Theory
QuantumTheoryField
Geodesic
Approximation
truncate
quantize
truncate
???quantize
Wavefunction ψ : R×Mn→ C
i∂ψ
∂t=
12∆ψ
Spectral geometry of Mn
![Page 100: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/100.jpg)
Open questions: Quantization
ClassicalField Theory
QuantumTheoryField
Geodesic
Approximation
truncate
quantize
truncate
???quantize
Wavefunction ψ : R×Mn→ C
i∂ψ
∂t=
12∆ψ
Spectral geometry of Mn
![Page 101: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/101.jpg)
Open questions: Validity
Geodesic approximation based on physical intuition. Can weprove it works, i.e. rigorously bound errors?
Proto-theorem: consider one-parameter family of IVPs for fieldequation with initial data ϕ(0) = ϕ0 ∈Mn, ϕt(0) = εϕ1 ∈ Tϕ0Mn,ε > 0. Define time-rescaled field
ϕε(τ) = ϕ(τ/ε).
Then there exists T > 0 such that ϕε : [0,T ]×Σ→ N convergesuniformly, as ε→ 0, to ψ(τ), the geodesic in Mn with initial data(ϕ0,ϕ1).
Proved for vortices, monopoles (Stuart), CP1 lumps on T 2 (JMS)
![Page 102: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/102.jpg)
Open questions: Validity
Geodesic approximation based on physical intuition. Can weprove it works, i.e. rigorously bound errors?
Proto-theorem: consider one-parameter family of IVPs for fieldequation with initial data ϕ(0) = ϕ0 ∈Mn, ϕt(0) = εϕ1 ∈ Tϕ0Mn,ε > 0. Define time-rescaled field
ϕε(τ) = ϕ(τ/ε).
Then there exists T > 0 such that ϕε : [0,T ]×Σ→ N convergesuniformly, as ε→ 0, to ψ(τ), the geodesic in Mn with initial data(ϕ0,ϕ1).
Proved for vortices, monopoles (Stuart), CP1 lumps on T 2 (JMS)
![Page 103: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/103.jpg)
Open questions: Validity
Geodesic approximation based on physical intuition. Can weprove it works, i.e. rigorously bound errors?
Proto-theorem: consider one-parameter family of IVPs for fieldequation with initial data ϕ(0) = ϕ0 ∈Mn, ϕt(0) = εϕ1 ∈ Tϕ0Mn,ε > 0. Define time-rescaled field
ϕε(τ) = ϕ(τ/ε).
Then there exists T > 0 such that ϕε : [0,T ]×Σ→ N convergesuniformly, as ε→ 0, to ψ(τ), the geodesic in Mn with initial data(ϕ0,ϕ1).
Proved for vortices, monopoles (Stuart), CP1 lumps on T 2 (JMS)
![Page 104: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/104.jpg)
Concluding remarks
Geodesic approximation provides a beautiful link betweengeometry and physics
Contributions from diverse sources: Atiyah, Hitchin + · · · ,Gibbons, Manton + · · · , Ward + · · · , Witten, Sen
Mixes Riemannian, symplectic, algebraic geometry, geometricanalysis, particle physics
Lots of interesting, accessible problems
Further reading: “Topological Solitons” Manton and Sutcliffe
![Page 105: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/105.jpg)
Concluding remarks
Geodesic approximation provides a beautiful link betweengeometry and physics
Contributions from diverse sources: Atiyah, Hitchin + · · · ,Gibbons, Manton + · · · , Ward + · · · , Witten, Sen
Mixes Riemannian, symplectic, algebraic geometry, geometricanalysis, particle physics
Lots of interesting, accessible problems
Further reading: “Topological Solitons” Manton and Sutcliffe
![Page 106: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/106.jpg)
Concluding remarks
Geodesic approximation provides a beautiful link betweengeometry and physics
Contributions from diverse sources: Atiyah, Hitchin + · · · ,Gibbons, Manton + · · · , Ward + · · · , Witten, Sen
Mixes Riemannian, symplectic, algebraic geometry, geometricanalysis, particle physics
Lots of interesting, accessible problems
Further reading: “Topological Solitons” Manton and Sutcliffe
![Page 107: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/107.jpg)
Concluding remarks
Geodesic approximation provides a beautiful link betweengeometry and physics
Contributions from diverse sources: Atiyah, Hitchin + · · · ,Gibbons, Manton + · · · , Ward + · · · , Witten, Sen
Mixes Riemannian, symplectic, algebraic geometry, geometricanalysis, particle physics
Lots of interesting, accessible problems
Further reading: “Topological Solitons” Manton and Sutcliffe
![Page 108: The Geometry of Soliton Moduli Spaces · 2009. 12. 14. · Low energy dynamics reduces to geodesic motion in Mn! Soliton dynamics ←→Riemannian geometry. Plan Planar antiferromagnets](https://reader033.vdocuments.us/reader033/viewer/2022060708/6074c06e585af32cca008eee/html5/thumbnails/108.jpg)
Concluding remarks
Geodesic approximation provides a beautiful link betweengeometry and physics
Contributions from diverse sources: Atiyah, Hitchin + · · · ,Gibbons, Manton + · · · , Ward + · · · , Witten, Sen
Mixes Riemannian, symplectic, algebraic geometry, geometricanalysis, particle physics
Lots of interesting, accessible problems
Further reading: “Topological Solitons” Manton and Sutcliffe