new soliton solutions of anti-self-dual yang-mills (asdym ...hamanaka/schuang20200625.pdfnew soliton...
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New Soliton Solutions of Anti-Self-Dual Yang-Mills(ASDYM) Equations
Shan-Chi Huang
名大多元数理
2020 年 06 月 25 日
共同研究者：浜中 真志 (名大多元数理)C. R. Gilson (University of Glasgow)J. J. C. Nimmo (University of Glasgow)
参考文献：[HH] arXiv:2004.09248[GHHN] arXiv:2004.01718
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 1 / 29
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IntroductionBrief Content of this Talk
Darboux transformation solution of (Anti-self-dual Yang-Mils)ASDYM on Complex Space (G = GL(2)).
=⇒ Under the Reduction conditions
Soliton Solutions on 4D Real Spaces.(Euclidean(+,+,+,+), Minkowski(+,-,-,-), Ultrahyperbolic (+,+,-,-)).=⇒ 1-Soliton Solutions (Not Instantion)(take a reduced form to remove the singularities)=⇒ Domain Wall type Soliton (Interpretation)
Motivation, Difficulty, and Objective’tHooft Ansatz solution (G = SU(2)) =⇒ TrFµνFµν=0Atiyah-Ward Ansatz solution (G = GL(2)) =⇒ TrFµνFµν=0For G = GL(2), Action Density is Complex-Valued in general.G = U(2) (For physical purpose)多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 2 / 29
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Achievement
TrFµνFµν ∝(2sech2X − 3sech4X
), and
∫TrFµνFµνd4x = 0.
Nontrivial Action Denstiy:=⇒ New formulation of ASDYM for Darboux Transformation.Real-Valued Action Density for G = GL(2):=⇒ Suitable construction of Yang’s J-matrix.Pure 1-Soliton:=⇒ No Singularites, No Periodic Fluctuation.Domain Wall type SolitonG = U(2) =⇒ Ultrahyperbolic Space.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 3 / 29
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Outline
Introduction
Darboux Transformation Solution of ASDYM
Soliton Solutions of ASDYM on Complex Space
Soliton Solutions of ASDYM on 4D Real Spaces(Euclidean, Minkowski, and Ultrahyperbolic Signature)
Conditions of Unitary J-matrix on each Space
Discussion of Unitary Gauge Group
Conclusion and Future Work
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 4 / 29
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Outline
Introduction
Darboux Transformation Solution of ASDYM
Soliton Solutions of ASDYM on Complex Space
Soliton Solutions of ASDYM on 4D Real Spaces(Euclidean, Minkowski, and Ultrahyperbolic Signature)
Conditions of Unitary J-matrix on each Space
Discussion of Unitary Gauge Group
Conclusion and Future Work
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 5 / 29
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Darboux Transformation Solution of ASDYMASDYM on 4D Complex Space with Coordinates (z, z̃,w, w̃)
Metric :
ds2 = gmndzmdzn = 2(dzdz̃ − dwdw̃), gmn :=
0 1 0 01 0 0 00 0 0 −10 0 −1 0
Field Strength: Fmn := ∂mAn − ∂nAm + [Am,An]ASDYM
Fzw = [Dz,Dw] = 0, Fz̃w̃ = [Dz̃,Dw̃] = 0,
Fzz̃ − Fww̃ = [Dz,Dz̃]− [Dw,Dw̃] = 0.
Complex Action DensityTrF2 := TrFmnFmn = −2Tr(F2
ww̃ + F2zz̃ + 2Fz̃wFzw̃ + 2FzwFz̃w̃),
where Fmn := gmkgnlFkl.多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 6 / 29
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Usual Formulation of ASDYM
ASDYM equation holds
⇐⇒ The Lax pair L := Dw − ζDz̃, M := Dz − ζDw̃ s.t [L,M] = 0.
⇐⇒ Yang equation holds (J: Yang’s J-matrix)
∂z̃(J−1∂zJ)− ∂w̃(J−1∂wJ) = 0.
(That is, J-matrix of Yang equation ⇐⇒ solution of ASDYM.)
Then ASD gauge fields can be given in terms of the decompositions ofJ-matrix, J = h̃−1h :
Az = −(∂zh)h−1, Aw = −(∂wh)h−1, Az̃ = −(∂z̃h̃)h̃−1, Aw̃ = −(∂w̃h̃)h̃−1.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 7 / 29
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New Formulation of ASDYM (Nimmo-Gilson-Ohta 2000)
Let Lax pair Lϕ := Dwϕ− (Dz̃ϕ)ζ, Mϕ := Dzϕ− (Dw̃ϕ)ζ,
then LMϕ− MLϕ = 0 ⇐⇒ ASDYM equations holds.ζ : Right-multiplicative Constant Matrix
A special gauge h̃ = 1
Gauge fields become a simpler form in terms of J:
Az = J−1∂zJ, Aw = J−1∂wJ, Az̃ = Aw̃ = 0,
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 8 / 29
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Darboux Transformation for n = 1 (Nimmo-Gilson-Ohta 2000)
Let (ϕ, ζ) = (Q,Λ) be an eigenfunction-eigenvalue pair of linear system
Lϕ = Dwϕ− (∂z̃ϕ)ζ = (∂w + J−1∂wJ)ϕ− (∂z̃ϕ)ζ = 0,
Mϕ = Dzϕ− (∂w̃ϕ)ζ = (∂z + J−1∂zJ)ϕ− (∂w̃ϕ)ζ = 0.
Then this linear system is ”form-invariant” under the transformation:
J̃ = −QΛ−1Q−1J, ϕ̃ = ϕζ − QΛQ−1ϕ .
That is, J̃ , ϕ̃ satisfy the linear system
L̃ϕ̃ = (∂w + J̃−1∂wJ̃)ϕ̃− (∂z̃ϕ̃)ζ = 0,
M̃ϕ̃ = (∂z + J̃−1∂zJ̃)ϕ̃− (∂w̃ϕ̃)ζ = 0.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 9 / 29
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Introduction
Darboux Transformation Solution of ASDYM
Soliton Solutions of ASDYM on Complex Space
Soliton Solutions of ASDYM on 4D Real Spaces(Euclidean, Minkowski, and Ultrahyperbolic Signature)
Conditions of Unitary J-matrix on each Space
Discussion of Unitary Gauge Group
Conclusion and Future Work
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 10 / 29
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Soliton Solutions of ASDYM on Complex Space
Soliton solution for n = 1 Darboux transf and matrix size N=2
Take seed solution J0 = I2×2 for simplicity, Then the (ϕ, ζ) = (Q,Λ) mustsatisfy the linear system
∂wϕ = (∂z̃ϕ)ζ, ∂zϕ = (∂w̃ϕ)ζ.
Soliton Solution
Q =
(a1eL + a2e−L b1eM + b2e−M
c1eL + c2e−L d1eM + d2e−M
), Λ =
(λ 00 µ
),
L := λβz + αz̃ + λαw + βw̃, M := µδz + γz̃ + µγw + δw̃,
where a1, a2, b1, b2, c1, c2, d1, d2, α, β, γ, δ, λ, µ are complex constants.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 11 / 29
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Gauge Fields:
Am =2(µ− λ)
∆2
(pBD − qAC −pB2 + qA2
pD2 − qC2 −pBD + qAC
)
(p, q) := (αε0, γε̃0) if m = w,(p, q) := (βε0, δε̃0) if m = zε0 := a2c1 − a1c2, ε̃0 := b2d1 − b1d2
Complex-Valued Action Density
TrF2 = 8(λ− µ)2(αδ − βγ)2ε0ε̃0
2ε1ε̃1 sinh2 X1 − 2ε2ε̃2 sinh2 X2 − ε0ε̃0((ε1ε̃1)
12 cosh X1 + (ε2ε̃2)
12 cosh X2
)4
where
X1 := M + L +1
2log(ε1/ε̃1), X2 := M − L +
1
2log(ε2/ε̃2)
ε0 := a2c1 − a1c2, ε̃0 := b2d1 − b1d2,ε1 := a1d1 − b1c1, ε̃1 := a2d2 − b2c2,ε2 := a2d1 − b1c2, ε̃2 := a1d2 − b2c1.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 12 / 29
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Introduction
Darboux Transformation Solution of ASDYM
Soliton Solutions of ASDYM on Complex Space
Soliton Solutions of ASDYM on 4D Real Spaces(Euclidean, Minkowski, and Ultrahyperbolic Signature)
Conditions of Unitary J-matrix on each Space
Discussion of Unitary Gauge Group
Conclusion and Future Work
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 13 / 29
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Soliton Solutions of ASDYM on 4D Euclidean Space EEuclidean real slice condition: z̃ = z, w̃ = −w.
z =1√2(x0 − ix1), z̃ =
1√2(x0 + ix1), w =
−1√2(x2 − ix3), w̃ =
1√2(x2 + ix3),
satisfying Euclidean metric : ds2 = (dx0)2 + (dx1)2 + (dx2)2 + (dx3)2.
ASDYM on E : F01 + F23 = 0, F02 − F13 = 0, F03 + F12 = 0.
Soliton Solution:
Q =
(a1eL + a2e−L b1eL + b2e−L
−b1eL − b2e−L a1eL + a2e−L
), Λ =
(λ 0
0 −1/λ
),
whereL = (λβ)z + αz + (λα)w − βw
=1√2(α+ λβ, i(α− λβ), β − λα, i(β + λα)) · (x0, x1, x2, x3) := lµxµ
M = L =⇒ γ = λβ, δ = −λα, µ = −1/λ多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 14 / 29
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Action Density of this Soliton Solution is Real-Valued:
TrFµνFµν=
8[(|α|2+ |β|2)(|λ|2+ 1) |ε0|
]22ε1ε̃1 sinh2 X1 − 2 |ε2|2 sinh2 X2 − |ε0|2((ε1ε̃1)
12 cosh X1 + |ε2| cosh X2
)4,
where
Coefficients
ε0 = a1b2 − a2b1ε1 = |a1|2 + |b1|2 , ε̃1 = |a2|2 + |b2|2 ∈ Rε2 = a1a2 + b1b2
Variables
X1 = L + L +1
2log(ε1/ε̃1) : Real-valued function
=⇒ Solitonic wave ∼ sech2X1
X2 = L − L +1
2log(ε2/ε2) : Pure imaginary function
=⇒ cosh X2 = cos (ImX2) , sinh X2 = i sin (ImX2)=⇒ Periodic fluctuation
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 15 / 29
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Setting a2 = b1 = 0, a1 = a, b2 = b, or equivalently,
Q =
(aeL be−L
−be−L ae−L
).
Then we can obtain a reduced form of action density without the periodicfluctuation part : (Note that |ε0|2 = ε1ε̃1 = |ab|2 , |ε2|2 = 0.)
TrFµνFµν = 8[(|α|2 + |β|2)(|λ|2 + 1)
]2 (2sech2X − 3sech4X
),
where X = L + L + log(|a| / |b|).
Pure 1-Soliton (No sigularities, periodic fluctuation.)Integration is possible. In fact,
∫E TrFµνFµνd4x = 0.
Domain Wall on R4 :The peak of Action Density lies on a 3D hyperplaneX = L + L + log |a/b | = 0 with normal vector lµ + lµ.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 16 / 29
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多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 17 / 29
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Instantion number = 0
Set X = X0 and introduce 3 independent axes X1,X2,X3 in the directionsorthogonal to the X-axis (normal direction of the domain wall (DW)).
Xµ := Cµ0x0 + Cµ1x1 + Cµ2x2 + Cµ3x3 + dµ.Consider integration on a finite box −R ≤ xµ ≤ R. Then under the abovelinear transformation, the new box is
−aµR + dµ ≤ Xµ ≤ aµR + dµ , aµ :=∑3
ν=0 |Cµν |.∫ R
−R
∫ R
−R
∫ R
−R
∫ R
−R(2sech2X − 3sech4X)dx0dx1dx2dx3
= |J|∫−aiR+di≤xi≤aiR+di
dX1dX2dX3
∫ a0R+d0
−a0R+d0
(2sech2X − 3sech4X)dX
= −|J|∫−aiR+di≤xi≤aiR+di
dX1dX2dX3 (tanhX · sech2X)∣∣a0R+d0
−a0R+d0
=⇒∫E
TrFµνFµνd4x ∝∫
DWdX1dX2dX3 (tanhX · sech2X)
∣∣∞−∞ = 0.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 17 / 29
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Soliton Solutions of ASDYM on Minkowski Space MMinkowski real slice condition: z, z̃ ∈ R, w̃ = w
z =1√2(x0 − x1), z̃ =
1√2(x0 + x1), w =
1√2(x2 − ix3), w̃ =
1√2(x2 + ix3)
Minkowski metric ds2 = (dx0)2 − (dx1)2 − (dx2)2 − (dx3)2.
ASDYM on M : F01 + iF23 = 0, F02 − iF13 = 0, F03 + iF12 = 0.
Soliton Solution
Q =
(a1eL + a2e−L b1eL + b2e−L
−b1eL − b−L2 a1eL + a2e−L
), Λ =
(λ 00 µ
),
whereL = (λµα)z + αz̃ + (λα)w + (µα)w
=1√2((1 + λµ)α, (1− λµ)α, (µ+ λ)α, i(µ− λ)α) · (x0, x1, x2, x3)
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 18 / 29
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The condition M = L =⇒ relations β = µα, γ = α, δ = λα(Relation between λ and µ is not necessary.)
Action Density is Real-Valued:
TrFµνFµν = 8 |α(λ− µ)|4 |ε0|2
2ε1ε̃1 sinh2 X1 − 2 |ε2|2 sinh2 X2 − |ε0|2((ε1ε̃1)
12 cosh X1 + |ε2| cosh X2
)4
Reduced Form of Action Density:
TrFµνFµν = 8∣∣α2(λ− µ)
∣∣2 (2sech2X − 3sech4X)
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 19 / 29
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Soliton Solutions of ASDYM on Ultrahyperbolic Space UUltrahyperbolic real slice condition for U : z, z̃, w, w̃ ∈ R
z =1√2(x0 − x2), z̃ =
1√2(x0 + x2), w = − 1√
2(x1 − x3), w̃ =
1√2(x1 + x3). (1)
Ultrahyperbolic metric : ds2 = (dx0)2 + (dx1)2 − (dx2)2 − (dx3)2.
ASDYM on U2 : F01 + F23 = 0, F02 + F13 = 0, F03 − F12 = 0.
Soliton Solution
Q =
(a1eL + a2e−L b1eL + b2e−L
−b1eL − b2e−L a1eL + a2e−L
), Λ =
(λ 0
0 λ
),
whereL = (λβ)z + αz̃ + (λα)w + βw̃,
=1√2(α+ λβ, β − λα, α− λβ, β + λα) · (x0, x1, x2, x3).
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 20 / 29
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The condition M = L ⇒ γ = α, δ = β, µ = λ.
Action Density is Real-Valued:
TrFµνFµν=
8[(αβ−αβ)(λ−λ) |ε0|
]22ε1ε̃1 sinh2 X1 − 2 |ε2|2 sinh2 X2 − |ε0|2((ε1ε̃1)
12 cosh X1 + |ε2| cosh X2
)4,
Reduced form of Action Density :
TrFµνFµν = 8[(αβ − αβ)(λ− λ)
]2 (2sech2X − 3sech4X
),
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 21 / 29
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Introduction
Darboux Transformation Solution of ASDYM
Soliton Solutions of ASDYM on Complex Space
Soliton Solutions of ASDYM on 4D Real Spaces(Euclidean, Minkowski, and Ultrahyperbolic Signature)
Conditions of Unitary J-matrix on each Space
Discussion of Unitary Gauge Group
Conclusion and Future Work
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 22 / 29
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Conditions of Unitary Yang’s J-Matrix on each Space
Proposition
Consider
Q =
(A B−B A
), Λ =
(λ 00 µ
).
Then Yang’s J-matrix is J = QΛ−1Q−1
=−1
|A|2 + |B|2(
(1/λ) |A|2 + (1/µ) |B|2 (1/µ− 1/λ)AB(1/µ− 1/λ)AB (1/µ) |A|2 + (1/λ) |B|2
).
J ∈ U(2) ⇔ |λ| = |µ| = 1In fact, J is a double-parameter deformation of U(2) group.J ∈ SU(2) ⇔ µ = λ and |λ| = 1.In this case, J is a one parameter deformation of SU(2) group.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 23 / 29
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Euclidean Sapce E :J ∈ U(2) ⇔ Λ =
(eiθ 00 −eiθ
), θ ∈ R
J ∈ SU(2) ⇔ Λ = ±(
i 00 −i
)Minkowski Space M :J ∈ U(2) ⇔ Λ =
(eiθ1 00 eiθ2
), (θ1, θ2 ∈ R)
J ∈ SU(2) ⇔ Λ =
(eiθ 00 e−iθ
)Ultrahyperbolic Space U :
J ∈ U(2) ⇔ J ∈ SU(2) ⇔ Λ =
(eiθ 00 e−iθ
).
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 24 / 29
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Introduction
Darboux Transformation Solution of ASDYM
Soliton Solutions of ASDYM on Complex Space
Soliton Solutions of ASDYM on 4D Real Spaces(Euclidean, Minkowski, and Ultrahyperbolic Signature)
Conditions of Unitary J-matrix on each Space
Discussion of Unitary Gauge Group
Conclusion and Future Work
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 25 / 29
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Discussion of Unitary Gauge Group
Recall that
g ∈ U(2) ⇐⇒ Aµ : Anti-Hermitian ⇐⇒ Fµν : Anti-Hermitian.
Euclidean Space E
Fµν : Anti-Hermitian =⇒ eignvalues of Fµν are pure imaginaryTrFµνFµν = TrF2
µν < 0(However, our action density is not negative definite.)
Minkowski Space M
ASDYM : F01 + iF23 = 0, F02 − iF13 = 0, F03 + iF12 = 0.=⇒ Fµν can not be all Anti-Hermitian.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 26 / 29
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G = U(2) is realized on Ultrahyperbolic space U successfully.
Gauge fields Az and Aw are anti-hermitian on U naturally :
Am =2(λ− λ)
∆2
(pAB + pAB −pB2 + pA2
pA2 − pB2 −pAB − pAB
){
p := αε0, if m = w, p := βε0, if m = z,ε0 := a1b2 − a2b1,
From Ultrahyperbolic real slice condition for U,√2Az = A0+A2,
√2Az̃ = A0−A2,
√2Aw = A1+A3,
√2Aw̃ = A1−A3
Az̃ = Aw̃ = 0
=⇒ All gauge fields Aµ must be anti-hermitian.
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 27 / 29
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Conclusion and Future Work
ConclusionDoamin Wall type soliton are constructed on each 4D space.
Action density is Real-Valued.Topological charge = 0G = U(2) is realized in Ultrahyperbolic Space U.
Future Work
G = U(2) solution on Euclidean and Minkowski Space (under a moregeneral framework)G = GL(3) and G = SU(3) solution.Soliton type solutions in gravitational field.(n + 2 dimensional axisymmetric solution of Einstein’s vacuumequation ⇐⇒ J-matrix of G = GL(n) Yang equation)
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 28 / 29
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Thank you for your attendence
多弦数理物理 Seminar New Soliton Solutions of ASDYM 2020/06/25 29 / 29