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Numerical Nahm transform for 2-caloron solutions
Daichi Muranaka, Nobuyuki Sawado Department of Physics, Tokyo University of Science
Atsushi Nakamula Department of Physics, School of Science, Kitasato University
Kouichi Toda Department of Mathematical Physics, Toyama Prefectural
University
Self-introduction
• 東京理科大学 理工学研究科 物理学専攻 修士 2 年 澤渡研究室
• セミナーの経緯 -> 名古屋大博士課程の受験
• 澤渡研の研究分野 -> ソリトン解とそれらを用いた物理 ( ソリトン解によるブレーンの構築など )
• 村中の研究分野 -> SU(2) Yang-Mills 理論におけるソリトン解 (instanton, monopole, Skyrmion, caloron)
• 業績 -> Physics Letters B, International conference in Miami 2011
Abstract
D. Muranaka, A. Nakamula, N. Sawado, K. Toda, Phys.Lett. B703 (2011) 498-503
Numerical Nahm transform for 2-caloron solutions
This talk is based on
and the talk in Physics Conference in Miami 2011. And see also
http://www2.yukawa.kyoto-u.ac.jp/~hamanaka/muranaka_d10.pdf
Graduation thesis (in Einstein farm)
• caloron と Nahm 構成法について
• インスタントンチャージ 2 を持つ caloron の数値的な Nahm 変換に おいて新たに発見された手法，及び得られた計算結果について
+ Introduction
+ Nahm construction
+ Nahm transform for 2-caloron
Ou
tline
+ Summary
Ou
tline
− Introduction
What’s calorons?
+ Nahm transform for 2-caloron
+ Summary
Holonomy and AM construction
+ Nahm construction
What’s calorons?
R4 R3R S3
Quark confinementin GUT
monopolescaloronsinstantons
Field theory
Skyrmions
Relations to the solutions in Yang-Mills theory
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What’s calorons?
R4 R3R S3
Large scale limitLarge period limitQuark confinement
in GUT
monopolescaloronsinstantons
Field theory
Skyrmions
Relations to the solutions in Yang-Mills theory
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What’s calorons?
R4 R3R S3
Large scale limitLarge period limitQuark confinement
in GUT
Field theory
monopolescaloronsinstantons
Skyrmions
Relations to the solutions in Yang-Mills theory
Atiyah-Manton ->construction
Holonomy
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What’s calorons?
R4 R3R S3
Large scale limitLarge period limit
Atiyah-Manton ->construction
Lowtemperature
Hightemperature
Quark confinementin GUT
Finite temperature
Holonomy
Interpret the period as
monopolescaloronsinstantons
Skyrmions
Relations to the solutions in Yang-Mills theory
Field theory
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Harrington Shepard calorons
t
Ex. HS caloron from JNR instanton (with period )
t t
Large period limit Large scale limit
monopolescaloronsinstantons
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What’s calorons?Skyrmion
ex. Hedgehog solution (charge 1)
Lagrangian of Skyrmions
Correspondence to Nucleon
G.S. Adkins, C.R. Nappi, E. Witten, Nucl.Phys. B228 (1983) 552
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What’s calorons?Holonomy and Atiyah-Manton construction
Holonomy of calorons
Path ordered exponentialof the Gauge field
R S3
t
R3The integral
seems to be zero Trivial Holonomy
Non-zero case Non-Trivial Holonomy
ex. HS caloron Symmetric caloron (by Ward)
ex. (2,2)-caloron (by Nakamura et.al.)
Atiyah-Manton Ansatz
M.F. Atiyah, N.S. Manton, Phys.Lett. B222 (1989) 438-442
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What’s calorons?Examples of Atiyah-Manton construction
2-Skyrmion from JNR solution
A. Hosaka, S.M. Griffies, M. Oka, R.D. Amado, Phys.Lett. B251 (1990) 1-5
1-Skyrmion from HS caloron (Thermal Skyrmion)
K.J. Eskola, K. Kajantie, Z.Phys. C44 (1989) 347
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Ou
tline
+ Introduction
Nahm construction
+ Nahm transform for 2-caloron
+ Summary
History of Nahm construction
− Nahm construction
In this talk
Nahm construction
ADHM/Nahm construction for instantons/monopoles
Real space
Ansatz
ASD/Bogomolny eqn.
Gauge fields
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isto
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talk
Nahm construction
Moduli space
ADHM/Nahm construction for instantons/monopoles
Real space
Ansatz
Nahm transform (numerical work)
Dirac eqn.
ASD/Bogomolny eqn.
Gauge fields
ADHM/Nahm data
ADHM/Nahm eqn.
Weyl eqn.
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ctio
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talk
Nahm construction
Moduli space
ADHM/Nahm construction for instantons/monopoles
Real space
Ansatz
Nahm transform (numerical work)
Dirac eqn.
ASD/Bogomolny eqn.
Gauge fields
ADHM/Nahm data
ADHM/Nahm eqn.
Weyl eqn.
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Nahm construction
Nahm construction
Gauge fields
Nahm dataNahm eqn.
Weyl eqn.
Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction for instantons
M. F. Atiyah, et.al., Phys. Lett. A 65, 185 (1978)
Nahm construction for monopoles and caloronsW. Nahm, Springer Lecture Notes in Physics Vol. 201
(Springer, New York, 1984), pp. 189-200
Nahm transform
Applied to monopoles and calorons
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History of Nahm construction
B.J. Harrington, H.K. Shepard, Phys.Rev. D17 (1978) 2122
Harrington-Shepard (HS) caloron (k=1, by Ansatz)
R.S. Ward, Phys.Lett. B582 (2004) 203-210
Nahm data of Symmetric calorons (k=2,3,4)
D. Harland, J.Math.Phys. 48 (2007) 082905
Nahm data of calorons with Non-trivial holonomy (k=1,2)
A. Nakamula, J. Sakaguchi, J.Math.Phys. 51 (2010) 043503
General Nahm data of calorons with Non-trivial holonomy (k=2)
-> Some parameters are lost
-> Nahm transform is not performed
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History of Nahm construction
F. Bruckmann, D. Nogradi, P. van Baal, Nucl.Phys. B698 (2004) 233-254
Nahm transform for calorons with Non-trivial holonomy (k=2)
-> with Green’s function method
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In this talk
(2,2)-calorons16 parameters
2-calorons10 parameters
Massless
Can the Nahm transform be performed with all parameters?
= Dimension of moduli space limit Some are new parameters
D. Muranaka, A. Nakamula, N. Sawado, K. Toda, Phys.Lett. B703 (2011) 498-503
Nahm transform for calorons with Trivial holonomy (k=2)
• Another methods• from general (2,2)-calorons (have some new parameters)
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Ou
tline
+ Introduction
Nahm data
− Nahm transform for 2-caloron
+ Summary
Numerical Nahm transform
+ Nahm construction
Results of computation
Nahm construction for 2-calorons
Nahm equation for 2-caloron
Weyl equation for 2-caloron
Nahm transform for 2-caloron
Bulk
Boundary
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Nahm data of 2-calorons
Boundary data and matching condition
Bulk data
Atsushi Nakamula, Jun Sakaguchi, J.Math.Phys. 51 (2010) 043503
Nahm data of SU(2) calorons with Instanton charge k = 2
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Boundary data and matching condition
Nahm data of 2-calorons
Bulk data
Atsushi Nakamula, Jun Sakaguchi, J.Math.Phys. 51 (2010) 043503
Nahm data of SU(2) calorons with Instanton charge k = 2
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Numerical Nahm transform for 2-calorons
Bulk
Boundary
Weyl equation (for computation)
Nahm transform
Orthonormalization
Gauge field
Bulk data
Boundary data
-> 1-order linear differential eqn. with boundary condition
where
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Bulk Weyl equation
Numerical Nahm transform for monopoles
C.J. Houghton, P.M. Sutcliffe, Commun.Math.Phys. 180 (1996) 343-362
Decomposition of tensor product
-> 1-order differential equation (can be solved by Runge-Kutta method)
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Bulk Weyl equation
Series expansion of solutions at the poles
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Two independent zeromodes of monopoles
Linear combination of the solutions
Bulk Weyl equationIn
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Bulk Weyl equation
The idea of solving the Weyl equation with all parameters
1. Compute from both poles
Two independentsolutions
Arbitrary solutions of the bulk Weyl equation
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Boundary Weyl equation
Boundary Weyl eqn.
<- Over- determined!
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Boundary Weyl equation
Boundary Weyl eqn.
(2) L.H.S. :
(1) From upper two columns :
where
For lowers the linear combinations must satisfy
<- Now removed!
Constraint to remove the over-determined
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Summary of our method
The idea of solving the Weyl equation with all parameters
1. Compute from both poles 2. Estimate differences of L.H.S.
3. Determine the linear combinations
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Summary of our method
The idea of solving the Weyl equation with all parameters
1. Compute from both poles 2. Estimate differences of L.H.S.
3. Determine the linear combinations
Mezasi (Dried sardines with skewer through the eyes)
<- Each bulk solutions can be computed independently
Boundary solution ->unify the solutions
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Action density and visualization
-> The action density can be visualizedby Mathematica or MicroAVS
Field strength and Action density
Action density without time dependence
ASD conditionAnti-symmetric property
The time can be regarded as a parameter
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Geodesic motion of 2-caloron
Action isosurface plot with
<- Most symmetric case with
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Non-symmetric cases
The action densities on x1x3-plane with
Impose symmetry in the bulk data, but not in the boundary data
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Ou
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+ Introduction
+ Nahm transform for 2-caloron
− Summary
+ Nahm construction
Summary
• caloron の Nahm 変換はモノポールの Nahm 変換の拡張
• 境界での Weyl 方程式を解くには優決定を解消する 必要があり，それには bulk の Weyl 方程式の線形結合を 適当に取り直せばよい
• 非自明なホロノミーを持つ 2-caloron の一般解から 自明な場合の Nahm データを導くことで，いくつかの 新たなパラメーターを持つ caloron のゲージ場が得られた
• 今後の展望としては，非自明な場合への適用， higher charge, higher flavor への拡張， 得られたゲージ場による Atiyah-Manton 構成など
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