Transcript slides

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
• 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
This talk is based on
Numerical Nahm transform for 2-caloron solutions
D. Muranaka, A. Nakamula, N. Sawado, K. Toda, Phys.Lett. B703 (2011) 498-503
and the talk in Physics Conference in Miami 2011. And see also
Graduation thesis (in Einstein farm)
http://www2.yukawa.kyoto-u.ac.jp/~hamanaka/muranaka_d10.pdf
+ Nahm construction
+ Nahm transform for 2-caloron
+ Summary
Outline
+ Introduction
What’s calorons?
Holonomy and AM construction
+ Nahm construction
+ Nahm transform for 2-caloron
+ Summary
Outline
− Introduction
Relations to the solutions in Yang-Mills theory
R4
R3
S
R3
Quark confinement
in GUT
instantons
calorons
monopoles
Field theory
What’s calorons
Holonomy and AM construction
Skyrmions
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
What’s calorons?
Relations to the solutions in Yang-Mills theory
R3
R4
S
Large period limit
instantons
calorons
R3
Quark confinement
Large scale limit in GUT
monopoles
Field theory
What’s calorons
Holonomy and AM construction
Skyrmions
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
What’s calorons?
Relations to the solutions in Yang-Mills theory
R3
R4
Large period limit
instantons
R3
S
Quark confinement
Large scale limit in GUT
calorons
monopoles
Atiyah-Manton ->
construction
Skyrmions
Field theory
What’s calorons
Holonomy and AM construction
Holonomy
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
What’s calorons?
Relations to the solutions in Yang-Mills theory
R3
R4
Large period limit
instantons
R3
S
Quark confinement
Large scale limit in GUT
calorons
Low
temperature
monopoles
High
temperature
Atiyah-Manton ->
construction
Skyrmions
Field theory
Finite temperature
What’s calorons
Holonomy and AM construction
Holonomy
Interpret the period as
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
What’s calorons?
Ex. HS caloron from JNR instanton (with period
)
instantons
calorons
monopoles
t
t
t
Large scale limit
What’s calorons
Holonomy and AM construction
Large period limit
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Harrington Shepard calorons
Lagrangian of Skyrmions
ex. Hedgehog solution (charge 1)
G.S. Adkins, C.R. Nappi, E. Witten, Nucl.Phys. B228 (1983) 552
What’s calorons
Holonomy and AM construction
Correspondence to Nucleon
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
What’s
Skyrmion
calorons?
Holonomy of calorons
Path ordered exponential
of the Gauge field
3
R
S
3
R
The integral
seems to be zero
t
Trivial Holonomy
Non-zero case
Non-Trivial Holonomy
ex. (2,2)-caloron (by Nakamura et.al.)
Atiyah-Manton Ansatz
M.F. Atiyah, N.S. Manton, Phys.Lett. B222 (1989) 438-442
What’s calorons
Holonomy and AM construction
ex. HS caloron
Symmetric caloron (by Ward)
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Holonomy and
What’s
Atiyah-Manton
calorons? construction
2-Skyrmion from JNR solution
A. Hosaka, S.M. Griffies, M. Oka, R.D. Amado, Phys.Lett. B251 (1990) 1-5
K.J. Eskola, K. Kajantie, Z.Phys. C44 (1989) 347
What’s calorons
Holonomy and AM construction
1-Skyrmion from HS caloron (Thermal Skyrmion)
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Examples ofWhat’s
Atiyah-Manton
calorons?construction
− Nahm construction
Nahm construction
History of Nahm construction
In this talk
+ Nahm transform for 2-caloron
+ Summary
Outline
+ Introduction
ADHM/Nahm construction for instantons/monopoles
Real space
Ansatz
Gauge fields
Nahm construction
History of Nahm construction
In this talk
ASD/Bogomolny eqn.
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Nahm construction
ADHM/Nahm construction for instantons/monopoles
Real space
Dirac eqn.
ADHM/Nahm data
ADHM/Nahm eqn.
Ansatz
Gauge fields
Weyl eqn.
Nahm transform (numerical work)
Nahm construction
History of Nahm construction
In this talk
ASD/Bogomolny eqn.
Moduli space
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Nahm construction
ADHM/Nahm construction for instantons/monopoles
Real space
Dirac eqn.
ADHM/Nahm data
ADHM/Nahm eqn.
Ansatz
Gauge fields
Weyl eqn.
Nahm transform (numerical work)
Nahm construction
History of Nahm construction
In this talk
ASD/Bogomolny eqn.
Moduli space
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Nahm construction
Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction for instantons
M. F. Atiyah, et.al., Phys. Lett. A 65, 185 (1978)
Applied to
monopoles and calorons
Nahm construction
Nahm eqn.
Nahm data
Weyl eqn.
Gauge fields
Nahm transform
Nahm construction
History of Nahm construction
In this talk
Nahm construction for monopoles and calorons
W. Nahm, Springer Lecture Notes in Physics Vol. 201
(Springer, New York, 1984), pp. 189-200
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Nahm construction
Harrington-Shepard (HS) caloron (k=1, by Ansatz)
B.J. Harrington, H.K. Shepard, Phys.Rev. D17 (1978) 2122
-> Some parameters are lost
Nahm data of Symmetric calorons (k=2,3,4)
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
History of Nahm construction
R.S. Ward, Phys.Lett. B582 (2004) 203-210
D. Harland, J.Math.Phys. 48 (2007) 082905
General Nahm data of calorons with Non-trivial holonomy (k=2)
A. Nakamula, J. Sakaguchi, J.Math.Phys. 51 (2010) 043503
-> Nahm transform is not performed
Nahm construction
History of Nahm construction
In this talk
Nahm data of calorons with Non-trivial holonomy (k=1,2)
Nahm transform for calorons with Non-trivial holonomy (k=2)
F. Bruckmann, D. Nogradi, P. van Baal, Nucl.Phys. B698 (2004) 233-254
Nahm construction
History of Nahm construction
In this talk
-> with Green’s function method
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
History of Nahm construction
Nahm transform for calorons with Trivial holonomy (k=2)
D. Muranaka, A. Nakamula, N. Sawado, K. Toda, Phys.Lett. B703 (2011) 498-503
• Another methods
• from general (2,2)-calorons (have some new parameters)
Massless
2-calorons
10 parameters
= Dimension of moduli space
limit
Some are new parameters
Can the Nahm transform be performed with all parameters?
Nahm construction
History of Nahm construction
In this talk
(2,2)-calorons
16 parameters
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
In this talk
+ Nahm construction
− Nahm transform for 2-caloron
Nahm data
Numerical Nahm transform
Results of computation
+ Summary
Outline
+ Introduction
Nahm equation for 2-caloron
Bulk
Boundary
Nahm transform for 2-caloron
Nahm data
Numerical Nahm transform
Results of computation
Weyl equation for 2-caloron
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Nahm construction for 2-calorons
Nahm data of SU(2) calorons with Instanton charge k = 2
Atsushi Nakamula, Jun Sakaguchi, J.Math.Phys. 51 (2010) 043503
Bulk data
Nahm data
Numerical Nahm transform
Results of computation
Boundary data and matching condition
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Nahm data of 2-calorons
Nahm data of SU(2) calorons with Instanton charge k = 2
Atsushi Nakamula, Jun Sakaguchi, J.Math.Phys. 51 (2010) 043503
Bulk data
Nahm data
Numerical Nahm transform
Results of computation
Boundary data and matching condition
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Nahm data of 2-calorons
Weyl equation (for computation)
Bulk data
Bulk
Boundary data
Boundary
-> 1-order linear differential eqn. with boundary condition
Nahm transform
Orthonormalization
Gauge field
Nahm data
Numerical Nahm transform
Results of computation
where
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Numerical Nahm transform for 2-calorons
Numerical Nahm transform for monopoles
C.J. Houghton, P.M. Sutcliffe, Commun.Math.Phys. 180 (1996) 343-362
Decomposition of tensor product
Nahm data
Numerical Nahm transform
Results of computation
-> 1-order differential equation (can be solved by Runge-Kutta method)
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Bulk Weyl equation
Series expansion of solutions at the poles
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Bulk Weyl equation
Nahm data
Numerical Nahm transform
Results of computation
Linear combination of the solutions
Nahm data
Numerical Nahm transform
Results of computation
Two independent zeromodes of monopoles
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Bulk Weyl equation
The idea of solving the Weyl equation with all parameters
1. Compute from both poles
Arbitrary solutions of the bulk Weyl equation
Nahm data
Numerical Nahm transform
Results of computation
Two independent
solutions
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Bulk Weyl equation
Boundary Weyl eqn.
<- Overdetermined!
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Boundary Weyl equation
Nahm data
Numerical Nahm transform
Results of computation
Boundary Weyl eqn.
<- Now
removed!
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Boundary Weyl equation
(1) From upper two columns :
where
For lowers the linear
combinations must satisfy
Constraint to remove the over-determined
Nahm data
Numerical Nahm transform
Results of computation
(2) L.H.S. :
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
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Summary of our method
Nahm data
Numerical Nahm transform
Results of computation
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
Boundary solution ->
unify the solutions
<- Each bulk solutions
can be computed
independently
Nahm data
Numerical Nahm transform
Results of computation
Mezasi (Dried sardines with skewer through the eyes)
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Summary of our method
Field strength and Action density
Anti-symmetric property
ASD condition
The time can be regarded as a parameter
-> The action density can be visualized
by Mathematica or MicroAVS
Nahm data
Numerical Nahm transform
Results of computation
Action density without time dependence
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Action density and visualization
<- Most symmetric case with
Nahm data
Numerical Nahm transform
Results of computation
Action isosurface plot with
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Geodesic motion of 2-caloron
Impose symmetry in the bulk data,
but not in the boundary data
Nahm data
Numerical Nahm transform
Results of computation
The action densities on x1x3-plane
with
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Non-symmetric cases
+ Nahm construction
+ Nahm transform for 2-caloron
− Summary
Outline
+ Introduction
• 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
Introduction
Nahm construction
Nahm transform for 2-caloron
Summary
Summary