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
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