Matryoshka Skyrmions, Confined Instantons and (P,Q) Torus Knots. Muneto Nitta (Keio Univ.)

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1 Matryoshka Skyrmions, Confined Instantons and (P,Q) Torus Knots 20 th August 2013 Field theory and String YITP Muneto Nitta (Keio Univ.) 1[1] Phys.Rev.D86 (2012) [arxiv: ] [2] Phys.Rev.D87 (2013) [arxiv: ] [3] Nucl.Phys.B872 (2013) 62 71[arXiv: ] [4] Phys.Rev.D87 (2013) [arxiv: ] 2 With Michikazu Kobayashi (Kyoto Univ) [5] Phys.Rev.D87 (2013) [arxiv: ] [6] arxiv: [7] arxiv: [8] Nucl.Phys.B [arxiv: ] [9] Phys.Rev. D87 (2013) [arxiv: ]

2 Matryoshka (toward) Unified Skyrmions, Understanding Confined of Topological Instantons Solitons and (P,Q) and Torus Instantons Knots 20 th August 2013 Field theory and String YITP Muneto Nitta (Keio Univ.) 1[1] Phys.Rev.D86 (2012) [arxiv: ] [2] Phys.Rev.D87 (2013) [arxiv: ] [3] Nucl.Phys.B872 (2013) 62 71[arXiv: ] [4] Phys.Rev.D87 (2013) [arxiv: ] 2 With Michikazu Kobayashi (Kyoto Univ) [5] Phys.Rev.D87 (2013) [arxiv: ] [6] arxiv: [7] arxiv: [8] Nucl.Phys.B [arxiv: ] [9] Phys.Rev. D87 (2013) [arxiv: ]

3 Topological Solitons and Instantons Field theory & String theory - non-perturbative effects in (SUSY) field theories instanton counting in d=3+1 -> SW preporential vortex counting in d=1+1 -> twisted superpotential - D-branes (brane within brane, brane ending on brane) - QCD Vortices and Other Topological Solitons in Dense Quark Matter Eto,Hirono, MN &Yasui, arxiv: [hep-ph],ptep invited162 p Condensed matter physics - vortices in superconductors, superfluids - ultracold atomic BEC, BEC/BCS crossover - quantum turbulence, - phase ordering in non-equibrium - topological supercond, topological quantum computation - skyrmions in magnets - BKT transition Cosmology - monopole/domain wall problem - cosmic strings Nuclear&Astrophysics - neutron stars: neutron superfluid, proton supercond

4 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton d = D + 1 = 1+ SG kink domain wall 1 Order parameter is defined everywhere π D ( M ) Order parameter is not defined at the core π ( M ) D 1 R D R + D { } = S D M = S D 1 M

5 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton d lump = D + 1 = 2 + vortex 1 Order parameter is defined everywhere π D ( M ) Order parameter is not defined at the core π ( M ) D 1 R D R + D { } = S D M = S D 1 M

6 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons Defects Gauge theory 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Domain wall Vortex Monopole YM instanton What are relations among them? Dimensional reduction (old idea) - YM instantons -> carolons -> BPS monopoles Harrington-Shepard ( 78) etc - vortices -> domain walls Eto-Isozumi-MN-Ohashi-Sakai( 04) YM instantons on H S, R S, Σ S etc 2 2 -> vortices on Σ Witten( 77), Forgacs-Manton( 80) R, H,

7 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons Defects Gauge theory 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Domain wall Vortex Monopole YM instanton What are relations among them? Brane within brane (these ten years) Confined monopole Tong( 03), Hanany-Tong, Shifman-Yung ( 04) domain wall (kink) on a vortex = monopole in d=3+1 bulk vortex Numerical solution by Fujimori

8 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton What are relations among them? Instanton inside a vortex lump (2D Skyrmion) on a vortex = instanton-particle in d=4+1 bulk Hanay-Tong, Eto-Isozumi-MN-Ohashi-Sakai( 04) vortex Amoeba Fujimori-MN-Ohta- Sakai-Yamazaki( 08)

9 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton What are relations among them? wall Domain wall Skyrmion (3D) Skyrmion on a domain wall = instanton-particle in d=4+1 bulk Physical realization for Atiyah-Manton construction of Skyrmion from instanton holonomy Eto,MN,Ohashi&Tong, PRL( 05)

10 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton What are relations among them? Other examples? Today s topic

11 Topological Solitons and Instantons model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton What are relations among them? wall Josephson vortex [1] Phys.Rev.D86 (2012) [arxiv: ] S N S Sine-Gordon kink (1D Skyrmion) on a domain wall = vortex in d=2+1 bulk

12 Topological Solitons and Instantons model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton What are relations among them? S N Josephson vortex S Sine-Gordon kink (1D Skyrmion) on a domain wall = vortex in d=2+1 bulk wall [1] Phys.Rev.D86 (2012) [arxiv: ]

13 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons Defects Gauge theory 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Domain wall Vortex Monopole YM instanton What are relations among them? 3D Skyrmion as 2D Skyrmion Lump (baby Skyrmion) on a domain wall = 3D Skyrmion in d=3+1 bulk wall [2] Phys.Rev.D87 (2013) [arxiv: ]

14 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons Defects Gauge theory 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Domain wall Vortex Monopole YM instanton What are relations among them? Matryoshka Skyrmions Wall on wall on wall.. = N Dim Skyrmion in d=n+1 bulk wall [3] Nucl.Phys.B872 (2013) 62 71[arXiv: ]

15 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Defects Gauge theory Domain wall Vortex Monopole YM instanton What are relations among them? monopole Confined instanton Sine-Gordon kink on a monopole-string = instanton-particle in d=4+1 bulk [4] Phys.Rev.D87 (2013) [arxiv: ]

16 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Topological Solitons and Instantons Defects Gauge theory 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Domain wall Vortex Monopole YM instanton What are relations among them? Brane within brane (these ten years) Summary so far wall wall vortex monopole

17 Plan of My Talk 1 Introduction 2 Josephson Vortices and Matryoshyka Skyrmions 3 Confined Instantons 4 (P,Q) Torus Knots 5 Summary and Discussion

18 Plan of My Talk 1 Introduction 2 Josephson Vortices and Matryoshyka Skyrmions 3 Confined Instantons 4 (P,Q) Torus Knots 5 Summary and Discussion

19 Lower dimensional version of Matryoshka U(1) gauge theory with two Higgs fields ~ ( φ 1,2, φ ( Σ, F ) µν 1,2 = 0) two hypermultiplets U(1) vector multiplet d=3+1, =2 supersymmetry vacua 1 2 ( φ, φ ) = ( v,0) : S semi-local vortex ( A φ θ 1, = φ 2 γ ) = v( f ( r) / r ( r) e iθ, ( f, g, γ ) 3 g( r)) (1,0,1), r

20 Lower dimensional version of Matryoshka U(1) gauge theory with two Higgs fields ~ ( φ 1,2, φ ( Σ, F ) µν 1,2 = 0) two hypermultiplets U(1) vector multiplet d=3+1, =2 supersymmetry vacua 1 2 SUSY preserving mass deformation ( φ, φ ; Σ) = ( v,0; m ), (0, v; m 1 2 domain wall semi-local vortex shrinks to Abrikosov-Nielsen-Olesen vortex e 2 2 v 2 v ) evr

21 Lower dimensional version of Matryoshka U(1) gauge theory with two Higgs fields ~ ( φ 1,2, φ ( Σ, F ) µν 1,2 = 0) two hypermultiplets U(1) vector multiplet d=3+1, =2 supersymmetry 2 SUSY preserving mass deformation 2 ( *1 2 *2 1) φ φ φ φ β Φ σ Φ = β + x Josephson coupling SUSY is broken

22 In the limit e CP 1 model = SUSY preserving mass deformation Josephson deformation O(3) NLSM Anisotropic ferromagnets with 2 easy axes

23 β = 0 No Josephson Domain wall solution u = 0 y Abraham-Townsend( 92) Arai-Naganuma-MN-Sakai( 02) + Y : translational zero mode ϕ : internal U(1) zero mode Effective theory on a domain wall y u = Arai-Naganuma- MN-Sakai( 02) finite e

24 β 0 << m Josephson Eff. theory on domain wall = sine-gordon model Sine-Gordon kink BPS equation 1-kink solution mass T = 4β ~ m

25 What does this SG kink look like in the bulk? Lump charge π ( C P 1 ) = 2 Z Lump is semi-local vortex for finite e PRD86 (2012) [arxiv: ] for k SG-kinks sine-gordon kink in d=1+1 CP 1 wall w.v. =CP 1 lump in d=2+1 bulk(cp 1 instanton in d=2+0)

26 Numerical solutions With M.Kobayashi, PRD87 (2013) [arxiv: ] Total energy density Topological lump charge density Josephson energy density

27 Jewels on wall ring with M.Kobayashi PRD87 (2013) [arxiv: ] 4 derivative(skyrme) term (Baby Skyrme model) Josephson energy density d = 2 + 1, m 0, β 0 Total energy density Lump charge density

28 The dynamics of domain wall Skyrmions P. Jennings, P. Sutcliffe. arxiv:

29 O(N+1) NLSM in d=n+1 dimensions Nucl.Phys.B872 (2013) [arxiv: ]

30 Plan of My Talk 1 Introduction 2 Josephson Vortices and Matryoshyka Skyrmions 3 Confined Instantons 4 (P,Q) Torus Knots 5 Summary and Discussion

31 Non-Abelian Phys.Rev.D87(2013) [ arxiv: ] d=4+1 bulk 1 m ( gv) 1 β 1 Potential SUSY mass deformation NA Josephson deformation non-abelian vortex membrane (d=2+1 world-volume) Instanton-particle in d=4+1 bulk = CP 1 lump in d=2+1 vortex w.v. = sine-gordon kink in d=1+1 CP 1 wall w.v. monopole-string in d=4+1 bulk = CP 1 wall in d=2+1 vortex w.v. Instantons confined by monopole strings

32 U(2) gauge theory with two Higgs fields in fund.rep. ~ ( H, H = ( Σ, F ) µν 0) 2 hypermultiplets in fund.rep. U(2) vector multiplet H : 2 2 H = v1 2 vacuum, Σ = Color-flavor locked vacuum U(2) C SU(2) F SU(2) C+F SUSY preserving mass deformation non-abelian Josephson term M = m m

33 β = 0, m = 0 No Josephson Non-Abelian vortex ANO We can embed the ANO solution H ( z ANO ANO H ( z z0) F12 ( z z0) H =, F = 12 v 0 This solution breaks SU(2) C+F U(1) The moduli space of Nambu-Goldstone modes: SU(2) C C+ F C CP 1 C U(1) Hanany-Tong, Konishi et.al ( 03) ), F ANO 12 S ( z) 2 Translation z 0 Internal symmety

34 β = 0, m 0 << v The vortex effective theory is the No Josephson 1 CP model vacuum state fluctuation of zero modes The monopole effective theory

35 β 0 << m The vortex effective theory is the With Josephson 1 CP model Josephoson term induced in the vortex eff. theory The monopole effective theory Sine-Gordon pot. induced in the monopole eff. theory +

36 Non-Abelian Phys.Rev.D87(2013) [ arxiv: ] d=4+1 bulk 1 m ( gv) 1 β 1 Potential SUSY mass deformation NA Josephson deformation non-abelian vortex membrane (d=2+1 world-volume) Instanton-particle in d=4+1 bulk = CP 1 lump in d=2+1 vortex w.v. = sine-gordon kink in d=1+1 CP 1 wall w.v. monopole-string in d=4+1 bulk = CP 1 wall in d=2+1 vortex w.v. Instantons confined by monopole strings

37 Plan of My Talk 1 Introduction 2 Josephson Vortices and Matryoshyka Skyrmions 3 Confined Instantons 4 (P,Q) Torus Knots 5 Summary and Discussion

38 d = 3+1 β 0 with M.Kobayashi arxiv: (P,Q) Torus Knots = Hopfions 4 deriv.(skyrme) term (Faddeev-Skyrme model) S3 S2 π 3 (S 2 ) = Z 2563

39 Winding Hopfion on R 2,1 S 1 Pontrjagin s homotopy M.Kobayashi & MN NPB[arXiv: ] 2 1 S S S 2

40 Plan of My Talk 1 Introduction 2 Josephson Vortices and Matryoshyka Skyrmions 3 Confined Instantons 4 (P,Q) Torus Knots 5 Summary and Discussion

41 model dim d=1+1 d=2+1 d=3+1 d=4+1 Textures NLSM Defects Gauge theory Topological Solitons and Instantons Topological Solitons and Instantons 1D Skyrmion (SG kink) 2D Skyrmion (lump) 3D Skyrmion 4D Skyrmion Domain wall Vortex Monopole YM instanton What are relations among them? Brane within brane (these ten years) Hopfion = sine-gordon kink on a curved (toroidal) domain wall wall wall vortex monopole ご清聴ありがとうございました