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Non-Abelian Vortices — Five Years Since the Discovery — RIKEN. Non-Abelian Vortices — Five Years Since the Discovery — Towards New Developments in Field and String Theories 12/22/2008 @ RIKEN Muneto Nitta (Keio U. @ Hiyoshi) 0 Collaborators TITech Soliton Group Norisuke Sakai(Tokyo Woman Ch.), Keisuke Ohashi(DAMTP), Youichi Isozumi, Toshiaki Fujimori(D3), Takayuki Nagashima(D2) Pisa Group Ken-ichi Konishi, Minoru Eto, Giacomo Marmorini, Walter Vinci, Sven Bjarke Gudnason Other Institutes Kazutoshi Ohta(Tohoku), Naoto Yokoi(Komaba), Masahito Yamazaki(Hongo), Koji Hashimoto(RIKEN), Luca Ferretti(Trieste), Jarah Evslin(Trieste), Takeo Inami(Chuo), Shie Minakami(Chuo), Hadron Physics Eiji Nakano, Taeko Matsuura, Noriko Shiiki Condensed Matter Physics Masahito Ueda, Yuki Kawaguchi, Michikazu Kobayashi (Hongo) Anyone is welcome to join us anytime ! 1 x1. Introduction: What are Vortices? Vortices are topological solitons ² of codimension 2: point-like in d = 2 + 1, string in d = 3 + 1, ² to exist when symmetry is broken G ! H with ¼1(G=H) ' ¼0(H) ' H=H0 6= 0 for simply connected G, ² formed via the Kibble-Zurek mechanism or rotation of media, ² carrying magnetic flux or circulation which is quantized. Defects Textures Gauge Structure ¼n codim n + 1 codim n codim n + 1 ¼0 domain walls(kinks) ¼1 vortices nonlinear kinks(sine-Gordon) ¼2 monopoles lumps(2D skyrmions) ¼3 Skyrmions (textures) YM instantons 1 They appear in various area of physics: 1. condensed matter physics ² superconductor (Abrikosov lattice) Abrikosov(’57) ² superfluid 4He Onsager(’49), Feynman(’55) superfluid 3He ² (skyrmions in) quantum Hall effects ² (Bloch line in) Ferromagnets ² atomic gas Bose-Einstein condensation (cold atom) (’01-) ² quantum turbulence (Kolmogorov law) MIT [Abo-Shaer et.al, Science 292 (2001) 476] 2 2. cosmology and astrophysics ² a candidate of cosmic strings Phase transition occurs in the early Universe. ) vortices must form (Kibble mechanism) Kibble (’76) (cf: monopoles ) monopole problem Preskill, Guth(’79)) Suggested as a source of structure formation (’80s – early’90) ) ruled out by Cosmic Microwave Background (’98 - ’01) ² vortex-ring(=vorton): candidate of dark matter, ultra high energy cosmic ray ² Recent revivals of cosmic strings (’03 - present): (a) cosmic superstrings (F/D-strings) in string theory, brane inflation Dvali-Tye, Polchinski etc (’04) (p,q) string network (b) possible detection of cosmic strings by CMB, gravitational lensing, gravitational wave 3 3. high energy physics ² magnetic flux tube confining monopoles Nielsen-Olesen(’73) = dual superconductor ’tHooft, Nambu, Mandelstam (’74) quark monopole anti-quark anti-monopole () dual Meissner effect electric flux magnetic flux ² The center vortex mechanism ’tHooft, Cornwall etc (’79) trying to extend it to color(non-Abelian) gauge symmetry lattice sim. Ambjorn et.al (’00) ² Supersymmetric QCD Hanany-Tong, Konishi group(Pisa), Shifman-Yung(Minnesota), TITech (’03-) ² Weinberg-Salam, Nambu(’77), Vachaspati(’92) ² SO(10) GUT Kibble (’82), SUSY GUTs Jeannerot et al (’03) 4 4. hadron physics ² proton vortices and neutron vortices in hadronic phase of neutron stars ) pulsar glitch Anderson-Itoh(’75) ² color superconductivity (core of neutron stars) Iida-Baym etc(’01), Balachandran-Digal-Matsuura(’05), Nakano-MN-Matsuura(’07) ² chiral phase transition Brandenberger(’97), Balachandran-Digal(’01), MN-Shiki,Nakano-MN-Matsuura(’07) ² YM plasma Chernodub-Zakharov, Liao-Shuryak(’07-) T RHIC QGP = color superconducting crystal? gas liq CFL nuclear compact star µ Alford et.al Hatsuda et.al 5 Abelian Vortices Vortices appear when U(1) local sym. is spontaneously broken. The Abelian Higgs model [(gauged) Laudau-Ginzburg model] Z · ¸ 1 ¸¡ ¢ H = d2x (E2 + B2) + j(r ¡ iA)Áj2 + jÁj2 ¡ c 2 (1) 2e2 |4 {z } V (Á) p e: gauge coupling, ¸: Higgs scalar coupling, v = hÁi = c local(=gauge) symmetry: Á(x) ! ei®(x)Á(x),A ! A + r®(x) 6 Magnetic flux is quantized to be integer. Vortex(winding) #(=vorticity) is given by 1st homotopy class: Z 2 d xB3 = 2¼c k; k 2 ¼1[U(1)] = Z: Abrikosov(’57) and Nielsen-Olesen(’73) (ANO vortices). B3⋆ g2c | | E 2 H⋆ √c g2c 2 g√c r g√c r 0 2 4 6 8 0 2 4 6 8 U(1) gauge symmetry is recovered in the core 7 e: gauge coupling, ¸: Higgs scalar coupling, v: VEV of scalar p p ¡1 ¡1 gauge mass: mv ' 2ev ) penetration depth: rv = mv ' ( 2ev) p ¡1 ¡1 scalar mass: ms ' ¸v ) coherence length: rs = ms ' (¸v) type range static force stability under B 2 type I rv < rs (2e > ¸) attractive force unstable 2 type II rv > rs (2e < ¸) repulsive force stable Abrikosov lattice 2 critical rv = rs (2e = ¸) non (! moduli dynamics) p a 1 2 type I type II 8 Critical coupling (Bogomol’nyi-Prasad-Sommerfield = BPS) Z · ¸ 1 ¸ ³ ´2 H = d2x B2 + j(r ¡ iA)Áj2 + jÁj2 ¡ c (2) 2e2 z 4 ¸ = 2e2 (critical) (à realized by Supersymmetry) Z · ¸ 2 2 1 2 2 2 2 H = d x j(@x ¡ iAx)Á + i(@y ¡ iAy)Áj + 2fBz + e (jÁj ¡ c) g Z 2e 2 +c d xBz Z 2 ¸ c d xBz = 2¼c k; k 2 Z (3) “=” , Bogomol’nyi bound (energy minimum) The most stable for a fixed vortex number k. The BPS equation (vortex equation) 2 2 (Dx + iDy)Á = 0;Bz + e (jÁj ¡ c) = 0 (4) 9 BPS solitons allow the moduli space Mk. 1. All possible configurations. 2. Dynamics/scattering = geodesic motion on the moduli space (geodesic/Manton approx.). 3. Collective coordinate quantization. 4. Integration over the instanton moduli space (Nekrasov). 5. Topological invariants (mathematics) The moduli space of ANO(Abelian) vortices E.Weinberg (’79) The index theorem counting zero modes: dim Mk = 2k. Taubes (’80) Rigorous proof of the existence and uniqueness of multiple vortex solutions. k The moduli space is symmetric product: Mk = C =Sk. Samols (’92) The moduli space metric. The right-angle (90 degree) scattering in head-on collisions. 10 The moduli space ) Dynamics If solitons move slowly there appear force between them. The moduli space describes classical dynamics of solitons, the scattering of solitons. The moduli (geodesic, Manton’s) approx. Soliton Scattering , Geodesics in Moduli Space ex.) For instance, a scattering of two BPS monopoles is described by a geodesic on the Atiyah-Hitchin metric. 11 Reconnection(intercommutation, recombination) of vortex-strings (in d = 3 + 1) is very important. 1. Essential process for (quantum) turbulence (Kolmogorov law) 2. superconductor, superfluid 4He. 3. Cosmic Strings When two cosmic strings collide with angle they may reconnect. Reconnection probability P is very important. P » 1 =) # density of strings is low. P » 0 =) # density is high (contradict to observation). 12 Many computer simulations have been performed: 1. local strings in the Abelian-Higgs model P » 1 (’80s) 2. semi-local strings P » 1 Laguna, Natchu, Matzner and Vachaspati, PRL[hep-th/0604177] Two different sizes vary to concide with each other. ) 3. non-intercommutation in high speed collision, P 6= 1 Achucarro and de Putter, PRD[hep-th/0605084] ) ) 13 analytical argument Right angle scattering of vortex-particles in head-on collisions m Copeland-Turok, Shellard (’88) Reconnection of vortex-strings A A0 final 0 0 DC BA B )( B0 C C0 0 A0 D D D B0 C 0 B C0 0 A D A A initial initial N J B0 B C C0 0 0 ABCD 0 final D D 14 interlude : How “non-Abelian” are non-Abelian vortices?? ¼1(G=H) ' ¼0(H) (5) Different definitions of “non-Abelian” vortices: (3 ) 2 ) 1) 1. G is non-Abelian ex) G = SU(N) with N adjoint Higgs H ' ZN: Abelian, ¼1(G=H) ' ZN: Abelian 2. H is non-Abelian à Our definition 3. ¼1(G=H) is non-Abelian ex1) biaxial nematics: SO(3) with 5 (sym.tensor) real Higgs SO(3)=K ' SU(2)=Q8 (Q8: quaternion), ¼1 ' Q8 ex2) spinor BEC (F = 2), cyclic phase: SO(3) £ U(1) with 5 (sym.tensor) complex Higgs [SO(3) £ U(1)]=T (T : tetrahedral) Kobayashi, Kawaguchi, MN and Ueda [arXiv:0810.5441] 15 a model for (p; q) web of cosmic strings Kobayashi, Kawaguchi, MN and Ueda [arXiv:0810.5441] 16 2 Knot soliton: ¼3(S ) ' Z Kawaguchi, MN and Ueda PRL [arXiv:0802.1968] cover 17 Plan of My Talk x1. Introduction: What are Vortices? (14+3 pages) x2. Non-Abelian Vortices: Review (13+5 pages) x3. Moduli Matrix Formalism (16+1 pages) x4. Conclusion / Discussion (2 pages) 18 x2. Non-Abelian Vortices: Review The non-Abelian extension has been discovered recently. Hanany-Tong (’03), Konishi et.al (’03) ² Vortices in the color-flavor locking vacuum. ² Each carries a non-Abelian magnetic flux. ² It is characterized by non-Abelian orientational moduli CP N¡1 (U(2) gauge ) CP 1 ' S2: sphere). ² Half properties of Yang-Mills instantons (on a NC R4). We call these non-Abelian vortices . 19 The non-Abelian Higgs model (bosonic part of N = 2 SUSY) U(N) gauge theory with N Higgs in the fund. rep. H (N £ N): " # 2 ³ ´ 1 ¹º ¹ y g y 2 L = Tr ¡ F¹ºF ¡D¹HD H ¡ c1 ¡ HH (6) NC 2g2 4 NC U(N) color(local) £ SU(N) flavor(global) symmetry. ¡1 H ! gC(x)HgF ;F¹º ! gC(x)F¹ºgC(x) (7) gC(x) 2 U(N); gF 2 SU(N) (8) p The system is in the color-flavor locking vacuum: H = c1N . U(N)C £ SU(N)F ! SU(N)C+F U(N) £ SU(N) U(1) £ SU(N) OPS : C F ' SU(N)C+F ZN 20 Vortex Equations The Bogomol’nyi bound for vortices: Z 1 2 2 E = dx dx (r.h.s of BPS eqs.) + Tvortices (9) Z ¸ Tvortices = ¡c dzdz¯ Tr F12 = 2¼c k; (10) k 2 N+ = ¼1[U(N)]: (11) The BPS equations (vortex equations): 0 = (D1 + iD2)H; (12) g2 0 = F + (c1 ¡ HHy): (13) 12 2 N cf.
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