Non-Abelian vortices and lumps; their moduli spaces and substructures

Transcription

1 Non-Abelian vortices and lumps; their moduli spaces and substructures Sven Bjarke Gudnason (Università di Pisa & INFN Sezione di Pisa) Theories of fundamental interactions Perugia June 25, 2010 In collaboration with: Roberto Auzzi, Minoru Eto, Toshiaki Fujimori, Yunguo Jiang, Kenichi Konishi, Takayuki Nagashima, Muneto Nitta, Keisuke Ohashi and Walter Vinci. arxiv: SBG arxiv: Konishi,SBG arxiv: SBG arxiv: Eto,Fujimori,SBG,Konishi,Nagashima,Nitta,Ohashi,Vinci arxiv: Eto,Fujimori,SBG,Konishi,Nagashima,Nitta,Ohashi,Vinci arxiv: Eto, Fujimori, SBG, Nitta, Ohashi

2 Plan of the talk Introductory part motivation crash course on vortices summary of recent results in the field Results SO, USp theories NLσM lumps fractional vortices Chern-Simons vortices

3 Motivation Confinement of quarks in Yang-Mills theory ( t Hooft- Mandelstam) Electric-Magnetic duality (Dirac, Montonen-Olive) Non-Abelian monopoles and GNOW-duality (Goddard- Nuyts-Olive-Weinberg) Condensed matter systems quantized vortices Fractional quantum Hall effect Chern-Simons theory Neutron stars Cosmic strings etc.

4 Dynamical Abelianization? Softly broken N = 2 pure SU(2) in Seiberg-Witten theory Abelian Generic N = 2 super-yang-mills (SYM) with quarks non-abelian Seiberg 1994, Seiberg-Witten 1994, Argyres-Plesser-Seiberg 1996 Hanany-Oz 1996, Carlino-Murayama-Konishi 2000 Question: magnetic monopoles of QCD are of Abelian or non-abelian type?

5 Non-Abelian monopoles Symmetry-breaking pattern: G H U(1), (1) Field strength tensor: F ij ɛ ijk x k r 3 (β T Cartan ). Charge quantization: 2β α = Z, (2) α is the root vector of H. Solution: β is any weight vector of the dual group H which has root vectors α = α α α. (3) The GNOW conjecture: non-abelian monopoles form multiplets of the dual group H. H U(N) SU(N) SO(2N) SO(2N + 1) SU(N) H U(N) Z Spin(2N) USp(2N) N

6 Unfortunately, there are some obstacles: 1. topological obstruction 2. non-normalizable zero-modes going to zero as r 1 2 we cannot quantize the non-abelian monopole The transformations in H and H, respectively are non-local. Problematic: Coulomb phase and (dual) Higgs phase. Strategy: H Higgs phase H confinement phase

7 Benchmark model G Λ H monopole Exact homotopy sequence:... π 2 (G) π 2 ( G H ) µ vortex 1, Λ µ. (4) π 1 (H) π 1 (G)... (5) Example: ( ) SU(3) 1 = π 2 (SU(3)) π 2 (6) SU(2) U(1) π 1 (SU(2) U(1)) π 1 (SU(3)) = 1.

8 Complete system no monopoles no vortices Paradox? G 1, (7) π 1 (G) = 1, (8) Solution: Vortices confine the monopoles Idea: vortex transformation = monopole transformation Auzzi-Bolognesi-Evslin-Konishi 2004 Auzzi-Bolognesi-Evslin-Konishi-Yung 2003

9 Plan of the talk Introductory part motivation crash course on vortices summary of recent results in the field Results SO, USp theories NLσM lumps fractional vortices Chern-Simons vortices

10 Abrikosov-Nielsen-Olesen vortex π 1 (S 1 ) = Z k : winding number = vorticity ξ : Fayet-Iliopoulos parameter theory on the Higgs branch

11 Derrick s theorem No finite energy scalar field configuration in more than one spatial dimension, other than the vacuum can have a stationary point d = 1 : domain wall OK d = 2 : vortex stabilized by flux V = 0 sigma model lumps harmonic maps J 0 Q-lumps

12 The vortex stabilized by magnetic flux Competition of forces: scalar field attractive force magnetic field repulsive force β m Higgs m γ classifies the vortices into: β < 1 : type I not experimentally observable as the flux attracts and breaks the superconducting phase β > 1 : type II Abrikosov lattice β = 1 : BPS supersymmetry preserving

13 Abrikosov lattice

14 Non-Abelian embedding U(N) theory with N F = N flavors: q 11 q 12 q 1N q q = 21 q 22 q 2N....., q = 1 N (9) q N1 q NN Transformation U flavor = U color global symmetry. Embedding of ANO in U(N) theory: q ANO ξ 0 q =....., A i = 0 ξ q U color qu flavor, (10) i A ANO

15 A whole family of solutions appears!

16 Non-Abelian moduli Color-flavor rotation: q ANO ξ 0 q = U..... U, (11) 0 ξ A ANO i A i = U..... U. 0 0 U G color+flavor global symmetry. Non-Abelian moduli parametrize: SU(N) SU(N 1) U(1) CP N 1. (12) effective world-sheet symmetry.

17 Moduli space the non-abelian vortex U(N) theory: M k = ( C CP N 1 ) k S k Hanany-Tong 2003, Auzzi-Bolognesi-Evslin-Konishi-Yung 2003, Shifman-Yung 2004

18 Plan of the talk Introductory part motivation crash course on vortices summary of recent results in the field Results SO, USp theories NLσM lumps fractional vortices Chern-Simons vortices

19 Summery of recent results A duality between 4 dim SYM and a 2 dim non-linear σ model, realized by the non-abelian vortex SO(5) U(2) 1: k = 2 vortices. k = 2 moduli space of U(2) vortices: Hanany-Tong 2003, Shifman-Yung 2004 regular monopoles are confined by CP 1 CP 1 W CP 1 2,1,1, Hashimoto-Tong 2005, Auzzi-Shifman-Yung 2005, Eto-Konishi-Marmorini-Nitta-Ohashi-Vinci-Yokoi 2006 The vortex-monopole system transforms under the representation of SU(2). Weinberg 1982, Lee-Weinberg-Yi 1996, Kampmeijer-Slingerland-Schoers-Bais 2008

20 Group theory of vortices: single vortex : k vortices k i=1. The full moduli space: Reconnection of cosmic strings In preparation Eto-Isozumi-Nitta-Ohashi-Sakai 2005 Hanany-Hashimoto 2004, Hashimoto-Tong 2005, Eto-Hashimoto-Marmorini-Nitta-Ohashi-Vinci 2006 D-brane solitons in field theory, e.g. an instanton-monopolevortex-domain wall system Tong 2005, Eto-Isozumi-Nitta-Ohashi-Sakai 2006, Shifman-Yung 2007 A Seiberg-like duality of non-abelian semi-local vortices Eto-Evslin-Konishi-Marmorini-Nitta-Ohashi-Vinci-Yokoi 2007

21 Non-BPS non-abelian vortex interactions: distancedependent forces (type I/I and type II/II ) Non-Abelian vortices on a torus: Non-Abelian vortices in dense QCD: Auzzi-Eto-Vinci 2007 Lozano-Marques-Schaposnik 2007 Nakano-Nitta-Matsuura 2007, Eto-Nitta 2009, Eto-Nakano-Nitta 2009, Eto-Nitta-Yamamoto 2009, Yasui-Itakura-Nitta 2001 A model of non-abelian vortices without dynamical Abelianization The stability of non-abelian semi-local vortices Dorigoni-Konishi-Ohashi 2008 Auzzi-Eto-SBG-Konishi-Vinci 2008

22 Multi-layer structure of non-abelian vortices: Non-Abelian global vortices: Eto-Fujimori-Nitta-Ohashi-Sakai 2009 Vortex description of quantum Hall ferromagnets Eto-Nakano-Nitta 2009 Kimura 2009 Non-Abelian vortex-string dynamics from non-linear realization: Non-Abelian vortices in mass-deformed ABJM model: Liu-Nitta 2009 Kim-Kim-Kwon-Nakajima 2009, Auzzi-Kumar 2009 Quantum Phases of a vortex string in N = 1 : Auzzi-Kumar 2009

23 Non-Abelian Chern-Simons vortices with generic gauge groups: Moduli space metric for non-abelian vortices Compact Riemann surfaces: SBG 2009 Well-separated vortices: Baptista 2010 Fujimori-Marmorini-Nitta-Ohashi-Sakai 2010 Low-energy U(1) U Sp(2M) gauge theory from simple high-energy gauge group: SBG-Konishi 2010

24 D-branes in rotating phase-separated two-component Bose- Einstein condensates Kasamatsu-Takeuchi-Nitta-Tsubota 2010

25 Plan of the talk Introductory part motivation crash course on vortices summary of recent results in the field Results SO, USp theories NLσM lumps fractional vortices Chern-Simons vortices

26 The model general gauge group H theory: N = 2 (8 supercharges), U(1) G super-yang-mills (4 dimensions) with N F massless hyper multiplets. { } vector multiplet : V, Φ { hyper multiplets : Q, Q} Truncated model, formally N = 1, U(1) G SYM with N F quarks Q. [ L = Tr d 4 θ ( Q e V Q + ξv ) + 1 ] g 2 d 2 θ (W W + h.c.), ξ > 0 FI parameter. L = 1 4e 2F µνf 0 0,µν 1 4g 2F µνf a a,µν + Tr D µ H (D µ H) e2 2 Tr HH t 0 t a generators of G. v2 2 2N g2 2 Tr HH t a 2, (13)

27 Bogomol nyi bound: BPS equations T v2 2N saturated by BPS-equations DH = 0, F 0 12 = e 2 ( Tr HH t 0 ν topological charge. C F 0 12 = 2πv 2 ν, (14) ) v2 2N, (15) F a 12 = g 2 Tr HH t a, (16)

28 Moduli matrix Solution: holomorphic matrix + complexified gauge transformations: H = S 1 (z, z) H 0 (z), (17) with z = x 1 +ix 2, holomorphic N N F matrix: moduli matrix, encodes all the moduli. S = ss, s C, S G C. (18) DH = 0, Ā = is 1 S. (19) Residual symmetry V -equivalence: (S, H 0 ) V (z) (S, H 0 ). (20)

29 Master equations DH = 0 solved. Ω = ωω = SS, Ω 0 = H 0 (z)h 0 (z): For SU(N) [ ] log ω = e2 1 4N ω Tr Ω 0Ω 1 v 2, (21) (Ω ) [ ] Ω 1 = g2 Ω 0 Ω 1 1 N 4ω N Tr Ω 0Ω 1, (22) SO, USp: (Ω ) Ω 1 = [Ω ( ) ] g2 0 Ω 1 J Ω 0 Ω 1 T J. (23) 8ω Existence and uniqueness is assumed.

30 Asymptotic behavior: s(z, z) z ν. (24) Holomorphic G -invariants: I i G ( H = s 1 S 1 H0 ) = s n i I i G (H 0 ), (25) Boundary conditions: Single valuedness condition Solution: I i G (H 0 ) z = I i vevz νn i, (26) νn i Z +, (27) ν = k n 0, k Z +, (28) n 0 gcd { n i I i vev 0 }. G = U(1) G Z n0. (29)

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32 A few groups G C G ν SU(N) Z N k/n SO(2N + 1) 1 k USp(2N) Z 2 k/2 SO(2N) Z 2 k/2 E 6 Z 3 k/3 E 7 Z 2 k/2 E 8 1 k F 4 1 k G 2 1 k

33 Examples Holomorphic invariant conditions: G = SU(N): det H 0 (z) = z k + O ( z k 1), (30) G = SO(2N), USp(2N): meson field: Invariant tensor: M = H T JH, (31) J SO(2N) σ 1 1 N, J USp(2N) iσ 2 1 N, (32) Holomorphic invariant conditions: G = SO(2N + 1): H T 0 JH 0 = z k J + O ( z k 1). (33) H T 0 JH 0 = z 2k J + O ( z 2k 1). (34)

34 Special points H 0 (z) = z ν1 N+ν α H α U(1) C G C, (35) H α : Cartan generators of g, α = 1,..., rank(g ) Single valued condition: ν = k n 0. (ν1 N + ν α H α ) ii Z 0 i, ν + ν α µ (j) α Z 0 i, µ (j) α : weight vector of G.

35 Quantization with respect to the dual group Quantization condition: ν α (j) Z, α (j) : root vectors of G. Formally Goddard-Nuyts-Olive-Weinberg quantization condition. Solution: ν = 2 µ, (36) µ: any weight vector of dual group, dual group s root vectors α = α α α. (37) No conceptual problems in quantizing zero-modes due to exact color+flavor symmetry.

36 Z 2 parity for G = SO(N) First homotopy group ( ) U(1) SO(N) π 1 Z n0 = Z Z 2, Moduli space is disconnected: { n0 = 1, N odd n 0 = 2, N even (38)

37 Special points for k = 1 vortex with G = SO, USp SO(2) USp(2) SO(3) ( 1 2, 1 2 ) ( 1 2, 1 2 ) (1, 1) (0, 0) (1, 0) SO(4) USp(4) SO(5)

38 ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 ) SO(6) USp(6) Dark grey points have positive Z 2 -charge, white ones have negative.

39 Moduli space of vortices The moduli space of vortices: completely described by moduli matrix Some examples: k = 1 M USp(2M) = C USp(2M) U(M) ( M SO(2M) = C SO(2M) ) U(M) M = {H 0 (z)} /U(N) C. (39), (40) + ( C SO(2M) ) U(M), (41)

40 Connectedness of the moduli space SO(2) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1 2, 1 2 ) SU(2)/U(1) CP 1 USp(2) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1 2, 1 2 ) SU(2)/U(1) + SO(4) USp(4) ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 ) ( 1 2, 1 2, 1 2 )

41 The k = 1 odd SO k = 2 even: (1, 1, 1) 1 0 SO(2) USp(2) 1 (0, 0, 0) (1, 1, 0) (1, 0, 0) (1, 1) (1, 1) SO(6) (1, 0) (1, 0) (0, 0) (0, 0) (1, 1, 1) SO(4) USp(4) (0, 0, 0) (1, 1, 0) (1, 0, 0) USp(6)

42 Differences ( 1, 1) (0, 1) (1, 1) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1, 0) (0, 0) (1, 0) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1, 1) (0, 1) (1, 1) SO(4) SO(5) k = 2, SO(2) k = 2, USp(2) k = 1, SO(3) (CP 1 ) 2 /S 2 CP 1

43 SO(4) higher windings k = 4 k = 2 k = 3 k = 5 k = 1

44 Plan of the talk Introductory part motivation crash course on vortices summary of recent results in the field Results SO, USp theories NLσM lumps fractional vortices Chern-Simons vortices

45 Existence and uniqueness 1. Result of our index calculation with generic gauge group # bosonic zero-modes # moduli in H 0 (z) 2. Strong gauge coupling limit non-linear σ-model analytic solution is obtained

46 Strong gauge coupling limit SU(N): 0 = 1 ω Tr Ω 0Ω 1 v 2, (42) SO, USp: 0 = Ω 0 Ω 1 1 N N Tr Ω 0Ω 1, (43) ( ) 0 = Ω 0 Ω 1 J Ω 0 Ω 1 T J, (44) Solution for SU(N): ω = N v 2 (det Ω N 0) 1, Ω = (det Ω 0 ) N 1 Ω 0, (45) Solution for SO, USp: (M H T 0 (z)jh 0 ) ω = 1 v 2Tr M M, 1 N Ω = H 0 (z) M M H 0 (z). (46)

47 Lumps Low energy NLσM stringy lumps H 0 (z) target space High energy YM-Higgs semi-local vortices H 0 (z) Higgs branch The NLσM is integrable. Except for the local vortex which is mapped to a point small lump singularity.

48 Kähler quotient U(N): SO, USp: L = Tr L = Tr d 4 θ { QQ e V + ξv }, (47) d 4 θ ξ log det QQ, (48) } d 4 θ {QQ e V e V e + ξv e, (49) e V SO, USp. Difficult calculation.

49 Kähler quotient for SO, USp Relax the algebra e V SL(N, C), introduce Lagrange multipliers λ. { ( ) } L = Tr d 4 θ QQ e V e V e + λ e V T Je V J + ξv e d 4 θ ξ log Tr MM, (50), M = Q T JQ is the meson field. A similar construction has been made for the hyper-kähler case (N = 2).

50 Plan of the talk Introductory part motivation crash course on vortices summary of recent results in the field Results SO, USp theories NLσM lumps fractional vortices Chern-Simons vortices

51 Fractional vortices of the first type

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53 Fractional vortices of the second type target space (vacuum moduli) deform lump configuration split

54 The droplet model U(1) A 2 B 1 H = ( ) A B. (51) B = 0 ϕ = ϕ vev 1 v 1 v2 2 φ(z) = φ vev (z z S 1)(z z S 2) (z z N 1 )2. (52) ϕ vev 1 v 1 v2 2 ϕ = 0 A = 0

55 Ú Ú ¾ Ú Ú ¾ µ ¼ ½µ Ú Ú ¾ Ú Ú ¾ µ ½ µ Ú Ú ¾ Ú Ú ¾ µ ½ ½ ¾µ

56 Ú Ú ¾ Ú Ú ¾ µ ½ µ Ú Ú ¾ Ú Ú ¾ µ ½ ¾ ½ ½¼¼¼¼µ Ú Ú ¾ Ú Ú ¾ µ ½ ¾ ¼µ

57 Example: SO(6) theory

58 Plan of the talk Introductory part motivation crash course on vortices summary of recent results in the field Results SO, USp theories NLσM lumps fractional vortices Chern-Simons vortices

59 Yang-Mills-Chern-Simons-Higgs theory N = 2 (4 supercharges) in d = dimensions with the gauge group G = U(1) G, where G is a simple group. L YMCSH = 1 ( ) F a 2 1 ( ) 4g 2 µν F 0 2 κ 4e 2 µν 8π ɛµνρ A 0 µ ν A 0 ρ µ 8π ɛµνρ ( A a µ ν A a ρ 1 3 f abc A a µa b νa c ρ + 1 2g 2 (D µφ a ) e 2 ( µ φ 0) 2 + Tr (Dµ H) (D µ H) Tr φh Hm 2 g2 2 e2 2 ) ( Tr ( HH t a) µ 4π φa) 2 ( Tr ( HH t 0) κ 4π φ0 1 2N ξ) 2,

60 Integrate out the adjoint fields Strong gauge coupling limit e, g : φ a = 4π µ Tr ( HH t a), (53) φ 0 = 4π κ 1 2N [ Tr ( HH ) ξ ]. (54)

61 Chern-Simons-Higgs theory L CSH = µ 8π ɛµνρ ( A a µ ν A a ρ 1 ) 3 f abc A a µa b νa c ρ κ ( ) 8π ɛµνρ A 0 µ ν A 0 ρ + Tr (D µ H) (D µ H) { 4π 2 Tr 1N ( ( Tr HH ) ξ ) + 2 Nκ µ Tr ( HH t a) } t a H (55) 2,

62 Master equations [Ω Ω 1 ] = 2π2 Nκµ ( Tr ( Ω0 Ω 1) ξ ) Ω 0 Ω 1 J + π2 µ 2 (Ω0 Ω 1) 2 J, (56) log ω = 4π2 N 2 κ 2Tr ( Ω 0 Ω 1) ( Tr ( Ω 0 Ω 1) ξ ) 2π2 Nκµ Tr ( Ω 0 Ω 1 Ω 0 Ω 1 J ). (57) X J X J X T J.

63 F 0 12 c b a c b a r r e (ψ+χ)/ e (ψ χ)/ r κ=4, µ=2 κ=2, µ=2 κ=1, µ=2 Energy density F 12 NA r κ=4, µ=2 κ=2, µ=2 κ=1, µ= r κ=4, µ=2 κ=2, µ=2 κ=1, µ=2

64 E r r κ=4, µ=2 κ=4, µ=2 κ=2, µ=2 κ=1, µ=2 E r NA κ=4, µ=2 κ=2, µ=2 κ=1, µ= r κ=1, µ=2 Magnetic field κf 12 NA µf 12 Magnetic field κf 12 0 µf 12 NA r r

65 Abelian magnetic field Non Abelian magnetic field Abelian magnetic field Non Abelian magnetic field

66 Opposite sign of couplings F 0 12 c b a c b a e (ψ χ)/ r r e (ψ+χ)/ r κ= 4, µ=2 κ= 2, µ=2 κ= 1, µ=2 Energy density F 12 NA κ= 4, µ=2 κ= 2, µ=2 κ= 1, µ= r κ= 4, µ=2 κ= 2, µ=2 κ= 1, µ= r

67 E r κ= 4, µ=2 κ= 2, µ=2 κ= 1, µ= E r NA κ= 4, µ=2 κ= 2, µ=2 κ= 1, µ= r r

68 Fractional Chern-Simons vortex SO(2M): H 0 = z z 1 c z z M c M, (58) Ω = diag ( e χ 1,..., e χ M, e χ 1,..., e χ M ), (59) ω = e ψ. (60)

69 δχ m = 2 c m 2 z 4 m 2 µ Asymptotic profiles [ + 2 c m 2 ( zm m 2 3 µ z + z ) m 1 M ( zn z M z + z ) ] n z 4 + O ( z 6) z n=1 ( δψ = 1 M ( zn Mm c n 2) 1 M 2 ) z n z 4 κ M n=1 n=1 [ M ( zn Mm c n 2) ( z n κ 2 z + z ) n z n=1 ( 1 M z 2 M M M ) n z n 2 z n + z n 2M z z + 1 M n=1 ( M n=1 + O ( z 6), n =1 n=1 ( zn c n 2) 1 M n =1 M n=1 z n 2 ) M n =1 ( zn z + z ) ] n z 4 z

70 Effective size c effective 2 = 1 2M M n=1 ( zn c n 2) 1 2M 2 M n=1 z n 2,

71 Figure fractional Chern-Simons vortex

72 Future developments monopole-vortex systems the GNOW-duality group theory of vortices (in preparation) Yang-Mills-Chern-Simons-Higgs (in preparation) non-bps corrections / stability Q-lumps in SO, USp theories quantum corrections to the N = 1 Kähler quotients Ricci-flat Calabi-Yau metrics D-brane constructions knotted solitons quantized vortices domain wall systems in SO, USp theories the mass deformed theories etc.

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