Nonabelian Gauge Theories
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(2) M-Theory S-D ua T-d lity ual ity nes. AD. S-. CF. T. Bra. 4D Field-Theory EM-Duality Di. CFT. Statistical Mechanics. Maxwell Equations ra. c. High T-Low T Duality Kink- Spin. Nonabelian Gauge Theories. QCD, Electroweak Theory Integrable systems. Supersymmetry. Exact SW Solution / Quantum Monopoles. Confinement. 2.
(3) EM Duality in 4D Gauge Theories. 2.
(4) Electro-Magnetic Duality ∇ · (E + i B) = 0;. ∇ × (E + i B) = i. ∂ (E + i B). ∂t. • Invariant under (E + i B) → ei φ (E + i B). (1). • Include E → B, B → −E ; • EM duality broken by charges ∇ · E = g δ 3(r);. ∇ · B = 0;. g + i gm → ei φ (g + i gm) • Magnetic monopoles possible if (Dirac, 1931) n g gm = , n = 0, ±1, ±2, . . . 2. (cfr. Eq.(2)). • Origin of the electric charge quantization ? • Dyons (gm 1, ge 1) , (gm 2, ge 2) are also possible if ge 1 gm 2 − ge 2 gm 1 = 3. n . 2. (2). (3).
(5) Nonabelian Gauge Symmetry • Invariance under ψ(x) → ei α(x) ψ(x),. ei α(x) ∈ U (1). leads to Electrodynamics (Gauge Principle). ∂µ ⇒ Dµ = ∂µ − i e Aµ(x),. 1 Aµ → Aµ + ∂µα e. • Invariance under ψi(x) → [ ei α. a (x)T a. ]ij ψj (x),. ei α. a (x)T a. =U ∈G. G = SU (2), SU (3), SO(10) etc., lead to Yang-Mills theories (Yang, Mills 1954) 1 2 ¯ L = − Fµν + ψ i γµDµ ψ 4. ∂µ ⇒ Dµ = ∂µ −i g Aaµ(x) ta,. 1 Aµ → U (Aµ + ∂µ)U † g. • G = SU (3), ψi = quarks: Quantum Chromodynamics (Theory of strong interactions): Asymptotic freedom, Confinement, Dynamical Symmetry Breaking 4.
(6) Spontaneously Broken Symmetries • Sp. Broken global symmetry ⇒ Nambu-Goldstone particles (pions ) • SSB of local gauge symmetry ⇒ Higgs mechanism L = |Dµφ|2 + . . . ⇒ g 2|φ|2 A2µ + . . . ,. MA = gφ. (4). • ⇒ Electroweak theory of Glashow-Weinberg-Salam • Standard Model of fundamental interactions (1974) SU (3)QCD × (SU (2)L × U (1))GW S • Spectrum of the theory ∼ elementary quanta and composite hadrons (γ, W ±, Z, leptons, Higgs, mesons and baryons ) • LHC (≥ 2007): Higgs, Extra Dimensions, Grand Unification, etc. • In more general Spont. Broken Gauge Theories (’t Hooft-Polyakov 1974) Spectrum of the theory ∼ elementary quanta and soliton magnetic monopoles. 5.
(7) Regular Monopoles in Nonabelian Gauge Theories Giorgi-Glashow model. φ=0. SU (2) −→ U (1) 1 1 H = d3x (Fija )2 + (Diφa)2 + V (φ) 4 2 has finite energy soliton solutions (’t Hooft-Polyakov): r→∞. Dφ −→ 0,. ⇒. φ ∼ U · φ · U. −1. ;. • Mass and a quantized magnetic charge φ M≥ = gm φ, g. Aai. ∼ (∂i n × n) ,. 1 gm = , g. a. ra n (r) = r a. ⇔ (4). (5). • Bogomolnyi-Prasad-Sommerfield (BPS) limit: Fija − &ijk Dk φa = 0, • For (BPS) dyons. 2 = v |g + i g | M (g, gm) = v g 2 + gm m. ( cfr. Eq.(2), Eq.(4) ) Symmetric for elementary quanta and solitons! 6. (6).
(8) Soliton. Elementary particles. They are actually not so dissimilar! • Symmetry [ solitons ⇔ elementary excitations ] in 4D particularly elegant in theories with Supersymmetry 7.
(9) Topology (Mapping: Space → G) • Charged particle ψ(x) in a monopole field Ai dxi → exp ige dS · H = exp 4πige gm exp ige S2. ∂Ω. 2 ge gm = n,. n ∈ Z,. (H = ∇. gm ). r. Π1(U (1)) = Z. • U (1) in a nonabelian theory G (Wu, Yang): monopole ∼ Π1(G) = ∅. • ’t Hooft-Polyakov monopoles: Π2(SU (2)/U (1)) = Π1(U (1)) = Z; φ=0. • G −→ H : similar → monopoles with nonabelian charges if Π2(G/H) = ∅. 8.
(10) Topology and Confinement: SU (N ) YM. Q. Q ZN. Confinement. M. ZN. SU(N)/ZN) ~ ZN. 9.
(11) Phases of SU (N ) YM (M ). (E). • (ZN , ZN ) classification (’t Hooft): If field with x = (a, b) condense, particles X = (A, B) with x, X ≡ a B − b A = 0 (mod N ) are confined. (e.g. φ(0,1) = 0 → Higgs phase.) 0 → Confinement: What is χ ? • If χ(a,0) = QCD as Dual Superconductor (’t Hooft) (1981) • ∃ Elementary/soliton monopoles • Monopoles as topological singularities (lines in 4D) of Abelian gauge fixing, SU (3) → U (1)2: Gauge dependence? • Dynamical SU (3) → U (1)2 Breaking? • In Nature and in QCD, Meson ∼ N i=1 | qi q¯i : no dynamical abelianization; • Dynamics hard to analyse 10.
(12) Supersymmetry. 11.
(13) Supersymmetry Symmetry under ¯ Q,Q. |Boson ⇐⇒ |F ermion ¯ α˙ } = 2E δαα˙ {Qα , Q • H ≥ 0 ⇒ Solution to Λcosm 0 ? • Solution of the Naturalness problem ( MMW. P lanck. ∼ 10−16). • No Experimental Evidence Yet (MSSM → LHC (≥ 2007) ? ) • Extra dimensions ? • Solvable Theories of Strong Interactions (cfr. Ising model) ⇒ Mechanism of Confinement and Dynamical Symmetry Breaking • “A truely beautiful idea never really dies... ” (Nambu) • Many exact nonperturbative results (1985-1994). 12.
(14) Complex Structure of Susy 4D Gauge Theories • Chiral superfields ¯ = φ(y) + Φ(x, θ, θ). √. 2 θ ψ(x) + θθF (y),. y = x + iθσ θ¯. • Verctor superfields V † = V, 1 ¯ 2 −V µ Wα = − D e Dα eV = −iλ + (σ µ σ ¯ ν )βα Fµν θβ + . . . 4 2 • Supersymmetric Lagrangian ( dθ1 θ1 = 1, etc) 1 4 † V 2 1 α Im τcl d θ Φ e Φ + d θ Wα W + d2 θ W (Φ) L= 8π 2 • W (Φ) = superpotential;. τcl =. θ 2π. +. 4πi g2. • Potential (F term and D term). 2. ∂W 2 1 . ∗ a. φ t φ. Vsc =. ∂φ + 2. mat a mat. Vacuum Degeneracy (Space of Vacua) • Non-renormalization Theorem (perturbative) (cfr. Anomaly) • Q: Superpotential Dynamically Generated ? 13.
(15) Phases of SQCD; SCFT and Seiberg’s Duality ˜ i = (˜ W = (Aµ, λ), Qi = (qi, ψq i), Q qi, ψq˜ i) Nf. Deg.Freed.. Eff. Gauge Group. Phase. Symmetry. 0 (SYM). -. -. Confinement. -. 1 ≤ Nf < Nc. -. -. no vacua. -. Nc. ˜ M, B, B ˜ M, B, B. -. Confinement. U (Nf ). -. Confinement. Unbroken. q, q˜, M. ˜c ) SU (N ˜c ) or SU (Nc ) SU (N. Nc + 1 Nc + 1 < Nf < 3Nc 2. 3Nc 2. Nf = 3Nc. ˜ q, q˜, M or Q, Q ˜ Q, Q. Nf > 3Nc. ˜ Q, Q. < Nf < 3Nc. Free-magnetic Unbroken SCFT. Unbroken. SU (Nc ). SCFT (finite). Unbroken. SU (Nc ). Free Electric. Unbroken. ˜ c ≡ Nf − N c N. g* g*D. 14.
(16) Seiberg-Witten Solution (1994) of N = 2 Theories SU(2). u. Massless (1,1) dyon. u. Massless (1,0) monopole. u. = -. u = < Tr. 2>. =. Semi-classical. u. • W = (Aµ, λ), Φ = (φ, ψ): vacua at Φ = Lef f. a. 0. =.
(17). ⇒ −a 2 (A) F (A) ∂F 1 ∂ p p = Im d4θ A¯ Wα W α + d2θ 2 ∂A 2 ∂A 0. • Shape parameter of auxiliary curve ∼ τef f = 15. dAD dA. =. ∂ 2 Fp (A) ∂A2.
(18) Breakthrough: • Exact mass formula (BPS) ( N = 2 Susy algebra ) : √ cf r. (6) Mnm,ne = 2 | nmAD + neA | ˜; • At the singularities u = ±Λ2, massless monopoles M, M • Theory invariant under SL(2, Z) ( ad − bc = 1 ): ( ⊃ EM Duality ).
(19) δL
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(21).
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(23) δL
(24). a b a b AD AD + + δFµν δFµν → → , + + c d A c d A Fµν Fµν • Dirac’s condition (3) ⇒ τD · τ = −1, where τ =. θ 2π. + 4πi g2. • Complete nonperturbative (instanton) effects encoded in the curve dx dA D y 2 = (x − u)(x + Λ2)(x − Λ2), = , etc. du y α • Mass term (µ) for Φ induces M =. . µΛ. Confinement of “electric” charges (dual ANO Vortex) 16.
(25) Seiberg-Witten Curves of SU (N ), U Sp(2N ), SO(N ) • SU (N ) with Nf Quarks Nf N. y2 = (x − φi)2 − Λ2N −Nf (x + ma); a=1. i=1. • SO(N ) with Nf Quarks in Vector Reprentation. Nf. (x − φ2i )2 − 4Λ2(N −2−Nf )x2+& (x − m2a),. [N/2]. y2 = x. a=1. i=1. where & = 1 if N is even; & = 0 if N is odd.. Genus R(= Rank of Gauge Group) Hypertorus. • “After Seiberg-Witten, everybody is more intelligent than used to be ....” 17.
(26) Vacua of N = 2 Theories SU(n) Theory with nf Quarks. USp(2n) Theory with nf Quarks Higgs Branches. (Non-baryonic) Higgs Branches. Special Higgs Branch. Baryonic Higgs Branch <Q> <Q>. r = nf /2. -- -. SCFT SCFT. r=1 r=0. Non Abelian monopoles. Dual Quarks. Non Abelian monopoles. Dual Quarks. Abelian monopoles. Coulomb Branch. Coulomb Branch N=1 Confining vacua (with N=1 vacua (with. perturbation). perturbation) in free magnetic phase. • Gef f ∼ SU (r) × U (1)nc−r−1; Confining for r ≤. nf 2 ;. SCFT at r =. nf 2. • U Sp(2nc): All confining vacua ( with µ Φ2 ) are SCFT, with global SO(2 nf ) → U (nf ) Symmetry Breaking (cfr ψ¯ ψ(QCD) = 0) 18.
(27) Phases of Softly Broken N = 2 Gauge Theories ˜ i = (˜ W = (Aµ, λ), Φ = (φ, ψ), Qi = (qi, ψq i), Q qi, ψq˜ i) label (r). Deg.Freed.. Eff. Gauge Group. Phase. Global Symmetry. 0 (NB). monopoles. U (1)nc −1. Confinement. U (nf ). 1 (NB). monopoles. U (1)nc −1. Confinement. U (nf − 1) × U (1). SU (r) × U (1)nc −r. Confinement. U (nf − r) × U (r). n −1 2, .., [ f2 ]. (NB) NA monopoles. nf /2 (NB). rel. nonloc.. -. Confinement. U (nf /2) × U (nf /2). BR. NA monopoles. SU (˜ nc ) × U (1)nc −˜nc. Free Magnetic. U (nf ). Table 1:. 1st Group. Phases of SU (nc ) gauge theory with nf flavors. n ˜ c ≡ nf − nc .. Deg.Freed.. Eff. Gauge Group. Phase. Global Symmetry. rel. nonloc.. -. Confinement. U (nf ). 2nd Group dual quarks U Sp(2˜ nc ) × U (1)nc −˜nc Free Magnetic Table 2:. SO(2nf ). Phases of U Sp(2nc ) gauge theory with nf flavors with mi → 0. n ˜ c ≡ nf − nc − 2.. 19.
(28) Confining Vacua in Softly-Broken N = 2 Theories • Abelian dual superconductor (’t Hooft, Mandelstam), always accompanied by Dynamical abelianization SU (N ) → U (1)N −1 ( enrichment of the meson spectrum ); • Nonabelian dual superconductor (r - vacua with Gef f = SU (r)×U (1)N −r ). In particular, “dual quarks” = nonabelian monopoles (cfr. Seiberg’s dual quarks) • Nonabelian dual superconductor with relatively nonlocal dyons. The theory is near a strongly-coupled superconformal point (SCFT). Nonlocal theory: difficult to analyse • Both at generic r - vacua and at the SCFT vacua, Miα = δαi v = 0,. (α = 1, 2, . . . , r;. (“Color-Flavor-Locking” phase ). 20. i = 1, 2, . . . , Nf ).
(29) “Almost-Superconformal” Confining Vacua Auzzi, Grena, Konishi ’02. An example: “r = 2” Vacua of N = 2, SU (3) theory with Nf = 4 • Gef f = SU (2) × U (1) • Low energy effective deg. freedom = 4 monopole doublets, 1 dyon doublet, 1 electric doublet • Nonlocal cancellation of beta function (τ ∗ = 0 • µ = 0 → M1 M4 = Matrix. gm1, gm2; ge1, ge2. M1, M4 (±1, 1, 0, 0)4 A3 , A6. (±2, −2, ±1, 0). M3, M6 (0, 2, ±1, 0) Table 3:. 21. −1+i 2 )..
(30) Duality and Dynamics Instantons g( ). ~ bD log Asymptotic Freedom Quark loops. SCFT ~ gD(. 1 b log. 1 bD log. Monopole loops (IR-freedom). • cfr. Old puzzle ge(µ) · gm(µ) = n2 ,. ∀µ. • EM duality two different languages to describe the same theory in appropriate regions, rather than being a symmetry (cfr. N = 4 ) • Limiting case, SCFT. Hardest but might be most relevant for QCD 22.
(31) Non-Abelian Superconductor. QCD Vacuum. =. Non-Abelian Dual Superconductor. Definition • Both condensing entity ( e.g. monopoles∗ ) and confined entity ( e.g. quarks∗ ) carry non-Abelian gauge charges; • Confining ( e.g. electric∗ ) vortex also carries a non-Abelian flux •. ∗. Refer to dual superconductors; in the non-Abelian generalization of the standard superconductors in Nature, monopoles ⇔ quarks,. electric ⇔ magnetic. 23.
(32) Quantum Nonabelian Monopoles. 24.
(33) Nonabelian Monopoles Basic results. Goddard-Nuyts-Olive-E.Weinberg φ=0. G −→ H φ ∼ U · φ · U −1 ∼ Π2(G/H) = Π1(H); rj Aai ∼ U · ∂iU † → &aij 3 βk Tk , Ti ∈ Cartan S.A. of G r Topological quantization =⇒ 2 α · β ⊂ Z (α = root vectors of G) r→∞. Dφ −→ 0,. ⇒. (cfr. Eq.(3)) ˜ (= dual of H). βi = weight vectors of H Dual Group: Def. α ˜ = α/α · α SU (N )/ZN ⇔ SU (N ) SO(2N ) ⇔ SO(2N ) SO(2N + 1) ⇔ U Sp(2N ). ˜ • Monopoles “live” in H 25. (7).
(34) Simple Example: SU (2) × U (1) SU (3)−→ , Z2 0 1 1 S = 0 2 1. 0 0 0. 1 0; 0. . 0 1 2 S = 0 2 i. − 12 v. φ(r) = 0 0. v 0. 0. φ = 0 v. φ. 0 0 0. −i 0 ; 0. . 0 0 0 −2v. 1 1 3 S = 0 2 0. 0 0 0. 0 0 −1. 0. 0. ˆ v 0 + 3 v S · rˆφ(r), 0 − 12 v. @ A(r) = Sˆ ∧ rˆA(r), φ(r) and A(r) are BPS- ’t Hooft’s fnc with φ(∞) = 1, φ(0) = 0, A(∞) = −1/r. ⇒ two degenerate SU (3) solutions (1) )=Z Topology: Π1( SU (2)×U Z2. 26. (8).
(35) Quantum Numbers of NA Monopoles G. H. Dual Group. Irrep. U (1). SU (N + 1). SU (N ) × U (1)/ZN. SU (N ) × U (1). N. 1/N. U Sp(2N + 2). U Sp(2N ) × U (1). SO(2N + 1) × U (1). 2N + 1. 1. U Sp(2N ) × U (1). 2N. 1. SO(2N + 3) SO(2N + 1) × U (1) SO(2N + 2). SO(2N ) × U (1). SO(2N ) × U (1). 2N. 1. U Sp(2N ). SU (N ) × U (1)/ZN. SU (N ) × U (1). N. 1/N. SO(2N ). SU (N ) × U (1)/ZN. SU (N ) × U (1). N (N − 1)/2 2/N. SO(2N + 1). SU (N ) × U (1)/ZN. SU (N ) × U (1). N (N + 1)/2 2/N. Table 4:. 27.
(36) Do Nonabelian Monopoles Really Exist? • No “Colored dyons” exist. (e.g. SUGU T (5) → SUcolor (3) × SU (2) × U (1)). ˜ group • Monopoles ∼ multiplets of the dual H ˜ • The no-go theorem → Ggauge = H ⊗ H • NA monopoles never really semi-classical, even if φ ' ΛH : ˜ - ONLY if H unbroken ⇒ nonabelian monopoles in irreps of H. • They appear in the r vacua of N = 2 SQCD with Gef f = SU (r)×U (1)nc−r+1; • The r-vacua only for r < (dual). b0. nf 2. ⇐ Sign-flip of the beta function:. ∝ −2 r + nf > 0,. b0 ∝ −2 nc + nf < 0.. • When sign flip not possible (e.g., pure N = 2 YM ) ⇒ Dynamical Abelianization ! • Moral: Quantum mechanical NA monopoles require massless fermions • Existence proof: NONABELIAN VORTEX. 28.
(37) Quantum Nonabelian Vortices. 29.
(38) Vortices in Gauge Theories Laudau-Ginzburg ( U (1) Higgs ) Model 1 1 H = d3x (Fij )2 + |(∂i − i e Ai) φ|2 + V (φ) 4 2. A. B. B~0 ei. 30. |v|.
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(40) Vortices in Nonabelian Theories φ=0. H −→ C i Ai ∼ U (ϕ)∂iU †(ϕ); g Quantization. =⇒. Vortex ∼ Π1(H/C). φA ∼. (0) U φA U †,. U (ϕ) = exp i. r. βj Tj ϕ. j. α · β ⊂ Z: ˜ (= dual of H). βi = weight vectors of H. • For H = U (1) (Abrikosov-Nielsen-Olesen vortices) Π1(U (1)) = Z; • ZN vortices: SU (N )/ZN ⇒ ∅ ( Π1(SU (N )/ZN ) = ZN ) Non BPS : difficult to analyse. Tension ratios. Tk ∝ sin πNk. k quarks. ?. TC + Tm < TC+m ?. Tk k antiquarks. 31.
(41) Non-Abelian Vortices (’03) Auzzi, Bolognesi, Evslin, Konishi, Yung;. φ=0. φ* =0. G −→ H −→ ∅,. Hanany, Tong. φ ' φ*,. • An exact global symmetry HC+F ⊂ H ⊗ GF (not spontaneously broken), but broken by the vortex to G0 ⇒ Vortex zero modes (moduli) ∼ HC+F /G0 • SU (N + 1) →. SU (N )×U (1) ZN. → ∅ with 2N + 2 > Nf ≥ 2N. ⇒ Vortex with 2(N − 1) - parameter family of zeromodes SU (N )F +C ∼ CPN −1. SU (N − 1) × U (1) • Monopoles of G/H are confined by magnetic vortices of H → ∅; • Monopoles and vortices are “incompatible” (as static configurations); • Q-M’ly, NA vortices also requires massless quarks! • Proof of nonabelian monopoles 32.
(42) Flux matching Plane Vortex. Sphere (rv 2 <<1). B. Monopole. Figure 1:. 33.
(43) New Development in Susy Gauge Theories. 34.
(44) New Results in N = 1/N = 2 Susy Gauge Theories ˜ i = (˜ W = (Aµ, λ), Φ = (φ, ψ), Qi = (qi, ψq i), Q qi, ψq˜ i) gk k+1 • U (N ) theory with superpotential W (Φ) = k+1 Tr Φ U (N ) → U (N1) × . . . U (Nn) →. Ui(1). i. where diag Φ = (a1, a1, . . . , a1, a2, . . . , a2, a3, . . .) classically (W *(ai) = 0.) • 4D Functional-integral effectively replaced by matrix-integrals (DijkgraafVafa ’02). . Nˆ d M exp − Tr W (M ) gm ˆ2 N. • Field-theoretic rederivation/explanation by Cachazo-Douglas-SeibergWitten (’02) by using: Symmetry, Holomorphy and Generalized Konishi Anomaly; • Elegant determination of all chiral ring operator VEV TrΦn,. TrWα Φn, 35. TrW α Wα Φn..
(45) Key Formulae 1 T (z) = Tr , z−Φ. Wα wα (z) = Tr , z−Φ. Wα W α R(z) = Tr ; z−Φ. ˜ M (z) = Q. 1 Q. z−Φ. • “Loop equation” from Generalized Konishi Anomaly (W *(Φ) − W *(z))Wα W α f (z) = Tr ; z−Φ * 2 R(z) = W (z) − W *(z)2 + f (z). 1 R(z) = W (z) R(z) + f (z), 4 2. *. • Effective superpotential 1 Si = (Wα W α )i = 2. . 1 dz R(z) = 2 Ci. . W *(z)2 + f (z). Ci. • Matrix model curve y 2 = W *(z)2 + f (z) • Solve for M (z) and T (z) defined on a double sheeted Riemann surface. 36.
(46) Phases and Multiplication Maps (Cachazo-Seiberg-Witten (’03)) • Precise relation between N = 1 superpotential W(Φ) and singularities of N = 2 curve (E). • Confinement index = the smallest possible ZN for which Wilson loop displays no area law e.g., SU (N ) YM: r = N completely confining; r = 1 totally Higgs • Multiplication Map U (N ) and U (tN ) theories with the same W(Φ) Tr. 1 1 = t Tr x−Φ x − Φ0. • All confining vacua with r = t in the U (tN ) theory, arise from the Coulomb vacua of U (N ) theory • Confinement Index r gets simply multiplied by t. • New types of duality 37.
(47) Summary. 38.
(48) Lessons from Susy World • Abelian superconductor always accompanied by dynamical abelianization; • Confining vacua typically nonabelian superconductor (condensation of non-abelian monopoles ) • Also, Confining vacua but with mutually nonlocal monopoles and dyons • Quantum mechanical nonabelian monopoles and nonabelian vortices exist - both require massless fermions in the underlying theory; • Many exact results and relations among vacua in different theories.. Electromagnetic Duality • Powerful method of solution, helping understanding physics, rather than being an exact symmetry. 39.
(49) QCD • No dynamical abelianization • QCD with nf flavor (˜ nc = 2, 3, nf = 2, 3) b0 = 11 nc − 2 nf. ⇒. ˜b0 = 11 n ˜ c − nf. No sign flip (no weakly-coupled nonabelian monopoles) • Strongly-interacting nonabelian superconductor? • Hint from r-vacua and from the almost SCF vacua MiL,α = δαi vR = 0,. MαR,i = δiα vL = 0,. (α = 1, 2, . . . n ˜ c; i = 1, 2, . . . nf ). • A better picture? MiL,α MαR,j = const. δji = 0; for n ˜ c = 2, nf = 2 GF = SUL(2) × SUR (2) ⇒ SUV (2) 40.
(50) Symmetry, Quantization, Phase Factor “Melodies of Theoretical Physics of the 20th Century”. C.N. Yang (TH2002). Theoretical Physics of the 21st Century:. What new melody ?. 1.
(51) Nonabelian Vortices v. 1 SU (N + 1) −→. Auzzi-Boplognesi-Evslin-Konishi-Yung, Hanany-Tong ’03. SU (N ) × U (1) v2 −→ ∅, ZN. v1 ' v2 ,. (9). • HE theory has Monopoles; LE (monopoles heavy) has Vortices e.g. SU (3), N m 1 Φ = −√ 0 2 0. = 2 with 4 ≤ nf ≤ 5 with bare mass m ( adj mass µ Φ2): ) 0 0 √ kA ¯˜kA >= ξ 1 0 , < q ξ = >=< q µm , m m 0 2 0 1 0 −2m. Set Φ = Φ; q = q˜†; and q → 12 q: 2 2 . 1 1 g g 2 2 2 2 2 a 8 S = d4x , Fµν + 2 Fµν + ∇µq A + 2 q¯Aτ aq A + 1 q¯Aq A − 2ξ 2 4g2 4g1 8 24 3
(52) 2 1 (a) g2 qAτ aq A &ij T = d2x Fij ± (¯ 2g2 4 a=1 2 . 1 (8) g1 A 2 1. ξ A A 2 + Fij ± √ |q | − 2ξ &ij + ∇i q ± i&ij ∇j q ± √ F˜ (8)) 2g1 2 4 3 2 3 42.
(53) Non-Abelian Bogomolny equations 1 (a) g2 Fij ± (¯ qAτ aq A)&ij = 0, 2g2 4. (a = 1, 2, 3);. 1 (8) g1 Fij ± √ |q A|2 − 2ξ &ij = 0, 2g1 4 3. ∇i q A + iε&ij ∇j q A = 0,. A = 1, 2.. Abelian (particular) solutions of SU (3) → U (1) × U (1) by e.g. setting A1µ = A2µ = 0, and with squark fields of the 2 × 2 color-flavor diag. form: kA q kA(x) = q¯˜ (x) = 0, for k = A = 1, 2. inϕ 0 e φ1(r) , q kA(x) = 0 ei k ϕφ2(r) √ xj xj 3 8 Ai (x) = −ε&ij 2 ((n − k) − f3(r)) , Ai (x) = − 3 ε&ij 2 ((n + k) − f8(r)) r r where. d 1 d 1 r φ2(r) − (f8(r) − f3(r)) φ2(r) = 0, φ1(r) − (f8(r) + f3(r)) φ1(r) = 0, dr 2 dr 2 1 d g12 1 d g22 2 2 − − f8(r)+ φ1(r) + φ2(r) − 2ξ = 0, f3(r)+ φ1(r)2 − φ2(r)2 = 0. r dr 6 r dr 2 r. 43.
(54) with boundary conds for the gauge fields: f3(0) = εn,k (n − k) ,. f8(0) = εn,k (n + k) ,. f3(∞) = 0,. f8(∞) = 0. and the requirement that the squark fields be everywhere regular. Also φ1(∞) = ξ, φ2(∞) = ξ 1 0.8 0.6 0.4 0.2. 2. 4. 6. 8. 10. Figure 2: Vortex profile functions φ1 (r) and φ2 (r) of the (1, 0)-string. Note φ1 (0) = 0.. 44.
(55) 1 0.8 0.6 0.4 0.2. 2. 4. 6. 8. 10. Figure 3: The profile functions f3 (r) and f8 (r) for the (1, 0)-string.. • Vortex flux (SU N )) @ = ∇ ∧ A, @ B. Fv =. R2. dS ·. Tr φ B = 2π · √1 (Tr φφ)1/2 2. . 2(N + 1) . N. (10). • Exact Symmetry The SU (3) → SU (2) × U (1) → ∅ theory has unbroken global symmetry, SU (2)C+F . SU (2)C+F broken (to a U (1)) by a vortex configuration ⇒ zeromodes: continuous family of vortex solutions (moduli) of SU (2)/U (1) = S 2 = CP1 45.
(56) Remarks v. 1 SU (N + 1) −→. SU (N ) × U (1) v2 −→ 0, ZN. v1 ' v2 ,. (11). ˜ fluxes match exactly . • Monopole and Vortex H • Monopoles (HE theory ) and vortices (LE theory) are solutions of different effective Lagrangians, valid at different mass scales and with different effec. deg. freedom. • Actually they are incompatible - as static configurations: SU (N +1) HE theory (v2 = 0): monopoles ∼ Π2( SU (N )×U (1)/Z ); Π2 (SU (N + 1)) = ∅ N. LE theory (v1 → ∞): vortices ∼ Π1( SU (NZ)×U (1) ); Π1(SU (N + 1)) = ∅. N. • Physically, perfectly OK to consider both together: “mesons” ∼ Monopole-vortex-antimonopole, rotating and dynamically stable • Mesons in QCD are similar • In N = 2 models v1 ∼ m; v2 ∼. √. µm. 46.
(57) Minimum vortex of generic orientation: q kA = U. iϕ. e φ1(r). 0. 0. φ2(r). . i. U −1 = e 2 ϕ(1+n. aτ a). U. φ1(r). 0. 0. φ2(r). . xj τ 3 xj 1 Ai(x) = U [− &ij 2 [1 − f3(r)]]U −1 = − naτ a&ij 2 [1 − f3(r)], 2 r 2 r √ xj A8i (x) = − 3 &ij 2 [1 − f8(r)] r where U ∈ SU (2)C+F na = (sin α cos β, sin α sin β, cos α), The tension T = 2πξ independent of U .. 47. U = e−iβ τ3/2 e−iα τ2/2.. U −1,.
(58) Remarks • Reduction of the vortex spectrum (meson spectrum): (Fig) Π1 ( to Π1 (. U (1) × U (1) ) = Z2 Z2. SU (2) × U (1) )=Z Z2. • Transition from abelian (mi = mj ) to nonabelian (mi = m) superconductivity reliably and quantum mechanically known; • (Indirect) solution to the “existence problem” of nonabelin monopoles; • Dynamics of vortex zero modes n → n(z, t) 1 Sσ(1+1) = β d2x (∂ na)2 + fermions. 2 O(3) = CP1 sigma model! Dual (Shifman et.al.; Vafa-Hori) to a chiral theory with two vacua → . No spontaneous breaking of SU (2)C+F ⇔ confining, dual SU (2) (Witten index = 2).. 48.
(59) Reduction of the vortex spectrum (meson spectrum) (0,2). (2,0). (1,1). (0,1). (1,0). (1,-1). Figure 4: Lattice of (n, k) vortices in the theory SU (3) → U (1)2 .. (0,2). (1,1). (0,1). (1,0). (2,0). Level 2. Level 1. Figure 5: Reduced lattice of Z vortices SU (3) → SU (2) × U (1).. 49.
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