Solitons in the SU(3) Faddeev-Niemi Model
|
|
- Adam Owen
- 5 years ago
- Views:
Transcription
1 Solitons in the SU(3) Faddeev-Niemi Model Yuki Amari Tokyo University of Science Based on arxiv:1805,10008 with PRD 97, (2018) In collaboration with Nobuyuki Sawado (TUS) Russia, 29 May 2018
2 Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook
3 Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook
4 Knots as particles Atoms are knotted vortex tubes in ether. Lord Kelvin
5 Knots as particles Atoms are knotted vortex tubes in ether. Lord Kelvin The mass spectrum of glueballs R. V. Buniy, T. W. Kephart, A model of glueballs PLB 576, 127 C. J. Morningstar and M. J. Peardon PRD 60, (1999)
6 SU(2) Yang-Mills fields in the IR limit L. D. Faddeev & A. J. Niemi,PRL 82, 1624 (1999) QCD in UV limit asymptotic freedom QCD in IR limit color confinement Very different phases! In the IR limit with monopole condensation, some other order parameter could play a central role rather than the gauge field A μ.
7 SU(2) Yang-Mills fields in the IR limit L. D. Faddeev & A. J. Niemi,PRL 82, 1624 (1999) QCD in UV limit asymptotic freedom QCD in IR limit color confinement Very different phases! In the IR limit with monopole condensation, some other order parameter could play a central role rather than the gauge field A μ. Cho-Faddeev-Niemi (CFN) decomposition A a μ = C μ n a + ε abc μ n b n c + ρ μ n a + σε abc μ n b n c C μ : Abelian gauge field, n = n 1, n 2, n 3 with n = 1, ρ, σ: scalar field If n = Ԧr r, C μ = ρ = σ = 0, it gives the Wu-Yang monopole.
8 Effective model in confinement phase In the confinement phase φ = 0 where φ = ρ + iσ, we get SU(2) Faddeev-Niemi model S eff = නd 4 x Λ 2 μ n 2 + μ n ν n 2 For static configurations, n defines a mapping S 3 S 2. Preimage of the Hopf map OPS = S 2 Phys. sp. = R 3 ~S 3 π 3 S 2 = Z Knot solitons
9 CFN decomposition of SU(3) gauge fields Decomposition L. D. Faddeev and A. J. Niemi, PLB 449, 214 (1999) A μ = C μ a n a + i n a, μ n a + ρ ab n a, μ n b + iσ ab n a, μ n b n a = Uh a U U SU 3 SU(3) Faddeev-Niemi model 2 S eff = නd 4 x M 2 Tr μ n a μ n a + 1 e 2 F μν a F a μν a=1 F μν a = i 2 Tr n a μ n b, ν n b OPS= SU 3 /U 1 2 = F 2 π 3 F 2 = Z Knot solitons Cf. S 2 = SU(2)/U(1)
10 Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook
11 The SU(2) Faddeev-Niemi model J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) The static energy L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n e 2 in j n 2 The Hopfions P. Sutcliffe, Proc. Roy. Soc. Lond. A 463, 3001 (2007) R. A. Battye & P. M. Putcliffe, PRL 81, 4798 (1998)
12 The SU(2) Faddeev-Niemi model J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) The static energy L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n e 2 in j n 2 Stereographic projection S 2 CP 1 n = 1 Δ u + u, i u u, u 2 1 Δ = 1 + u 2
13 The SU(2) Faddeev-Niemi model J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) The static energy L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n e 2 in j n 2 Stereographic projection S 2 CP 1 n = 1 Δ u + u, i u u, u 2 1 Δ = 1 + u 2 At the end of my talk, we see the e.o.m. of the SU 3 model reduces to that of the SU 2 case with the stereographic projection.
14 Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook
15 The action density The F 2 nonlinear σ-model S = Z A μ Z B 2 + ZB μ Z C 2 + ZC μ Z A 2
16 The action density The F 2 nonlinear σ-model S = Z A μ Z B 2 + ZB μ Z C 2 + ZC μ Z A 2 D μ (a) μ Z a μ Z a Copies of the CP 2 NLσ-model A. D'Adda et.al., NPB 146, 63 (1978) S = න d 2 x D μ A Z A 2 + Dμ C Z C 2 ZC μ Z A 2
17 The action density The F 2 nonlinear σ-model S = Z A μ Z B 2 + ZB μ Z C 2 + ZC μ Z A 2 D μ (a) μ Z a μ Z a Copies of the CP 2 NLσ-model A. D'Adda et.al., NPB 146, 63 (1978) S = න d 2 x D A 2 μ Z A + C 2 Dμ Z C 2 ZC μ Z A 2π N A + N C න d 2 x Z 2 C μ Z A Winding number of a map S 2 CP 2 N a = i 2π නd2 x ε μν D μ a Z a Dν (a) Za N A + N B + N C = 0
18 Parametrization Isomorphism SU 3 /U 1 2 SL 3, C /B u u 2 u 3 1 = c 1, c 2, c 3 SL 3, C 6 DOF = dim F 2 Gram-Schmidt orthogonalization process U = Z Α, Z Β, Z C SU 3
19 Parametrization Isomorphism SU 3 /U 1 2 SL 3, C /B u u 2 u 3 1 = c 1, c 2, c 3 SL 3, C 6 DOF = dim F 2 Gram-Schmidt orthogonalization process U = Z Α, Z Β, Z C SU 3 Z A = 1 Δ 1 1 u 1 u 2 Z B = 1 Δ 1 Δ 2 u 1 u 2 u 3 1 u 1 u 2 u 3 + u 2 2 u 1 u 2 + u 3 + u 3 u 1 2 Z C = 1 u 1 u 3 u 2 u 3 Δ 2 1 Δ 1 = 1 + u u 2 2 Δ 2 = 1 + u u 1 u 3 u 2 2
20 The 2D instantons The BPS-like bound S 2π N A + N C න d 2 x Z 2 C μ Z A satisfied if and only if u j = u j x 1 ± ix 2 for j = 1,2,3.
21 The 2D instantons The BPS-like bound S 2π N A + N C න d 2 x Z 2 C μ Z A satisfied if and only if u j = u j x 1 ± ix 2 for j = 1,2,3. Stationary points in the bound correspond to the instantons. π N A + N C න d 2 x Z 2 C μ Z A 0 u 1 = u 3 = 0 u 2 = u 2 x 1 ± ix 2 H. T. Ueda, Y. Akagi & N. Shannon, PRA. 93, (R) (2016) Embedding type
22 The 2D instantons The BPS-like bound S 2π N A + N C න d 2 x Z 2 C μ Z A satisfied if and only if u j = u j x 1 ± ix 2 for j = 1,2,3. Stationary points in the bound correspond to the instantons. π N A + N C න d 2 x Z 2 C μ Z A 0 u 1 = u 3 = 0 u 2 = u 2 x 1 ± ix 2 H. T. Ueda, Y. Akagi & N. Shannon, PRA. 93, (R) (2016) Embedding type u k = u k x 1 ± ix 2 u 3 μ u 1 μ u 2 = 0 Y.A. & N. Sawado, PRD 97, (2018) Genuine type k = 1,2
23 Contents 1. Background & motivation 2. Hopfions in the SU(2) Faddeev-Niemi model 3. 2D instantons in the F 2 nonlinear σ-model 4. Hopfions in the SU(3) Faddeev-Niemi model 5. Summary & Outlook
24 L = The SU(3) Faddeev-Niemi model The Lagrangian 2 a=1 M 2 Tr μ n a μ n a F μν a = i 2 Tr n a μ n b, ν n b 1 e 2 F μν a F a μν For simplicity, we set M = e = 1. n a = Uh a U U = Z Α, Z Β, Z C SU 3 The Cartan-Weyl basis h 1 = λ 3 / 2, h 2 = λ 8 / 2 e 1 = , e 2 = , e 3 = e p = e p T
25 Equations of motion Cartan decomposition U μ U = ia μ a h a + ij μ p e p The EL eq. is equivalent to conservation of the Noether current K μ associated to U gu, g SU(3). μ K μ = 0 K μ = UB μ U μ B μ + U μ U, B μ = 0 B μ = i J p μ iα p a F a μν J pν p e p
26 Equations of motion Cartan decomposition U μ U = ia μ a h a + ij μ p e p The EL eq. is equivalent to conservation of the Noether current K μ associated to U gu, g SU(3). μ K μ = 0 K μ = UB μ U μ B μ + U μ U, B μ = 0 The equations of motion μ J p μ ig p μν J pν B μ = i + ir pμ J p μ ig p μν J p ν +G pμν J p 1 μ J p+1 ν = 0 p J μ p iα a p F μν a J pν e p p = 1,2,3 R μ p = α a p A μ a, G p μν = α a p F μν a α a p = Tr e p h a, e p
27 Trivial embedding The embedding can be realized if two of the scalars vanish. u 1 = u 3 = 0, u 2 = u U = 1 Δ 1 0 u 0 Δ 0 u 0 1 Δ = 1 + u 2
28 Trivial embedding The embedding can be realized if two of the scalars vanish. u 1 = u 3 = 0, u 2 = u U = 1 Δ 1 0 u 0 Δ 0 u 0 1 Δ = 1 + u 2 We have J μ 1 = J μ 3 = 0, J μ 2 = i Δ μu The e.o.m for p = 1, 3 are automatically satisfied and for p = 2 reduces to μ μ u ig μν ν u + ir μ μ log Δ μ u ig μν ν u = 0 R μ = i Δ u μ u u μ u G μν = 2i Δ 2 μu ν u μ u ν u
29 Trivial embedding The embedding can be realized if two of the scalars vanish. u 1 = u 3 = 0, u 2 = u U = 1 Δ 1 0 u 0 Δ 0 u 0 1 Δ = 1 + u 2 We have J μ 1 = J μ 3 = 0, J μ 2 = i Δ μu The e.o.m for p = 1, 3 are automatically satisfied and for p = 2 reduces to μ μ u ig μν ν u + ir μ μ log Δ μ u ig μν ν u = 0 R μ = i Δ u μ u u μ u G μν = 2i Δ 2 μu ν u μ u ν u The EL eq. of the SU 2 Faddeev-Niemi model
30 Genuine solution Our strategy (i) We impose J μ 2 u 3 μ u 1 μ u 2 = 0. u 2 = f u 1, u 3 = f u 1 (ii)el eq. for p = 1, 3 are proportional to each other. 1 + u u 2 2 / 1 + u u 1 u 3 u 2 2 = const.
31 Genuine solution Our strategy (i) We impose J μ 2 u 3 μ u 1 μ u 2 = 0. (ii)el eq. for p = 1, 3 are proportional to each other. u 1 = u 3 = u 2 = u 2 u 2 = f u 1, u 3 = f u u u 2 2 / 1 + u u 1 u 3 u 2 2 = const. 2u, U = 1 Δ 1 2u u 2 2u 1 u 2 2u u 2 2u 1 The e.o.m for p = 2 is automatically satisfied and for p = 1, 3 reduce to μ μ u ig μν ν u + ir μ μ log Δ μ u ig μν ν u = 0
32 Genuine solution Our strategy (i) We impose J μ 2 u 3 μ u 1 μ u 2 = 0. (ii)el eq. for p = 1, 3 are proportional to each other. u 1 = u 3 = u 2 = u 2 u 2 = f u 1, u 3 = f u u u 2 2 / 1 + u u 1 u 3 u 2 2 = const. 2u, U = 1 Δ 1 2u u 2 2u 1 u 2 2u u 2 2u 1 The e.o.m for p = 2 is automatically satisfied and for p = 1, 3 reduce to μ μ u ig μν ν u + ir μ μ log Δ μ u ig μν ν u = 0 The EL eq. of the SU 2 Faddeev-Niemi model!!
33 Summary We confirmed the existence of Hopfions in the SU 3 Faddeev-Niemi model. The model is an effective model of the SU(3) pure YM theory. We found an ansatz of nonembedding type which the e.o.m. reduces to the SU 2 Faddeev-Niemi model. Outlook Stability of the Hopfions Implication of the constraint Estimation of mass spectrum of glueballs
34 Summary We confirmed the existence of Hopfions in the SU 3 Faddeev-Niemi model. The model is an effective model of the SU(3) pure YM theory. We found an ansatz of nonembedding type which the e.o.m. reduces to the SU 2 Faddeev-Niemi model. Outlook Stability of the Hopfions Implication of the constraint Estimation of mass spectrum of glueballs Thank you for your attention!
35
36 Reformulation Cartan decomposition of the Murer-Cartan form U μ U = ia μ a h a + ij μ p e p p = ±1, ±2, ±3. Under the gauge transformation U U exp iθ a h a, A μ a A μ a + μ θ a, The energy functional 3 E = නd 3 x q=1 J μ p J μ p e iθa α a p J q i J q i 1 4 J q [ij q j] J q+1 q+1 [i J 2 j] J [i q J j] q J i q J j q J j q J i q The energy represents the gauge fixing functional for a nonlinear maximal Abelian gauge.
37 χ χ 2 χ 4 0 χ 3 χ 5 1 χ 1 χ 4 Parametrization = c 1, c 2, c 3 χ 1 0, χ dof Gramm-Schmit orthogonalization process χ 2 /χ 1 = u 1, χ 3 /χ 1 = u 2, χ 5 /χ 4 = u 3 U = Z A e iθ 1, Z Β e iθ 4, Z C e i θ 1+θ 4 θ i = arg χ i 8 dof = dim SU(3) W = Z Α, Z Β, Z C 6 dof = dim F 2 Z A = 1 Δ 1 1 u 1 u 2 Z B = 1 Δ 1 Δ 2 u 1 u 2 u 3 1 u 1 u 2 u 3 + u 2 2 u 1 u 2 + u 3 + u 3 u 1 2 Z C = 1 u 1 u 3 u 2 u 3 Δ 2 1 Δ 1 = 1 + u u 2 2 Δ 2 = 1 + u u 1 u 3 u 2 2
38 The Hopf invariant M. Kisielowski, J. Phys. A 49, 17, (2016) Exact sequence 0 π 3 SU 3 π 3 SU 3 /U Isotropy π 3 F 2 π 3 SU 3 Hopf invariant =Winding number of U: S 3 SU(3) =CS term for the SU 3 pure gauge R H = 1 24π 2 නTr U du 3 = 1 16π 2 න Tr R dr + 2 Tr R R R 3 The Hopf invariant H = 1 8π 2 නd3 x ε ijk A i a F jk a R = U du + 1 8π 2 නd3 x ε ijk Im J i 1 J j 2 J k 3
39 CFN decomposition of SU(3) gauge fields There are two options of the decomposition: SB pattern Order parameter space Homotopy group SU 3 U 1 2 F 2 = SU 3 /U 1 2 π 3 F 2 = Z SU 3 U 2 CP 2 = SU 3 /U 2 π 3 CP 2 = 0 Decomposition L. D. Faddeev and A. J. Niemi, PLB 449, 214 (1999) A μ = C μ a n a + i n a, μ n a + ρ ab n a, μ n b + iσ ab n a, μ n b SU(3) Faddeev-Niemi model 2 S eff = නd 4 x M 2 Tr μ n a μ n a + 1 e 2 F μν a F a μν a=1 n a = Uh a U U SU 3 F μν a = i 2 Tr n a μ n b, ν n b
40 Effective models of the SU(2) YM theory (1) Coulomb phase The original Yang-Mills action at low energy (2) Higgs phase φ 0 φ = ρ + iσ S eff = නd 4 x G 2 μν + D μ φ φ 2 2 G μν = μ C ν ν C μ, D μ = μ + ic μ Abelian Higgs model (3) Confinement phase φ = 0 S eff = නd 4 x Λ 2 μ n 2 + μ n ν n 2 SU(2) Faddeev-Niemi model
41 The SU(2) Faddeev-Niemi model The static energy J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n e 2 in j n 2 n const. as x n R 3 ~S 3 S 2 π 3 S 2 = Z The top. charge can be defined. Hopf map Derrick s theorem Scaling x λx E = E 2 + E 4 E n; λ = λ 1 E 2 + λe 4 λ E n; 1 = E 2 + E 4 = 0 λ 2 E n; 1 = 2E 2 > 0 If E 4 is absent, there are no stable solitons.
42 The SU(2) Faddeev-Niemi model J. Gladikowski & M. Hellmund, PRD 56, 5194 (1997) The static energy L. D. Faddeev & A. J. Niemi, Nature 387, 58 (1997) E = නd 3 x M 2 i n e 2 in j n 2 Stereographic projection S 2 CP 1 n = 1 Δ u + u, i u u, u 2 1 Δ = 1 + u 2 The complex scalar u is convenient to solve the EL eq.. But, the topological charge cannot be defined by n nor u. We need to introduce Z = Z 0, Z 1 T where u = Z 1 /Z 0. Hopf invariant π 3 S 2 H = 1 4π නd3 x A da = Z A = iz dz
43 Applications in condensed matter physics There are many suggestions that knot solitons appear in condensed matter physics; Triplet superconductor Nematic liquid crystal E. Babaev, PRL 88, (2002) P. J. Ackerman & I. I. Smalyukh, PRX 7, (2017) SU(3) analogue Color superconductor Spin-nematic described by the SU 3 Heisenberg model. OPS of the SU 3 AFH model on the Cubic lattice = F 2 Continuum limit The SU(3) Faddeev-Niemi (like) model
Localized objects in the Faddeev-Skyrme model
Localized objects in the Faddeev-Skyrme model Department of Physics and Astronomy, University of Turku FIN-20014 Turku, Finland IWNMMP, Beijing, China, July 15-21,2009 Introduction Solitons: Topological
More informationThe Role Of Magnetic Monopoles In Quark Confinement (Field Decomposition Approach)
The Role Of Magnetic Monopoles In Quark Confinement (Field Decomposition Approach) IPM school and workshop on recent developments in Particle Physics (IPP11) 2011, Tehran, Iran Sedigheh Deldar, University
More informationMonopoles and Skyrmions. Ya. Shnir. Quarks-2010
Monopoles and Skyrmions Ya. Shnir Quarks-2010 Коломна, Россия, 8 Июня 2010 Outline Abelian and non-abelian magnetic monopoles BPS monopoles Rational maps and construction of multimonopoles SU(2) axially
More informationMatryoshka Skyrmions, Confined Instantons and (P,Q) Torus Knots. Muneto Nitta (Keio Univ.)
Matryoshka Skyrmions, Confined Instantons and (P,Q) Torus Knots 20 th August 2013 Field theory and String theory @ YITP Muneto Nitta (Keio Univ.) 1[1] Phys.Rev.D86 (2012) 125004 [arxiv:1207.6958] [2] Phys.Rev.D87
More informationThe SU(2) quark-antiquark potential in the pseudoparticle approach
The SU(2) quark-antiquark potential in the pseudoparticle approach Marc Wagner mcwagner@theorie3.physik.uni-erlangen.de http://theorie3.physik.uni-erlangen.de/ mcwagner 3 th March 26 Outline PP = pseudoparticle
More informationTopological Solitons from Geometry
Topological Solitons from Geometry Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge Atiyah, Manton, Schroers. Geometric models of matter. arxiv:1111.2934.
More informationExtended Soliton Solutions in an Ef fective Action for SU(2) Yang Mills Theory
Symmetry, Integrability and Geometry: Methods and Applications Vol. 006), Paper 016, 11 pages Extended Soliton Solutions in an Ef fective Action for SU) Yang Mills Theory Nobuyuki SAWADO, Noriko SHIIKI
More informationSolitons, Vortices, Instantons
Solitons, Vortices, Instantons Seminarvortrag von David Mroß Universität Bonn 1. Juni 2006 Seminar Theoretische Elementarteilchenphysik SS 06 Structure of the talk Solitons in a nutshell Recap: solitons
More informationBaby Skyrmions in AdS 3 and Extensions to (3 + 1) Dimensions
Baby Skyrmions in AdS 3 and Extensions to (3 + 1) Dimensions Durham University Work in collaboration with Matthew Elliot-Ripley June 26, 2015 Introduction to baby Skyrmions in flat space and AdS 3 Discuss
More informationSome selected results of lattice QCD
Some selected results of lattice QCD Heidelberg, October 12, 27 Kurt Langfeld School of Mathematics and Statistics University of Plymouth p.1/3 Glueball spectrum: no quarks (quenched approximation) 12
More informationInstanton constituents in sigma models and Yang-Mills theory at finite temperature
Instanton constituents in sigma models and Yang-Mills theory at finite temperature Falk Bruckmann Univ. of Regensburg Extreme QCD, North Carolina State, July 8 PRL (8) 56 [77.775] EPJ Spec.Top.5 (7) 6-88
More informationarxiv: v1 [hep-lat] 24 Nov 2009
KEK Preprint 29-28 CHIBA-EP-8 arxiv:9.433v [hep-lat] 24 Nov 29 Topological configurations of Yang-Mills field responsible for magnetic-monopole loops as quark confiner Computing Research Center, High Energy
More informationLecture 7: N = 2 supersymmetric gauge theory
Lecture 7: N = 2 supersymmetric gauge theory José D. Edelstein University of Santiago de Compostela SUPERSYMMETRY Santiago de Compostela, November 22, 2012 José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric
More informationLefschetz-thimble path integral for solving the mean-field sign problem
Lefschetz-thimble path integral for solving the mean-field sign problem Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN Jul 15, 2015 @
More informationInstantons and Monopoles in Maximal Abelian Projection of SU(2) Gluodynamics
ITEP-95-34 hep-th/9506026 arxiv:hep-th/9506026v2 11 Jun 1995 Instantons and Monopoles in Maximal Abelian Projection of SU(2) Gluodynamics M.N. Chernodub and F.V. Gubarev ITEP, Moscow, 117259, Russia and
More informationAbelian and non-abelian Hopfions in all odd dimensions
Abelian and non-abelian Hopfions in all odd dimensions Tigran Tchrakian Dublin Institute for Advanced Studies (DIAS) National University of Ireland Maynooth, Ireland 7 December, 2012 Table of contents
More informationLefschetz-thimble path integral and its physical application
Lefschetz-thimble path integral and its physical application Yuya Tanizaki Department of Physics, The University of Tokyo Theoretical Research Division, Nishina Center, RIKEN May 21, 2015 @ KEK Theory
More informationSuitable operator to test the Abelian dominance for sources in higher representation
Suitable operator to test the Abelian dominance for sources in higher representation Ryutaro Matsudo Graduate School of Science and Engineering, Chiba University May 31, 2018 New Frontiers in QCD 2018
More informationThe Phases of QCD. Thomas Schaefer. North Carolina State University
The Phases of QCD Thomas Schaefer North Carolina State University 1 Motivation Different phases of QCD occur in the universe Neutron Stars, Big Bang Exploring the phase diagram is important to understanding
More informationContinuity of the Deconfinement Transition in (Super) Yang Mills Theory
Continuity of the Deconfinement Transition in (Super) Yang Mills Theory Thomas Schaefer, North Carolina State University 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0 0.5 0.0 0.5 1.0 0.5
More informationSeminar presented at the Workshop on Strongly Coupled QCD: The Confinement Problem Rio de Janeiro UERJ November 2011
and and Seminar presented at the Workshop on Strongly Coupled QCD: The Problem Rio de Janeiro UERJ 28-30 November 2011 Work done in collaboration with: N.R.F. Braga, H. L. Carrion, C. N. Ferreira, C. A.
More informationHamiltonian approach to QCD: The Polyakov loop potential H. Reinhardt
Hamiltonian approach to QCD: The Polyakov loop potential H. Reinhardt H. R. & J. Heffner Phys. Lett.B718(2012)672 PRD(in press) arxiv:1304.2980 Phase diagram of QCD non-perturbative continuum approaches
More informationInfrared Propagators and Confinement: a Perspective from Lattice Simulations
Infrared Propagators and Confinement: a Perspective from Lattice Simulations Tereza Mendes University of São Paulo & DESY-Zeuthen Work in collaboration with Attilio Cucchieri Summary Lattice studies of
More informationGauge invariance of the Abelian dual Meissner effect in pure SU(2) QCD
arxiv:hep-lat/51127v1 15 Nov 25 Gauge invariance of the Abelian dual Meissner effect in pure SU(2) QCD Institute for Theoretical Physics, Kanazawa University, Kanazawa 92-1192, Japan and RIKEN, Radiation
More information2-Group Global Symmetry
2-Group Global Symmetry Clay Córdova School of Natural Sciences Institute for Advanced Study April 14, 2018 References Based on Exploring 2-Group Global Symmetry in collaboration with Dumitrescu and Intriligator
More informationt Hooft-Polyakov Monopoles on the Lattice
t Hooft-Polyakov Monopoles on the Lattice Davis,Kibble,Rajantie&Shanahan, JHEP11(2000) Davis,Hart,Kibble&Rajantie, PRD65(2002) Rajantie, in progress 19 May 2005 University of Wales, Swansea Introduction
More informationarxiv: v4 [hep-th] 17 Aug 2017
Microscopic structure of the QCD vacuum D.G. Pak, 1, 2, Bum-Hoon Lee, 2, Youngman Kim, 5 Takuya Tsukioka, 6 and P.M. Zhang 1 1 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 70000, China
More informationt Hooft Loops and S-Duality
t Hooft Loops and S-Duality Jaume Gomis KITP, Dualities in Physics and Mathematics with T. Okuda and D. Trancanelli Motivation 1) Quantum Field Theory Provide the path integral definition of all operators
More informationQCD and Instantons: 12 Years Later. Thomas Schaefer North Carolina State
QCD and Instantons: 12 Years Later Thomas Schaefer North Carolina State 1 ESQGP: A man ahead of his time 2 Instanton Liquid: Pre-History 1975 (Polyakov): The instanton solution r 2 2 E + B A a µ(x) = 2
More informationSkyrme model U : S 3 S 3. d 3 x. The Skyrme field: The topological charge: Q = 1. Rescaling: Sigma-model term Skyrme term Potential term.
Skyrme model The Skyrme field: U(r, t) r I U : S 3 S 3 L = f2 π 4 Tr µ U µ U + 1 32e 2 Tr U µ U,U ν U 2 f π = 186 MeV, m π = 136 MeV + m2 πf 2 π 8 Tr (U I) Sigma-model term Skyrme term Potential term The
More informationInstantons and Sphalerons in a Magnetic Field
Stony Brook University 06/27/2012 GB, G.Dunne & D. Kharzeev, arxiv:1112.0532, PRD 85 045026 GB, D. Kharzeev, arxiv:1202.2161, PRD 85 086012 Outline Motivation & some lattice results General facts on Dirac
More informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More information100 Years after Einstein A New Approach to Quantum Gravity
100 Years after Einstein A New Approach to Quantum Gravity Y. M. Cho Administration Building 310-4, Konkuk University and School of Physics and Astronomy College of Natural Science, Seoul National University
More informationHamilton Approach to Yang-Mills Theory Confinement of Quarks and Gluons
Hamilton Approach to Yang-Mills Theory Confinement of Quarks and Gluons H. Reinhardt Tübingen Collaborators: G. Burgio, M. Quandt, P. Watson D. Epple, C. Feuchter, W. Schleifenbaum, D. Campagnari, J. Heffner,
More informationSymmetries Then and Now
Symmetries Then and Now Nathan Seiberg, IAS 40 th Anniversary conference Laboratoire de Physique Théorique Global symmetries are useful If unbroken Multiplets Selection rules If broken Goldstone bosons
More informationLIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS
LIBERATION ON THE WALLS IN GAUGE THEORIES AND ANTI-FERROMAGNETS Tin Sulejmanpasic North Carolina State University Erich Poppitz, Mohamed Anber, TS Phys.Rev. D92 (2015) 2, 021701 and with Anders Sandvik,
More informationin collaboration with K.Furuta, M.Hanada and Y.Kimura. Hikaru Kawai (Kyoto Univ.) There are several types of matrix models, but here for the sake of
Curved space-times and degrees of freedom in matrix models 1 Based on hep-th/0508211, hep-th/0602210 and hep-th/0611093 in collaboration with K.Furuta, M.Hanada and Y.Kimura. Hikaru Kawai (Kyoto Univ.)
More informationarxiv: v1 [hep-lat] 6 Oct 2008
KEK Preprint 2008-36 CHIBA-EP-73 A new description of lattice Yang-Mils theory and non-abelian monopoles as the quark confiner arxiv:080.0956v [hep-lat] 6 Oct 2008 Akihiro Shibata Computing Research Center,
More informationSolitons and point particles
Solitons and point particles Martin Speight University of Leeds July 28, 2009 What are topological solitons? Smooth, spatially localized solutions of nonlinear relativistic classical field theories Stable;
More informationProperties of monopole operators in 3d gauge theories
Properties of monopole operators in 3d gauge theories Silviu S. Pufu Princeton University Based on: arxiv:1303.6125 arxiv:1309.1160 (with Ethan Dyer and Mark Mezei) work in progress with Ethan Dyer, Mark
More informationChiral symmetry breaking, instantons, and monopoles
Chiral symmetry breaking, instantons, and monopoles Adriano Di Giacomo 1 and Masayasu Hasegawa 2 1 University of Pisa, Department of Physics and INFN 2 Joint Institute for Nuclear Research, Bogoliubov
More informationCLASSIFICATION OF NON-ABELIAN CHERN-SIMONS VORTICES
CLASSIFICATION OF NON-ABELIAN CHERN-SIMONS VORTICES arxiv:hep-th/9310182v1 27 Oct 1993 Gerald V. Dunne Department of Physics University of Connecticut 2152 Hillside Road Storrs, CT 06269 USA dunne@hep.phys.uconn.edu
More informationThe Phases of QCD. Thomas Schaefer. North Carolina State University
The Phases of QCD Thomas Schaefer North Carolina State University 1 Plan of the lectures 1. QCD and States of Matter 2. The High Temperature Phase: Theory 3. Exploring QCD at High Temperature: Experiment
More information5 Topological defects and textures in ordered media
5 Topological defects and textures in ordered media In this chapter we consider how to classify topological defects and textures in ordered media. We give here only a very short account of the method following
More informationSolitons, Links and Knots
UKC/IMS/98-34 DAMTP-1998-110 Solitons, Links and Knots arxiv:hep-th/9811077v1 9 Nov 1998 Richard A. Battye and Paul M. Sutcliffe Department of Applied Mathematics and Theoretical Physics, University of
More informationCold atoms and AdS/CFT
Cold atoms and AdS/CFT D. T. Son Institute for Nuclear Theory, University of Washington Cold atoms and AdS/CFT p.1/20 What is common for strong coupled cold atoms and QGP? Cold atoms and AdS/CFT p.2/20
More informationAxions. Kerstin Helfrich. Seminar on Theoretical Particle Physics, / 31
1 / 31 Axions Kerstin Helfrich Seminar on Theoretical Particle Physics, 06.07.06 2 / 31 Structure 1 Introduction 2 Repetition: Instantons Formulae The θ-vacuum 3 The U(1) and the strong CP problem The
More informationSingular Monopoles and Instantons on Curved Backgrounds
Singular Monopoles and Instantons on Curved Backgrounds Sergey Cherkis (Berkeley, Stanford, TCD) C k U(n) U(n) U(n) Odense 2 November 2010 Outline: Classical Solutions & their Charges Relations between
More informationLattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures
Lattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures Yoshiyuki Nakagawa Graduate School of Science and Technology, Niigata University, Igarashi-2, Nishi-ku,
More informationVortices and other topological defects in ultracold atomic gases
Vortices and other topological defects in ultracold atomic gases Michikazu Kobayashi (Kyoto Univ.) 1. Introduction of topological defects in ultracold atoms 2. Kosterlitz-Thouless transition in spinor
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14.
As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14 Majorana spinors March 15, 2012 So far, we have only considered massless, two-component
More informationThe Infrared Behavior of Landau Gauge Yang-Mills Theory in d=2, 3 and 4 Dimensions
The Infrared Behavior of Landau Gauge Yang-Mills Theory in d=2, 3 and 4 Dimensions Markus Huber 1 R. Alkofer 1 C. S. Fischer 2 K. Schwenzer 1 1 Institut für Physik, Karl-Franzens Universität Graz 2 Institut
More informationA Review of Solitons in Gauge Theories. David Tong
A Review of Solitons in Gauge Theories David Tong Introduction to Solitons Solitons are particle-like excitations in field theories Their existence often follows from general considerations of topology
More informationHopf Fibrations. Consider a classical magnetization field in R 3 which is longitidinally stiff and transversally soft.
Helmut Eschrig Leibniz-Institut für Festkörper- und Werkstofforschung Dresden Leibniz-Institute for Solid State and Materials Research Dresden Hopf Fibrations Consider a classical magnetization field in
More informationExtremal black holes from nilpotent orbits
Extremal black holes from nilpotent orbits Guillaume Bossard AEI, Max-Planck-Institut für Gravitationsphysik Penn State September 2010 Outline Time-like dimensional reduction Characteristic equation Fake
More informationLectures April 29, May
Lectures 25-26 April 29, May 4 2010 Electromagnetism controls most of physics from the atomic to the planetary scale, we have spent nearly a year exploring the concrete consequences of Maxwell s equations
More informationNTNU Trondheim, Institutt for fysikk
NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 99890701 Allowed tools: mathematical tables Some formulas can be found on p.2. 1. Concepts.
More informationExact Results in D=2 Supersymmetric Gauge Theories And Applications
Exact Results in D=2 Supersymmetric Gauge Theories And Applications Jaume Gomis Miami 2012 Conference arxiv:1206.2606 with Doroud, Le Floch and Lee arxiv:1210.6022 with Lee N = (2, 2) supersymmetry on
More informationLecture 12 Holomorphy: Gauge Theory
Lecture 12 Holomorphy: Gauge Theory Outline SUSY Yang-Mills theory as a chiral theory: the holomorphic coupling and the holomorphic scale. Nonrenormalization theorem for SUSY YM: the gauge coupling runs
More informationAn origin of light and electrons a unification of gauge interaction and Fermi statistics
An origin of light and electrons a unification of gauge interaction and Fermi statistics Michael Levin and Xiao-Gang Wen http://dao.mit.edu/ wen Artificial light and quantum orders... PRB 68 115413 (2003)
More informationCold atoms and AdS/CFT
Cold atoms and AdS/CFT D. T. Son Institute for Nuclear Theory, University of Washington Cold atoms and AdS/CFT p.1/27 History/motivation BCS/BEC crossover Unitarity regime Schrödinger symmetry Plan Geometric
More informationtowards a holographic approach to the QCD phase diagram
towards a holographic approach to the QCD phase diagram Pietro Colangelo INFN - Sezione di Bari - Italy in collaboration with F. De Fazio, F. Giannuzzi, F. Jugeau and S. Nicotri Continuous Advances in
More information752 Final. April 16, Fadeev Popov Ghosts and Non-Abelian Gauge Fields. Tim Wendler BYU Physics and Astronomy. The standard model Lagrangian
752 Final April 16, 2010 Tim Wendler BYU Physics and Astronomy Fadeev Popov Ghosts and Non-Abelian Gauge Fields The standard model Lagrangian L SM = L Y M + L W D + L Y u + L H The rst term, the Yang Mills
More informationThink Globally, Act Locally
Think Globally, Act Locally Nathan Seiberg Institute for Advanced Study Quantum Fields beyond Perturbation Theory, KITP 2014 Ofer Aharony, NS, Yuji Tachikawa, arxiv:1305.0318 Anton Kapustin, Ryan Thorngren,
More informationInteracting non-bps black holes
Interacting non-bps black holes Guillaume Bossard CPhT, Ecole Polytechnique Istanbul, August 2011 Outline Time-like Kaluza Klein reduction From solvable algebras to solvable systems Two-centre interacting
More informationMath. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ.
Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for anti-self-dual connections
More informationVortices and vortex states of Rashba spin-orbit coupled condensates
Vortices and vortex states of Rashba spin-orbit coupled condensates Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University March 5, 2014 P.N, T.Duric, Z.Tesanovic,
More informationContinuity of the Deconfinement Transition in (Super) Yang Mills Theory
Continuity of the Deconfinement Transition in (Super) Yang Mills Theory Thomas Schaefer, North Carolina State University 0.0 0.0 0.0 0.0 0.0 0.0 with Mithat Ünsal and Erich Poppitz Confinement and the
More informationLattice Gauge Theory and the Maxwell-Klein-Gordon equations
Lattice Gauge Theory and the Maxwell-Klein-Gordon equations Tore G. Halvorsen Centre of Mathematics for Applications, UiO 19. February 2008 Abstract In this talk I will present a discretization of the
More informationDiffeomorphism Invariant Gauge Theories
Diffeomorphism Invariant Gauge Theories Kirill Krasnov (University of Nottingham) Oxford Geometry and Analysis Seminar 25 Nov 2013 Main message: There exists a large new class of gauge theories in 4 dimensions
More informationarxiv:hep-th/ v3 18 Jul 2001
Extended QCD versus Skyrme-Faddeev Theory W. S. Bae, Y.M. Cho, and S. W. Kimm Department of Physics, College of Natural Sciences, Seoul National University Seoul 151-742, Korea ymcho@yongmin.snu.ac.kr
More informationarxiv: v1 [hep-lat] 25 Apr 2012
arxiv:14.5586v1 [hep-lat] 5 Apr 1 Dual Meissner effect and non-abelian dual superconductivity in SU(3) Yang-Mills theory High Energy Accelerator Research Organization (KEK) E-mail: Akihiro.Shibata@kek.jp
More informationString calculation of the long range Q Q potential
String calculation of the long range Q Q potential Héctor Martínez in collaboration with N. Brambilla, M. Groher (ETH) and A. Vairo April 4, 13 Outline 1 Motivation The String Hypothesis 3 O(1/m ) corrections
More informationBaryon Physics in Holographic QCD Models (and Beyond) in collaboration with A. Wulzer
Baryon Physics in Holographic QCD Models (and Beyond) Alex Pomarol (Univ. Autonoma Barcelona) in collaboration with A. Wulzer One of the main goals of AdS/CFT: Find the string dual theory of QCD One of
More informationGeneralized Global Symmetries
Generalized Global Symmetries Anton Kapustin Simons Center for Geometry and Physics, Stony Brook April 9, 2015 Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries
More informationElectric Dipole Moment of Magnetic Monopole
479 Progress of Theoretical Physics, Vol. 117, No. 3, March 27 Electric Dipole Moment of Magnetic Monopole Makoto Kobayashi High Energy Accelerator Research Organization (KEK, Tsukuba 35-81, Japan and
More information8.821 String Theory Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.81 String Theory Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.81 F008 Lecture 1: Boundary of AdS;
More informationElectric-Magnetic Duality, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory
hep-th/9407087, RU-94-52, IAS-94-43 arxiv:hep-th/9407087v1 15 Jul 1994 Electric-Magnetic Duality, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory N. Seiberg Department
More informationSuperinsulator: a new topological state of matter
Superinsulator: a new topological state of matter M. Cristina Diamantini Nips laboratory, INFN and Department of Physics and Geology University of Perugia Coll: Igor Lukyanchuk, University of Picardie
More informationBPS states, permutations and information
BPS states, permutations and information Sanjaye Ramgoolam Queen Mary, University of London YITP workshop, June 2016 Permutation centralizer algebras, Mattioli and Ramgoolam arxiv:1601.06086, Phys. Rev.
More informationQuantum Information, the Jones Polynomial and Khovanov Homology Louis H. Kauffman Math, UIC 851 South Morgan St. Chicago, IL 60607-7045 www.math.uic.edu/~kauffman http://front.math.ucdavis.edu/author/l.kauffman
More informationMeson-Nucleon Coupling. Nobuhito Maru
Meson-Nucleon Coupling from AdS/QCD Nobuhito Maru (Chuo Univ.) with Motoi Tachibana (Saga Univ.) arxiv:94.3816 (Accepted in in EPJC) 7/9/9 workshop@yitp Plan 1: Introduction : Meson sector (review) 3:
More informationScale hierarchy in high-temperature QCD
Scale hierarchy in high-temperature QCD Philippe de Forcrand ETH Zurich & CERN with Oscar Åkerlund (ETH) Twelfth Workshop on Non-Perturbative QCD, Paris, June 2013 QCD is asymptotically free g 2 4 = High
More informationCHAPTER II: The QCD Lagrangian
CHAPTER II: The QCD Lagrangian.. Preparation: Gauge invariance for QED - 8 - Ã µ UA µ U i µ U U e U A µ i.5 e U µ U U Consider electrons represented by Dirac field ψx. Gauge transformation: Gauge field
More informationBadis Ydri Ph.D.,2001. January 19, 2011
January 19, 2011 Curriculum Vitae.. : Fuzzy. : Noncommutative Gauge Theory and Emergent Geometry From Yang-Mills Matrix Models.. Education-Pre Baccalaureat in Mathematics (June 1989), Moubarek El Mili
More informationFractionized Skyrmions in Dense Compact-Star Matter
Fractionized Skyrmions in Dense Compact-Star Matter Yong-Liang Ma Jilin University Seminar @ USTC. Jan.07, 2016. Summary The hadronic matter described as a skyrmion matter embedded in an FCC crystal is
More informationEffective theories for QCD at finite temperature and density from strong coupling
XQCD 2011 San Carlos, July 2011 Effective theories for QCD at finite temperature and density from strong coupling Owe Philipsen Introduction to strong coupling expansions SCE for finite temperature: free
More informationg abφ b = g ab However, this is not true for a local, or space-time dependant, transformations + g ab
Yang-Mills theory Modern particle theories, such as the Standard model, are quantum Yang- Mills theories. In a quantum field theory, space-time fields with relativistic field equations are quantized and,
More informationG2 gauge theories. Axel Maas. 14 th of November 2013 Strongly-Interacting Field Theories III Jena, Germany
G2 gauge theories Axel Maas 14 th of November 2013 Strongly-Interacting Field Theories III Jena, Germany Overview Why G2? Overview Why G2? G2 Yang-Mills theory Running coupling [Olejnik, Maas JHEP'08,
More informationInteger quantum Hall effect for bosons: A physical realization
Integer quantum Hall effect for bosons: A physical realization T. Senthil (MIT) and Michael Levin (UMCP). (arxiv:1206.1604) Thanks: Xie Chen, Zhengchen Liu, Zhengcheng Gu, Xiao-gang Wen, and Ashvin Vishwanath.
More informationA Brief Introduction to AdS/CFT Correspondence
Department of Physics Universidad de los Andes Bogota, Colombia 2011 Outline of the Talk Outline of the Talk Introduction Outline of the Talk Introduction Motivation Outline of the Talk Introduction Motivation
More informationQuantising Gravitational Instantons
Quantising Gravitational Instantons Kirill Krasnov (Nottingham) GARYFEST: Gravitation, Solitons and Symmetries MARCH 22, 2017 - MARCH 24, 2017 Laboratoire de Mathématiques et Physique Théorique Tours This
More informationAnalytical study of Yang-Mills theory from first principles by a massive expansion
Analytical study of Yang-Mills theory from first principles by a massive expansion Department of Physics and Astronomy University of Catania, Italy Infrared QCD APC, Paris Diderot University, 8-10 November
More informationKontsevich integral in topological models of gravity
Kontsevich integral in topological models of gravity Alexander Lobashev, MSU February 2, 2018 Alexander Lobashev, MSU Kontsevich integral in topological models of gravityfebruary 2, 2018 1 / 16 3d gravity
More informationGravity vs Yang-Mills theory. Kirill Krasnov (Nottingham)
Gravity vs Yang-Mills theory Kirill Krasnov (Nottingham) This is a meeting about Planck scale The problem of quantum gravity Many models for physics at Planck scale This talk: attempt at re-evaluation
More informationShow, for infinitesimal variations of nonabelian Yang Mills gauge fields:
Problem. Palatini Identity Show, for infinitesimal variations of nonabelian Yang Mills gauge fields: δf i µν = D µ δa i ν D ν δa i µ..) Begin by considering the following form of the field strength tensor
More informationGlobal phase diagrams of two-dimensional quantum antiferromagnets. Subir Sachdev Harvard University
Global phase diagrams of two-dimensional quantum antiferromagnets Cenke Xu Yang Qi Subir Sachdev Harvard University Outline 1. Review of experiments Phases of the S=1/2 antiferromagnet on the anisotropic
More informationPushing dimensional reduction of QCD to lower temperatures
Intro DimRed Center symm. SU(2) Outlook Pushing dimensional reduction of QCD to lower temperatures Philippe de Forcrand ETH Zürich and CERN arxiv:0801.1566 with A. Kurkela and A. Vuorinen Really: hep-ph/0604100,
More informationNon-Perturbative Thermal QCD from AdS/QCD
Issues Non-Perturbative Thermal QCD from AdS/QCD * Collaborators: B. Galow, M. Ilgenfritz, J. Nian, H.J. Pirner, K. Veshgini Research Fellow of the Alexander von Humboldt Foundation Institute for Theoretical
More informationPoS(LAT2005)324. D-branes and Topological Charge in QCD. H. B. Thacker University of Virginia
D-branes and Topological Charge in QCD University of Virginia E-mail: hbt8r@virginia.edu The recently observed long-range coherent structure of topological charge fluctuations in QCD is compared with theoretical
More information