Topological Solitons from Geometry

Transcription

1 Topological Solitons from Geometry Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge Atiyah, Manton, Schroers. Geometric models of matter. arxiv: Proc. R. Soc. Lond A (2012). MD. Abelian vortices from Sinh Gordon and Tzitzeica equations. arxiv: Phys. Lett. B. 710 (2012). MD. Skyrmions from gravitational instantons. arxiv: Proc. R. Soc. Lond. A (2013). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

2 Waves or Particles Here I will only say that I am emphatically in favour of the retention of the particle idea. Naturally, it is necessary to redefine what is meant. Max Born, Nobel Lecture, December 11, Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

3 Waves or Particles Here I will only say that I am emphatically in favour of the retention of the particle idea. Naturally, it is necessary to redefine what is meant. Max Born, Nobel Lecture, December 11, Waves: Dispersion, diffraction, superpositions. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

4 Waves or Particles Here I will only say that I am emphatically in favour of the retention of the particle idea. Naturally, it is necessary to redefine what is meant. Max Born, Nobel Lecture, December 11, Waves: Dispersion, diffraction, superpositions. Particles: mass, localisation. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

5 Waves or Particles Here I will only say that I am emphatically in favour of the retention of the particle idea. Naturally, it is necessary to redefine what is meant. Max Born, Nobel Lecture, December 11, Waves: Dispersion, diffraction, superpositions. QFT: Fourier transform, infinities, regularisation,..., string theory. Particles: mass, localisation. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

6 Waves or Particles Here I will only say that I am emphatically in favour of the retention of the particle idea. Naturally, it is necessary to redefine what is meant. Max Born, Nobel Lecture, December 11, Waves: Dispersion, diffraction, superpositions. QFT: Fourier transform, infinities, regularisation,..., string theory. Particles: mass, localisation. Solitons, classical non linear field equations, topological charges, stability. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

7 Solitons Solitons: Non singular, static, finite energy solutions of the field equations. ( 1 L = 2 φ2 t 1 2 φ2 x U(φ)) dx, where φ : R 1,1 R. R Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

8 Solitons Solitons: Non singular, static, finite energy solutions of the field equations. ( 1 L = 2 φ2 t 1 2 φ2 x U(φ)) dx, where φ : R 1,1 R. R Stable vacuum: U 0. Set U 1 (0) = {φ 1, φ 2,..., φ N }. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

9 Solitons Solitons: Non singular, static, finite energy solutions of the field equations. ( 1 L = 2 φ2 t 1 2 φ2 x U(φ)) dx, where φ : R 1,1 R. R Stable vacuum: U 0. Set U 1 (0) = {φ 1, φ 2,..., φ N }. Finite energy: φ(x, t) φ i as x. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

10 Solitons Solitons: Non singular, static, finite energy solutions of the field equations. ( 1 L = 2 φ2 t 1 2 φ2 x U(φ)) dx, where φ : R 1,1 R. R Stable vacuum: U 0. Set U 1 (0) = {φ 1, φ 2,..., φ N }. Finite energy: φ(x, t) φ i as x. φ(x, t) = φ 1 + δφ(x, t). Perturbative Scalar boson ( + m 2 )δφ = 0. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

11 Solitons Solitons: Non singular, static, finite energy solutions of the field equations. ( 1 L = 2 φ2 t 1 2 φ2 x U(φ)) dx, where φ : R 1,1 R. R Stable vacuum: U 0. Set U 1 (0) = {φ 1, φ 2,..., φ N }. Finite energy: φ(x, t) φ i as x. φ(x, t) = φ 1 + δφ(x, t). Perturbative Scalar boson ( + m 2 )δφ = 0. Kink: φ = φ 1 as x. φ = φ 2 as x. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

12 Solitons Solitons: Non singular, static, finite energy solutions of the field equations. ( 1 L = 2 φ2 t 1 2 φ2 x U(φ)) dx, where φ : R 1,1 R. R Stable vacuum: U 0. Set U 1 (0) = {φ 1, φ 2,..., φ N }. Finite energy: φ(x, t) φ i as x. φ(x, t) = φ 1 + δφ(x, t). Perturbative Scalar boson ( + m 2 )δφ = 0. Kink: φ = φ 1 as x. φ = φ 2 as x. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

13 Solitons and Topology Soliton stability Nontrivial topology Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

14 Solitons and Topology Soliton stability Nontrivial topology Complete integrability Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

15 Solitons and Topology Soliton stability Nontrivial topology Complete integrability Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

16 Solitons and Topology Soliton stability Nontrivial topology Complete integrability Static vortices: Abelian gauge potential + Higgs field φ : R 2 C. Boundary conditions: F = 0 and φ = 1 as r. φ : S 1 S 1. Vortex number π 1 (S 1 ) = Z. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

17 Solitons and Topology Soliton stability Nontrivial topology Complete integrability Static vortices: Abelian gauge potential + Higgs field φ : R 2 C. Boundary conditions: F = 0 and φ = 1 as r. φ : S 1 S 1. Vortex number π 1 (S 1 ) = Z. Skyrmions: U : R 3,1 SU(2). Static+finite energy. U : S 3 SU(2). Topological baryon number π 3 (S 3 ) = Z. B = 1 24π 2 R 3 Tr[(U 1 du) 3 ]. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

18 Abelian Higgs Model 2D Kähler manifold (Σ, g, ω). Hermitian line bundle L Σ with U(1) connection A and a global section C section φ : Σ L. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

19 Abelian Higgs Model 2D Kähler manifold (Σ, g, ω). Hermitian line bundle L Σ with U(1) connection A and a global section C section φ : Σ L. Ginsburg Landau energy E[A, φ] = 1 ( Dφ 2 + F (1 φ 2 ) 2 )vol Σ, Σ F = da Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

20 Abelian Higgs Model 2D Kähler manifold (Σ, g, ω). Hermitian line bundle L Σ with U(1) connection A and a global section C section φ : Σ L. Ginsburg Landau energy E[A, φ] = 1 ( Dφ 2 + F (1 φ 2 ) 2 )vol Σ, Σ Bogomolny equations Dφ = 0, F = 1 2 ω(1 φ 2 ) F = da Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

21 Abelian Higgs Model 2D Kähler manifold (Σ, g, ω). Hermitian line bundle L Σ with U(1) connection A and a global section C section φ : Σ L. Ginsburg Landau energy E[A, φ] = 1 ( Dφ 2 + F (1 φ 2 ) 2 )vol Σ, Σ Bogomolny equations Dφ = 0, F = 1 2 ω(1 φ 2 ) Taubes equation: z = x + iy, g = Ω(z, z) dzd z. Set φ = exp(h/2 + iχ), solve for A. 0 h + Ω Ωe h = 0, where 0 = 4 z z. F = da Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

22 Vortex from geometry. 0 h + Ω Ωe h = 0. Vortex number N = 1 2π Σ B vol Σ, h 2N log z z 0 + const Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

23 Vortex from geometry. 0 h + Ω Ωe h = 0. Vortex number N = 1 2π Σ B vol Σ, h 2N log z z 0 + const Vortices on R 2, i. e. Ω = 1. No explicit solutions. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

24 Vortex from geometry. 0 h + Ω Ωe h = 0. Vortex number N = 1 2π Σ B vol Σ, h 2N log z z 0 + const Vortices on R 2, i. e. Ω = 1. No explicit solutions. Make the Taubes equations integrable: Ω = exp ( h/2). 0 (h/2) = sinh (h/2), Sinh Gordon equation. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

25 Vortex from geometry. 0 h + Ω Ωe h = 0. Vortex number N = 1 2π Σ B vol Σ, h 2N log z z 0 + const Vortices on R 2, i. e. Ω = 1. No explicit solutions. Make the Taubes equations integrable: Ω = exp ( h/2). 0 (h/2) = sinh (h/2), Solitons from geometry: L = T Σ, Chern number=euler characteristic. Sinh Gordon equation. SO(2) = U(1), Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

26 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

27 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

28 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

29 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

30 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. In general, Painléve III equation. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

31 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. In general, Painléve III equation. Theorem (MD, 2012) Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

32 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. In general, Painléve III equation. Theorem (MD, 2012) 1 σ(λ) = 2π 1 arcsin (πλ) if 0 λ π 1. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

33 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. In general, Painléve III equation. Theorem (MD, 2012) 1 σ(λ) = 2π 1 arcsin (πλ) if 0 λ π 1. 2 There exists a one vortex solution with strength 4 2/π. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

34 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. In general, Painléve III equation. Theorem (MD, 2012) 1 σ(λ) = 2π 1 arcsin (πλ) if 0 λ π 1. 2 There exists a one vortex solution with strength 4 2/π. 3 CMC surface with deficit angle π near r = 0. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

35 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. In general, Painléve III equation. Theorem (MD, 2012) 1 σ(λ) = 2π 1 arcsin (πλ) if 0 λ π 1. 2 There exists a one vortex solution with strength 4 2/π. 3 CMC surface with deficit angle π near r = 0. Uses isomonodromy theory, and connection formulae (Kitaev). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

36 Geometry of Sinh Gordon vortex g = e h/2 dzd z. Constant mean curvature surface Σ R 2,1. Radial solutions, h = h(r), s.t. h 0 as r. Large r: Modified Bessel equation, h 8λK 0 (r) for a constant λ. Small r: Liouville equation, h = 4σ log r +... for a constant σ. In general, Painléve III equation. Theorem (MD, 2012) 1 σ(λ) = 2π 1 arcsin (πλ) if 0 λ π 1. 2 There exists a one vortex solution with strength 4 2/π. 3 CMC surface with deficit angle π near r = 0. Uses isomonodromy theory, and connection formulae (Kitaev). One more integrable case: Tzitzeica equation and affine spheres. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

37 Skyrmions Skyrme model of Baryons. Nonlinear pion field U : R 3,1 SU(2). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

38 Skyrmions Skyrme model of Baryons. Nonlinear pion field U : R 3,1 SU(2). Energy of static skyrmions. Set R j = U 1 j U su(2). E[U] = 1 ( Tr(R j R j ) + 1 ) 2 8 (Tr([R i, R j ][R i, R j ]) d 3 x > 12π 2 B. R 3 Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

39 Skyrmions Skyrme model of Baryons. Nonlinear pion field U : R 3,1 SU(2). Energy of static skyrmions. Set R j = U 1 j U su(2). E[U] = 1 ( Tr(R j R j ) + 1 ) 2 8 (Tr([R i, R j ][R i, R j ]) d 3 x > 12π 2 B. R 3 Finite energy, U : S 3 S 3 and B = deg(u) Z. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

40 Skyrmions Skyrme model of Baryons. Nonlinear pion field U : R 3,1 SU(2). Energy of static skyrmions. Set R j = U 1 j U su(2). E[U] = 1 ( Tr(R j R j ) + 1 ) 2 8 (Tr([R i, R j ][R i, R j ]) d 3 x > 12π 2 B. R 3 Finite energy, U : S 3 S 3 and B = deg(u) Z. Skyrmions from instanton holonomy (Atiyah+ Manton, 1989). ( ) U(x) = P exp A 4 (x, τ)dτ. Yang Mills potential A su(2) for a Yang Mills field F = da + A A on R 4. Baryon number = instanton number. R Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

41 Skyrmions Skyrme model of Baryons. Nonlinear pion field U : R 3,1 SU(2). Energy of static skyrmions. Set R j = U 1 j U su(2). E[U] = 1 ( Tr(R j R j ) + 1 ) 2 8 (Tr([R i, R j ][R i, R j ]) d 3 x > 12π 2 B. R 3 Finite energy, U : S 3 S 3 and B = deg(u) Z. Skyrmions from instanton holonomy (Atiyah+ Manton, 1989). ( ) U(x) = P exp A 4 (x, τ)dτ. Yang Mills potential A su(2) for a Yang Mills field F = da + A A on R 4. Baryon number = instanton number. R 4 S R R3 Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

42 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

43 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

44 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Small r: g flat metric on R 4. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

45 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Small r: g flat metric on R 4. Large r: Hopf bundle S 3 S 2, Chern class = 1. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

46 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Small r: g flat metric on R 4. Large r: Hopf bundle S 3 S 2, Chern class = 1. Asymptotically locally flat (ALF): M S 1 bundle over S 2 at. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

47 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Small r: g flat metric on R 4. Large r: Hopf bundle S 3 S 2, Chern class = 1. Asymptotically locally flat (ALF): M S 1 bundle over S 2 at. Particles = gravitational instantons. (Atiyah, Manton, Schroers). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

48 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Small r: g flat metric on R 4. Large r: Hopf bundle S 3 S 2, Chern class = 1. Asymptotically locally flat (ALF): M S 1 bundle over S 2 at. Particles = gravitational instantons. (Atiyah, Manton, Schroers). Charged particles = ALF instantons. Charge = Chern number of asymptotic S 1 bundle. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

49 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Small r: g flat metric on R 4. Large r: Hopf bundle S 3 S 2, Chern class = 1. Asymptotically locally flat (ALF): M S 1 bundle over S 2 at. Particles = gravitational instantons. (Atiyah, Manton, Schroers). Charged particles = ALF instantons. Charge = Chern number of asymptotic S 1 bundle. Neutral particles = compact instantons. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

50 Particles as Gravitational Instantons Gravitational instantons = Riemannian, Einsten four manifolds (M, g) whose curvature is concentrated in a finite region of a space-time. Taub NUT metric g = r + m dr 2 + (r 2 + mr)(dθ 2 + sin θ 2 dφ 2 ) + rm2 r r + m (dψ + cos θdφ)2, where θ [0, π), φ [0, 2π), ψ [0, 4π). Small r: g flat metric on R 4. Large r: Hopf bundle S 3 S 2, Chern class = 1. Asymptotically locally flat (ALF): M S 1 bundle over S 2 at. Particles = gravitational instantons. (Atiyah, Manton, Schroers). Charged particles = ALF instantons. Charge = Chern number of asymptotic S 1 bundle. Neutral particles = compact instantons. Proton, neutron, electron, neutrino. Atiyah Hitchin, CP 2, Taub NUT, S 4. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

51 Skyrmions from gravitational instantons Skyrmion from the holonomy of the spin connection (MD, 2012). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

52 Skyrmions from gravitational instantons Skyrmion from the holonomy of the spin connection (MD, 2012). Spin connection Λ 2 = so(4) = so(3) so(3). Taub NUT γ = 0. γ + su(2) self dual YM on (M, g) Skyrmion on M/SO(2). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

53 Skyrmions from gravitational instantons Skyrmion from the holonomy of the spin connection (MD, 2012). Spin connection Λ 2 = so(4) = so(3) so(3). Taub NUT γ = 0. γ + su(2) self dual YM on (M, g) Skyrmion on M/SO(2). A = (1/2)ε ijk γ ij t k, where [t i, t j ] = ε ijk t k. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

54 Skyrmions from gravitational instantons Skyrmion from the holonomy of the spin connection (MD, 2012). Spin connection Λ 2 = so(4) = so(3) so(3). Taub NUT γ = 0. γ + su(2) self dual YM on (M, g) Skyrmion on M/SO(2). A = (1/2)ε ijk γ ij t k, where [t i, t j ] = ε ijk t k. AH, Taub NUT, CP 2 are SO(3) invariant. Pick SO(2) SO(3). ( U(r, ψ, θ) = exp π 3 f j (r)n j t j ), j=1 where n = (cos ψ sin θ, sin ψ sin θ, cos θ) and f 1 = f 2 = r r + m, f 3 = r(r + 2m) (r + m) 2. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

55 Geometry of Taub NUT skyrmion Skyrmion on three space (B, h) = M/SO(2) = H 2 S 1 h = h H 2 + R 2 dψ 2, R : H 2 R. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

56 Geometry of Taub NUT skyrmion Skyrmion on three space (B, h) = M/SO(2) = H 2 S 1 h = h H 2 + R 2 dψ 2, R : H 2 R. Upper half plane: h = dx2 +dy 2 y 2, R 2 = x 2 +y 2 y 2 (m 1 x 2 +y 2 +1) 2 +x 2. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

57 Geometry of Taub NUT skyrmion Skyrmion on three space (B, h) = M/SO(2) = H 2 S 1 h = h H 2 + R 2 dψ 2, R : H 2 R. Upper half plane: h = dx2 +dy 2 y 2, R 2 = Poincare disc D H 2, x + iy = z i iz 1, x 2 +y 2 y 2 (m 1 x 2 +y 2 +1) 2 +x 2. S1 B D Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

58 Atiyah Hitchin skyrmion SO(3) invariant metrics. Let a i = a i (r), and dη 1 = η 2 η 3 etc. g = (a 2 /r) 2 dr 2 + a 1 η a 2 η a 3 η 3 2, Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

59 Atiyah Hitchin skyrmion SO(3) invariant metrics. Let a i = a i (r), and dη 1 = η 2 η 3 etc. g = (a 2 /r) 2 dr 2 + a 1 η a 2 η a 3 η 2 3, Self-dual spin connection A = f 1 (r)η 1 t 1 + f 2 (r)η 2 t 2 + f 3 (r)η 3 t 3. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

60 Atiyah Hitchin skyrmion SO(3) invariant metrics. Let a i = a i (r), and dη 1 = η 2 η 3 etc. g = (a 2 /r) 2 dr 2 + a 1 η a 2 η a 3 η 2 3, Self-dual spin connection A = f 1 (r)η 1 t 1 + f 2 (r)η 2 t 2 + f 3 (r)η 3 t 3. Topological degree Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

61 Atiyah Hitchin skyrmion SO(3) invariant metrics. Let a i = a i (r), and dη 1 = η 2 η 3 etc. Self-dual spin connection g = (a 2 /r) 2 dr 2 + a 1 η a 2 η a 3 η 3 2, A = f 1 (r)η 1 t 1 + f 2 (r)η 2 t 2 + f 3 (r)η 3 t 3. Topological degree 1 B T N = 2 (integration). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

62 Atiyah Hitchin skyrmion SO(3) invariant metrics. Let a i = a i (r), and dη 1 = η 2 η 3 etc. Self-dual spin connection g = (a 2 /r) 2 dr 2 + a 1 η a 2 η a 3 η 3 2, A = f 1 (r)η 1 t 1 + f 2 (r)η 2 t 2 + f 3 (r)η 3 t 3. Topological degree 1 B T N = 2 (integration). 2 B AH = 1. Set r = 2 π/2 0 1 sin (β/2) 2 sin (τ) 2 2 dτ. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

63 Outlook Solitons (vortices, skyrmions) induced by the geometry of space time. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

64 Outlook Solitons (vortices, skyrmions) induced by the geometry of space time. Topological charges (vortex number, baryon number)= Characteristic classes the Levi Civita connection (Euler characteristic, Hirzebruch signature). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

65 Outlook Solitons (vortices, skyrmions) induced by the geometry of space time. Topological charges (vortex number, baryon number)= Characteristic classes the Levi Civita connection (Euler characteristic, Hirzebruch signature). Explicit solutions. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

66 Outlook Solitons (vortices, skyrmions) induced by the geometry of space time. Topological charges (vortex number, baryon number)= Characteristic classes the Levi Civita connection (Euler characteristic, Hirzebruch signature). Explicit solutions. Particles as four manifolds. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13

67 Outlook Solitons (vortices, skyrmions) induced by the geometry of space time. Topological charges (vortex number, baryon number)= Characteristic classes the Levi Civita connection (Euler characteristic, Hirzebruch signature). Explicit solutions. Particles as four manifolds. Thank You. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December / 13