Topological 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. Proc. R. Soc. Lond A (2012). MD. Abelian vortices from Sinh Gordon and Tzitzeica equations. arxiv:1201.0105. Phys. Lett. B. 710 (2012). MD. Skyrmions from gravitational instantons. arxiv:1206.0016. Proc. R. Soc. Lond. A (2013). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 1 / 13

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, 1954. Waves: Dispersion, diffraction, superpositions. Particles: mass, localisation. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 2 / 13

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, 1954. Waves: Dispersion, diffraction, superpositions. QFT: Fourier transform, infinities, regularisation,..., string theory. Particles: mass, localisation. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 2 / 13

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, 1954. 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 2012 2 / 13

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 2012 3 / 13

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 2012 3 / 13

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 2012 3 / 13

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 2012 3 / 13

Solitons and Topology Soliton stability Nontrivial topology Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 4 / 13

Solitons and Topology Soliton stability Nontrivial topology Complete integrability Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 4 / 13

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 2012 4 / 13

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 2012 5 / 13

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 2 + 1 2 4 (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 2012 5 / 13

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 2 + 1 2 4 (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 2012 5 / 13

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 2012 6 / 13

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 2012 6 / 13

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 2012 6 / 13

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 2012 6 / 13

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 2012 7 / 13

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 2012 7 / 13

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 2012 7 / 13

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 2012 7 / 13

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 2012 7 / 13

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 2012 7 / 13

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 2012 7 / 13

Skyrmions Skyrme model of Baryons. Nonlinear pion field U : R 3,1 SU(2). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 8 / 13

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 2012 8 / 13

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 2012 8 / 13

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 2012 9 / 13

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 2012 9 / 13

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 2012 9 / 13

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 2012 9 / 13

Skyrmions from gravitational instantons Skyrmion from the holonomy of the spin connection (MD, 2012). Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 10 / 13

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 2012 10 / 13

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 2012 10 / 13

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 2012 11 / 13

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 2012 11 / 13

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 2012 11 / 13

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 1 2 + a 2 η 2 2 + a 3 η 3 2, Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 12 / 13

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 2 1 + a 2 η 2 2 + 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 2012 12 / 13

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 2 1 + a 2 η 2 2 + 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 2012 12 / 13

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 1 2 + a 2 η 2 2 + 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 2012 12 / 13

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 1 2 + a 2 η 2 2 + 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 2012 12 / 13

Outlook Solitons (vortices, skyrmions) induced by the geometry of space time. Dunajski (DAMTP, Cambridge) Topological Solitons from Geometry December 2012 13 / 13

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 2012 13 / 13

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 2012 13 / 13

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 2012 13 / 13

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 2012 13 / 13