Abelian 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 Generalities The Chern-Simons densities Chern-Simons densities of Abelian gauge fields Chern-Simons densities of non-abelian gauge fields Abelian Hopfions CP n sigma models on R 2n+1 Imposition of n-fold azimuthal symmetry The reduced (Abelian) Chern-Simons densities non-abelian Hopfions The 4n 2n Grassmannian sigma models on R 2n+1 Imposition of n-fold azimuthal symmetry The reduced (non-abelian) Chern-Simons densities

Generalities Hopfions are finite energy soliton like solutions to the field equations of some scalar field system. The lower bound on their energies rely on the Chern-Simons density as a topological charge density and hence can exist only in odd space dimensions. The Chern-Simons density is defined in terms of a gauge connection and a gauge curvature, which means that the scalar field system must be a complex sigma model enabling the necessary definitions of a composite connection and curvature. The Chern-Simons density is by construction, not a total divergence, even when the sigma model constraint is taken account of. Hence, it is not a candidadte for a topological charge density as it stands.

Only after imposition of suitable symmetries does it become a total divergence, qualifying it as a topological charge density. Our prescription here is to achieve this aim by imposing multi-azimuthal symmetries. This is the case with the familiar Skyrme-Fadde ev Hopfion on IR 3. Multi-azimuthal symmetry in IR 2n+1 eliminates n azimuthal angles, each in one of the n 2-dimensional subspaces (planes). It follows that after imposition of such symmetry, the residual subsystems will be (n + 1)-dimensional. The residual Chern-Simons density in a (n + 1)-dimensional space can be a total divergence if the Ansatz is parametrised by (n + 1) independant functions of the (n + 1) residual coordinates.

The residual Chern-Simons density will be a total divergence only when parametrised by functions that fulfill the sigma model constraint. Otherwise, it will be essentially total divergence in the sense that only when the constraint is imposed via a Lagrange multiplier will the resulting Euler-Lagrange equations be trivial. Chern-Simons densities composite connections and curvatures present candidates for topological charge densities only when subjected to suitable symmetries. This is in stark contrast to the topological charge densities of solitons (incl. instantons, monopoles, Skyrmions, vortices, etc.) whose topological charge densities are defined by Chern-Pontryagin densities (and their descendants) which are total divergence without imposition of any symmetry.

Abelian Chern-Simons densities Abelian curvature of Abelian connection A i F ij = i A j j A i Definition of Abelian (gauge invariant) Chern-Pontryagin density in 2n + 2 dimensions Ω CP = ε µ1 ν 1 µ 2 ν 2...µ n+1 ν n+1 F µ1 ν 1 F µ2 ν 2... F µn+1 ν n+1 = ε µ1 ν 1 µ 2 ν 2...µ n+1 ν n+1 µn+1 ( Aνn+1 F µ1 ν 1 F µ2 ν 2... F µnν n ) = µn+1 Ω µn+1 implies definition of Abelian (gauge variant) Chern-Simons density in 2n + 1 dimensions Ω CS = ε i1 j 1 i 2 j 2...i nj nj n+1 A jn+1 F i1 j 1 F i2 j 2... F inj n Ω CS is not a total divergence and leads to the variational equations which is gauge covariant. ε i1 j 1 i 2 j 2...i nj nj n+1 F i1 j 1 F i2 j 2... F inj n = 0

non-abelian Chern-Simons densities non-abelian curvature of Abelian connection A i F ij = i A j j A i + [A i, A j ] Definition of non-abelian (gauge invariant) Chern-Pontryagin density in 2n + 2 dimensions Ω CP = ε µ1 ν 1 µ 2 ν 2...µ n+1 ν n+1 Tr F µ1 ν 1 F µ2 ν 2... F µn+1 ν n+1 = µn+1 Ω µn+1 which is a total divergence and likewise implies definition of non-abelian (gauge variant) Chern-Simons density in 2n + 1 dimensions. For n = 1, D = 3, it is ( Ω (1) CS = ε ijktr A k F ij 2 ) 3 F if j. For n = 2, D = 5, it is Ω (2) CS = ε ijklm Tr A m ( F ij F kl F ij A k A l + 2 5 A ia j A k A l ).

The equations of motion these result in are which are gauge covariant. ε ijk F ij = 0 ε ijklm F ij F kl = 0 For n 3 there are multiple distinct definitions for the CS density, each characterised by the number of traces in the definition. For n = 3, D = 7, there are two possibilities; a definition with a single trace and another one with double trace. These are Ω (3) CS = ε ijklmnp Tr A p (F ij F kl F mn 4 5 F ijf kl A m A n 2 5 F ija k F lm A n + 4 5 F ija k A l A m A n 8 ) 35 A ia j A k A l A m A n, ( Ω (3) CS = ε ijklmnp Tr A p F mn 2 ) 3 A ma n (Tr F ij F kl )

The equations of motion these result in are ε ijklmnp F ij F kl F mn = 0 ε ijklmnp (Tr F ij F kl ) F mn = 0 which are gauge covariant.

CP n models on IR 2n+1 Described by complex n tuplets Z = z 1 z 2.... z n+1 z a ; Z = z 1 z 2.... z n+1 za a = 1, 2,..., n + 1, parametrised by 2n + 1 real independant functions subject to constraint and P = is a projection operator. Z Z z a z a = 1, ( 1I Z Z ) ( δ b a z a z b).

This constraint in invariant under the action of the local Abelian transformation g = e iλ(x) Z Z g leading to the definition of the Abelian composite connection B i = i Z i Z, i = 1, 2,..., 2n + 1 transforming like B i B i ± i Λ. There follow the definitions of the covariant derivative of Z and the Abelian curvature D i Z = i Z + i B i Z, G ij = i B j j B i, D i Z transforming covariantly under the action of g, and G ij invariantly.

The topological charge density of the Abelian Hopfion on IR 2n+1 is the Abelian Chern-Simons density on IR 2n+1 defined by replacing A i B i and F ij G ij Ω CS = ε i1 i 2...i 2n+1 B i2n+1 G i1 i 2 G i3 i 4... G i2n 1 i 2n. Subjecting the density to variations of Z (or Z) Ω CS + λ (1 Z Z) λ being the Lagrange multiplier yields the nontrivial gauge covariant equation ε i1 i 2...i 2n+1 D i2n+1 Z G i1 i 2 G i3 i 4... G i2n 1 i 2n = 0 which trivialises only under the appropriate symmetries, subject to which Ω CS becomes essentially total divergence, and hence a candidate for a topological charge density.

CP 1 system on IR 3 : mono-azimuthal (axial) symmetry The Ansatz for Z, featuring axial symmetry in the (x 1, x 2 ) plane, is Z = [ a + ib c e inϕ ] [ sin f 2 eiα cos f 2 einϕ ], a 2 + b 2 + c 2 = 1 a, b, c, f and α are functions of ρ = x α 2 and z x 3 ; α = 1, 2 n is the vorticity and ϕ is the azimuthal angle in the (x 1, x 2 ) plane. Subject to this Ansatz, the Abelian Chern Simons density reduces to Ω (3) CS = 4 n ρ c det a b c a,ρ b,ρ c,ρ a,z b,z c,z.

This expression of Ω (3) CS [a, b, c] is not a total divergence. Rather, Ω (3) CS [a, b, c] + λ(1 a2 + b 2 + c 2 ) is essentially total divergence and yields trivial equations of motion. (λ is a Lagrange multiplier.) Only when the trigonometric parametrisation, which satisfies the sigma model constraint, does [f, α] become explicitly the total divergence, Ω (3) CS where F (ρ, z) is the function Ω (3) CS = 4 n 3 ρ (F,ρ α,z (ρ, z)), F = cos 3 f 2. The topological charge can then be expressed as, Q = 4 3 2π Ω (3) CS ρ dρ dz = 4 3 2π n F [,ρ α,z] dρ dz

Denoting the coordinates in the half plane (ρ, z) = ξ i, i = 1, 2, 3, i.e., ( ) r sin ψ ξ i = r cos ψ with 0 ψ π, this integral can be rewritten as Q = 4 3 2π n ε ij i F j α d 2 ξ = 4 3 2π n (F ˆx i ε ij j α) r ˆξ i ds = 4 ψ=π 3 2π F ψ α r dψ ψ=0 where ψ is the polar angle in the (ρ, z) half plane. Requiring the field configurations have the asymptotic values the results is lim f (r, θ) = 0, lim r α(r, θ) = m π, r Q = 8 3 n m π2.

CP 2 system on IR 5 : bi-azimuthal symmetry The Ansatz for Z, featuring bi-azimuthal symmetry in the (x 1, x 2 ) and (x 3, x 4 ) planes, is a + ib sin 1 Z = c 1 e in 2 f eiα 1ϕ cos 1 c 2 e in 2 f sin g ein 1ϕ 2χ cos 1 2 f cos g ein 2χ a, b, c 1, c 2, f, g and α are functions of ρ = x α 2, σ = x A 2, and z x 5 ; α = 1, 2; A = 3, 4 n 1 and n 2 are the vorticities, and, ϕ and χ are the azimuthal angles in the (x 1, x 2 ) and (x 3, x 4 ) planes respectively. Subject to this Ansatz, the Abelian Chern Simons density on IR 5 reduces to Ω (5) CS = 32 n a b c 1 c 2 1n 2 ρσ c 1 c 2 det a,ρ b,ρ c 1,ρ c 2,ρ a,σ b,σ c 1,σ c 2,σ. a,z b,z c 1,z c 2,z

Using the trigonometric parametrisation of this Ansatz, in terms of the functions F = cos f + 1 cos 2f, G = cos 2g, 2 and denoting the coordinates in the quarter sphere (ρ, σ, z) = ξ i, i = 1, 2, 3, i.e., r sin ψ sin θ ξ i = r sin ψ cos θ r cos ψ with 0 ψ π and 0 θ π 2, the topological charge Q is given by the volume integral Q = (2π) 2 n 1 n 2 = (2π) 2 n 1 n 2 where ds = r 2 sin ψ dψ dθ, i.e. ε ijk i G j G k α d 3 ξ ε ijk (F j G k α) ˆξ i ds r

Q = 4 π 2 n 1 n 2 π ψ=0 π 2 θ=0 F ( ψ G θ α ψ α θ G) dψ dθ. r Here, ψ is the polar angle between the (ρ, σ)-plane and z. Requiring that the field configurations in question satisfy the asymptotic values lim f = 0, lim g = θ, lim α = m π, r r r the result is Ω (5) CS = 12 n 1 n 2 m π 3.

CP 3 system on IR 7 : tri-azimuthal symmetry The Ansatz for Z, featuring tri-azimuthal symmetry in the (x 1, x 2 ), (x 3, x 4 ) and (x 5, x 6 ) planes, is Z = a + ib c 1 e in 1ϕ c 2 e in 2χ c 3 e in 3ζ = sin 1 2 f eiα cos 1 2 f sin g cos h ein 1ϕ cos 1 2 f sin g sin h ein 2χ cos 1 2 f cos g ein 3ζ a, b, c 1, c 2, c 3, f, g, h and α are functions of ρ = x α 2, σ = x A 2, τ = x a 2, and z x 7 ; with α = 1, 2; A = 3, 4; a = 5, 6. n 1, n 2 and n 3 are the vorticities, and, ϕ, χ and ζ are the azimuthal angles in the (x 1, x 2 ), (x 3, x 4 ) and (x 5, x 6 ) planes respectively.

Subject to this Ansatz the Abelian Chern Simons density on IR 7 reduces to a b c 1 c 2 c 3 Ω (7) CS = 96 n 1 n 2 n 3 a,ρ b,ρ c 1,ρ c 2,ρ c 3,ρ c 1 c 2 c 3 det ρστ a,σ b,σ c 1,σ c 2,σ c 3,σ. a,τ b,τ c 1,τ c 2,τ c 3,τ a,z b,z c 1,z c 2,z c 3,z Using the trigonometric parametrisation of this Ansatz, in terms of the functions F = cos 6 f 2, G = 1 4 (1 cos 2g)2, H = cos 2h, and denoting the coordinates in the sextant of the hypersphere (ρ, σ, τ, z) = ξ i, i = 1, 2, 3, 4, i.e., r sin ψ sin θ 1 sin θ 2 ξ i = r sin ψ sin θ 1 cos θ 2 r sin ψ cos θ 1 r cos ψ with 0 ψ π, 0 θ 1 π 2 and 0 θ 2 π 2,

Q = 4 (2π) 3 n 1 n 2 n 3 = 4 (2π) 3 n 1 n 2 n 3 ε ijkl i F j G k H z α d 4 ξ ε ijkl (F j G k H l α) ˆξ i ds r where ds = r 3 sin 2 ψ sin θ 1 dψ dθ 1 dθ 2. Requiring the field configuration in question take the asymptotic values lim g = θ 1, lim h = θ 2, lim α = m π, r r r the result is Ω (5) CS = 64 n 1 n 2 n 3 m π 4.

Grassmannian models on IR 2n+1 These Grassmannian models are described by complex valued fields [ ] z1 Z = where z 1 and z 2 are complex 2n 2n matrices subject to the constraint Z Z = 1I 2n 2n. and the 4n 4n quantity Π = (1I Z Z ) is a projection operator. This constraint is invariant under the action of the local unitary non-abelian gauge transformation g acting on Z z 2 Z Z g, Z g Z.

Here, the unitary matrix g is chosen to be an element of SO(2n + 2) in the 2n 2n chiral Dirac matrix representation. The invariance of this constraint leads to the definition of the non-abelian (anti-hermitian) composite connection transforming like B i = Z i Z. B i g 1 B i g + g 1 i g. There follow the definitions of the covariant derivative of Z and the (composite) non-abelian curvature D i Z = i Z Z B i G ij = [i B j] + [B i, B j ] which under the action of g transform covariantly as D i Z D i Z g, G ij g 1 G ij g.

The non-abelian Chern-Simons densities on IR 2n+1, up to n = 3, are ( Ω (1) CS = ε ijktr B k G ij 2 ) 3 B ib j. Ω (2) CS = ε ijklm Tr B m ( G ij G kl G ij B k B l + 2 5 B ib j B k B l ). Ω (3) CS = ε ijklmnp Tr B p (G ij G kl G mn 4 5 G ijg kl B m B n 2 5 G ijb k G lm B n + 4 5 G ijb k B l B m B n 8 ) 35 B ib j B k B l B m B n, ( Ω (3) CS = ε ijklmnp Tr B p G mn 2 ) 3 B mb n (Tr G ij G kl ).

Subjecting these densities to variations of Z (or Z) Ω (n) CS + Λ (1I Z Z) Λ being the (now matrix valued) Lagrange multiplier, the resulting nontrivial gauge covariant equations in the above examples are ε ijk D k Z G ij = 0 ε ijklm D m Z G ij G kl = 0 ε ijklmnp D p Z G ij G kl G mn = 0 ε ijklmnp (Tr G ij G kl ) D p Z G mn = 0 which trivialise only under the appropriate symmetries, subject to which Ω (n) CS, rendering them candidates for topological charge densities.

2 4 Grassmannian on IR 3 : mono-azimuthal symmetry The Anstaz we for the field Z on IR 3 is [ ] [ ] z1 a 1I + 2 b Σ34 Z = = z 2 c n α, Σ α the functions (a, b, c) depending on (ρ, z) as in the Abelian case and with the unit vector n α ( ) n α cos n ϕ = sin n ϕ n being the winding (vortex) number in the (x 1, x 2 ) plane. The spin matrices Σ α = ( Σ 1, Σ 2 ), are the first two of the chiral Dirac matrices in 4 dimensions, the last two being ( Σ 3, Σ 4 ). The chiral representations of the SO(4) algebra are with i, j = α, 3, 4. Σ ij = 1 4 Σ [i Σ j],

It turns out that this Ansatz leads to an Abelian composite connection B i = (B α, B z ), so our prescription cannot supply a non-abelian Hopfion in three dimensions. B α = 2 [(a b,ρ b a,ρ ) ˆx α + nρ ] c2 (ˆxε) α Σ 12, B z = 2 (a b,z b a,z ) Σ 12, whose commutators [B i, B j ] = [B i, B z ] = 0. The composite curvature G ij is then Abelian, and coincides with the previously constructed Abelian case.

4 8 Grassmannian on IR 5 : bi-azimuthal symmetry The Anstaz we use for the field on IR 5 is Z = [ z1 z 2 ] [ = a 1I + 2 b Σ 56 c 1 n α 1 Σ α + c 2 n A 2 Σ A ], The functions (a, b, c 1, c 2 ) depending on (ρ, σ, z) as in the corresponding Abelian case, with α = 1, 2, A = 3, 4 and z x 5. ϕ and χ are the azimuthal angles in the (x 1, x 2 ) and (x 3, x 4 ) planes respectively, such that the unit vectors n α and n2 A are parametrised as n α 1 = ( cos n1 ϕ sin n 1 ϕ ) n A 2 = ( cos n2 χ sin n 2 χ ), (n 1, n 2 ) being the winding (vortex) numbers in each of the two planes respectively. The spin matrices ( Σ α, Σ A, Σ 5, Σ 6 ), are the (left) chiral Dirac matrices in 6 dimensions, in terms of which the chiral representations of the SO(6) algebra are constructed.

Subject to the bi-azimuthally symmetric Ansatz the Chern-Simons density reduces to Ω (5) CS = 96 i n a b c 1 c 2 1n 2 ρσ c 1 c 2 [ 4 + 3(c1 2 + c2 2 )] det a,ρ b,ρ c 1,ρ c 2,ρ a,σ b,σ c 1,σ c 2,σ. a,z b,z c 1,z c 2,z Note that this non-abelian density differs qualitatively from the corresponding Abelian one due to the appearance of the prefactor [ 4 + 3(c 2 1 + c 2 2 )]. Using the trigonometric parametrisation of this Ansatz, in terms of the functions F = 5 cos f + cos 2 f cos 3 f, G = cos 2g, and denoting the coordinates in the quarter sphere (ρ, σ, z) = ξ i, i = 1, 2, 3, i.e., as before in the Abelian case on IR 5 the topological charge Q is given by the volume integral

Q = 3 2 i (2π)2 n 1 n 2 = 3 2 i (2π)2 n 1 n 2 ε ijk i G j G k α d 3 ξ ε ijk (F j G k α) ˆξ i ds r where ds = r 2 sin ψ dψ dθ, and where we have applied Gauss Theorem, and Q = 6 i π 2 n 1 n 2 π ψ=0 π 2 θ=0 F ( ψ G θ α ψ α θ G) dψ dθ. r Requiring the field configurations in question have the asymptotic values the result is lim f = 0, lim g = θ, lim α = m π, r r r Ω (5) CS = 60 i n 1 n 2 m π 3.

8 16 Grassmannian on IR 7 : tri-azimuthal symmetry The Anstaz we use for the field on IR 7 is [ ] [ ] z1 a 1I + 2 b Σ 78 Z = = z 2 c 1 n1 α Σ α + c 2 n2 A Σ A + c 3 n3 A Σ, a The functions (a, b, c 1, c 2, c 3 ) depending on (ρ, σ, τ, z) as in the corresponding Abelian case, with α = 1, 2, A = 3, 4, a = 5, 6 and z x 7. The azimutal angles in the (x 1, x 2 ), (x 3, x 4 ) and (x 5, x 6 ) planes are denoted by ϕ, χ and ζ respectively, such that the unit vectors n1 α, na 2 and na 3 are parametrised as n α 1 = ( cos n1 ϕ sin n 1 ϕ ) n A 2 = ( cos n2 χ sin n 2 χ ), n3 a = ( cos n3 ζ sin n 3 ζ (n 1, n 2, n 3 ) being the winding (vortex) numbers in each of the three planes respectively. The spin matrices ( Σ α, Σ A, Σ a, Σ 7, Σ 8 ), are the (left) chiral Dirac matrices in 8 dimensions ),

Subject to the tri-azimuthally symmetric Ansatz in the Chern-Simons density reduces to a b c d e Ω (7) CS n 1 n 2 n 3 a,ρ b,ρ c 1,ρ c 2,ρ c 3,ρ c 1 c 2 c 3 Θ(c 1, c 2, c 3 ) det ρστ a,σ b,σ c 1,σ c 2,σ c 3,σ a,τ b,τ c 1,τ c 2,τ c 3,τ a,z b,z c 1,z c 2,z c 3,z where the prefactor Θ(c 1, c 2, c 3 ) is not yet calcuated. Using the trigonometric parametrisation of this Ansatz and requiring the appropriate asymptotic behaviours, as before, the resulting topological charge is calculated Ω (5) CS n 1 n 2 n 3 m π 4.