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arXiv:hep-th/0408068v5 24 Jan 2011

January 6, 2005

January 14, 2011 OU-HET 478/2004

hep-th/0408068

Generalized Monopoles

in Six-dimensional Non-Abelian Gauge Theory

Hironobu Kihara,1Yutaka Hosotani1and Muneto Nitta2

1Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

2Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

Abstract

A spherically symmetric monopole solution is found in SO(5) gauge theory with

Higgs scalar fields in the vector representation in six-dimensional Minkowski spacetime.

The action of the Yang-Mills fields is quartic in field strengths. The solution saturates

the Bogomolny bound and is stable.

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Long time ago Dirac showed that quantum mechanics admits a magnetic monopole of

quantized magnetic charge despite the presence of a singular Dirac string.[1, 2] A quantized

Dirac string is unphysical entity in the sense that it yields no physical, observable effect.

Much later ’t Hooft and Polyakov showed that such magnetic monopoles emerge as regu-

lar configurations in SO(3) gauge theory with spontaneous symmetry breaking triggered by

triplet Higgs scalar fields.[3]-[6] ’t Hooft-Polyakov monopoles emerge in grand unified theory

of electromagnetic, weak, and strong interactions as well. Although a monopole has not

been found experimentally as a single particle, the existence of such objects has far reaching

consequences. In the early universe, monopoles might have beed copiously produced, sig-

nificantly affecting the history of the universe since then. In strong intercations, monopole

configurations are believed vital for color and quark confinement.

In the superstring theory all matter and interactions including gravity are truely unified

in ten spacetime dimensions. Six extra dimensions may be compactified in a small size, or the

observed four-dimensional spacetime can be a brane immersed in ten dimensional spacetime.

It is important in this scenario to explore solitonic objects in higher dimensional spacetime,

which may play an important role in compactfying extra dimensions, or in producing and

stabilizing brane structures. Recent extensive study of domain walls in supersymmetric

theories, for instance, may have a direct link to the brane world scenario.[7] In this paper

we explore and establish solitons with finite energies in higher dimensional spacetime.

The energy of ’t Hooft-Polyakov monopoles is bound from below by a topological charge.

Monopole solutions saturate such bound, thereby the stability of the solutions being guar-

anteed by topology.[8] This observation prompts a question if there can be a monopole

solution in higher dimensions. Kalb and Ramond introduced Abelian tensor gauge fields

coupled to closed strings.[9] Nepomechie showed that a new type of monopole solutions

appear in those Kalb-Ramond antisymmetric tensor gauge fields.[10] Their implications to

the confinement[11] and to ten-dimensional Weyl invariant spacetime[12] has been explored.

Topological defects in six dimensional Minkowski space-time as generalization of Dirac’s

monopoles were also found.[13] Tchrakian has investigated monopoles in non-Abelian gauge

theory in higher dimensions whose action involves polynomials of field strengths of high

degrees.[14, 15] Further, it has been known that magnetic monopoles appear in the matrix

model in the gauge connections describing Berry’s phases on fermi states. In particular, in

the USp matrix model they are described by SU(2)-valued anti-self-dual connections.[16]

The purpose of this paper is to present regular monopole configurations with saturated

Bogomolny bound in SO(5) gauge theory in six dimensions. Although the existence of

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such solutions has been suspected by Tchrakian for a long time, the explicit construction of

solutions has not been given. We stress that the monopole solution presented below is the

first example of a soliton in non-Abelian gauge theory in higher dimensions which is regular

everywhere and has a finte energy.

Let us recall that in ’t Hooft-Polyakov monopoles in four dimensions, both SO(3) gauge

fields and scalar fields are in the vector representation. In three space dimensions the Bo-

gomolny equations for those fields match both in space indices and internal SO(3) indices.

This correspondence seemingly becomes obscure when space dimensions are greater than

three. A key to find correct Bogomolny equations is facilitated with the use of the Dirac or

Clifford algebra.

Consider SO(5) gauge theory in six dimensions. Gauge fields Aab

adjoint representation, whereas scalar fields φaare in the vector representation (a,b = 1 ∼ 5).

To interrelate these two, we introduce a basis {γa} of the Clifford algebra; {γa,γb} = 2δab

(a,b = 1 ∼ 5). We write φ ≡ φaγaand A = 1/2Aab

strength 2-form is given by F = F(A) ≡ dA + gA2where g is the gauge coupling constant.

Similarly, a covariant derivative 1-form of φ is given by DAφ ≡ dφ + g[A,φ]. Under a gauge

transformation, A → ΩAΩ−1+ (1/g)ΩdΩ−1, F → ΩFΩ−1, and DAφ → ΩDAφΩ−1, where

Ω = exp{εab(x)γab}

The action is given by

µ= −Aba

µare in the

µγabdxµwhere γab= 1/2[γa,γb]. The field

I≡

? ?1

8TrF2∗ F2+1

8TrDAφ ∗ DAφ − λ(φaφa− H2

0)2d6x

?

=

?

d6x

?

−

1

8 · 4!Tr(F2)µνρσ(F2)µνρσ−1

8{Fµν,Fρσ}dxµ∧ dxν∧ dxρ∧ dxσare given by

(F2)µνρσ= Te

2DµφaDµφa− λ(φaφa− H2

0)2

?

.(1)

Here the components of F2=1

µνρσγe− Sµνρσ

1

2 · 4!ǫabcde?

?

Te

µνρσ(A) =Fab

µνFcd

ρσ+ Fab

µρFcd

σν+ Fab

µσFcd

νρ

?

Sµνρσ(A) =1

4!

Fab

µνFab

ρσ+ Fab

µρFab

σν+ Fab

µσFab

νρ

?

,(2)

so that in the action

1

4Tr(F2)µνρσ(F2)µνρσ= Te

µνρσTµνρσ

e

+ 4SµνρσSµνρσ. The action of this

type has been considered in ref. [14]. The relations in (2) are special to SO(5) gauge theory.

The action is quartic in Fµν, but is quadratic in F0k. The Hamiltonian is positive semi-

definite and is bounded from below by a topological charge. To see it, first notice that

Te

0jkl= Fab

0iMab,e

i,jkl

,Mab,e

i,jkl=

1

2 · 4!ǫabcdeLcd

i,jkl

,

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S0jkl= Fab

0iNab

i,jkl

,Nab

i,jkl=1

4!Lab

i,jkl

,

Lcd

i,jkl= δijFcd

kl+ δikFcd

lj+ δilFcd

jk

.(3)

The canonical conjugate momentum fields are given by

Πab

i

=

δI

δ˙Aab

i

=1

3!Te

0jkl

δTe

δ˙Fab

0jkl

0i

+4

3!S0jklδS0jkl

δ˙Fab

0i

=

1

3(Mab,e

i,jklMcd,e

m,jkl+ Nab

i,jklNcd

m,jkl)Fcd

0m

≡ Uab,cd

i,mFcd

0m.(4)

U is a symmetric, positive-definite matrix. To confirm the positivity of the Hamiltonian, we

take the A0= 0 gauge in which Fab

0i=˙Aab

i. It immediately follows that

E =

?

d5x

?1

2ΠU−1Π +

1

2 · 4!

?

(Te

ijkl)2+ (Sijkl)2?

+ Hφ

?

≥ 0(5)

where Hφis the scalar field part of the Hamiltonian density.

In the A0= 0 gauge the energy becomes lowest for static configurations˙Aab

i =˙φa= 0.

It is given by

E=

?

d5x1

4!

?1

2(Te

ijkl∓ ǫijklmDmφe)2+1

2Sijkl2± ǫijklmTe

ijklDmφe+ λ(φaφa− H2

0)2

?

≥ ±

?

d5x1

4!ǫijklmTe

ijklDmφe= ±

?

TrDAφF2≡16π2

g2H0Q .(6)

As DAF = 0 and therefore TrDAφF2= d(TrφF2), Q can be expressed as a surface integral

g2

16π2H0

Q = ±

?

S4TrφF2,(7)

where S4is a space infinity of R5.

Q is a charge

(g2/16π2H0)TrDAφF2, which is conserved, d ∗ k = 0. Q can also be viewed as a topo-

logical charge associated with Abelian Kalb-Ramond 3-form gauge fields whose 4-form field

?d5xk0of a 6-dimensional current kµdefined by k = kµdxµ= ± ∗

strength G is given by [14]

G = Tr

?ˆφF2+1

F +1

2g

ˆφ(DAˆφ)2F +

1

16g2ˆφ(DAˆφ)4?

= Trˆφ

?

4g(DAˆφ)2?2

.(8)

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Hereˆφ = φ/|φ|, |φ| =√φaφaand DAˆφ = dˆφ + g[A,ˆφ].

It is the salient feature of G given in (8) that it can be written as

1

16g2Trˆφ(dˆφ)4,

G = dC +

C =

1

2gTrˆφ

?

(dˆφ)2A + g (dˆφAˆφA + dAA + AdA) + g2?

A3+1

3AˆφAˆφA

??

.(9)

C does not have a singularity of the Dirac string type where |φ| ?= 0. G and C are the

’t Hooft 4-form field strengths and the corresponding Kalb-Ramond 3-form fields in six

dimensions, respectively. The expression (9) is valid in the entire six-dimensional spacetime.

We remark that the Kalb-Ramond 3-form fields C in (9) is almost the same as those in

ref. [15] where A is replaced by the asymptotic one which is valid only at r → ∞ (on

S4). We also note that for configurations withˆφ = γ5, only gauge fields in the unbroken

SO(4),ˆA =

2

?4

Trγ5(dˆAˆA +ˆAdˆA) and Tr[ˆφA3+1

As DAˆφ = 0 on S4at space infinity for any configuration with a finite energy, G coinsides

with TrˆφF2on S4. Hence

1

a,b=1Aab

µγabdxµ, contribute in (8) and (9). Indeed, Trγ5(dAA + AdA) =

3(ˆφA)3] =1

6Tr{ˆφ,A}3=1

6Tr{ˆφ,ˆA}3.

Q =

g2

16π2H0

?

S4|φ|G =

1

256π2

?

S4Trˆφ(dˆφ)4.(10)

In the second equality we used the fact that C is regular in S4as |φ| ∼ H0. The quantity

appearing in the last equality in (10) is the winding number. The charge Q is thus regarded

as the magnetic charge associated with Abelian Kalb-Ramond field strengths G.

The Bogomolny bound equation is

∗5(F ∧ F) = ±DAφ(11)

where ∗5is Hodge dual in five-dimensional space. In components it is given by

ǫijklmTe

ijkl

= ±Dmφe,

Sijkl = 0 .(12)

Let us define e ≡ xaγa/r. We make a hedgehog ansatz[15]

φ = H0U(r)e,

A =

1 − K(r)

2g

ede.(13)

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