1.3 ’t Hooft-Polyakov monopoles

Here we summarize the features of magnetic monopoles which we will repeatedly quote in the rest of the lecture note. For a detailed exposition of topics discussed in this subsection, the readers should consult the reviews such as [2526], or the textbook [27]. The review by Coleman [28] is also very instructive.3

1.3.1 Classical features

Consider an SU(2) gauge theory with a scalar in the adjoint representation, with the action

S =d4x 1 g2 1 2trFμνFμν + trDμΦDμΦ . (1.3.1)

The field Φ is a traceless Hermitean 2 × 2 matrix.

Consider the vacuum where

Φ = a 0 0 a . (1.3.2)

When a0, the SU(2) gauge symmetry is broken to U(1). Indeed, the vev (1.3.2) commutes with a gauge field strength of the form

Fμν = FμνU(1) 0 0 FμνU(1) (1.3.3)

where FμνU(1) is a U(1) gauge field strength normalized as in Sec. 1.1. Note that the quanta of the scalar fields Φ has electric charge 2 under this U(1) field, as can be found by expanding the covariant derivative.

We are considering a gauge theory; therefore the field Φ does not have to be given exactly as in the right hand side of (1.3.2). Rather, we just need that Φ has eigenvalues ± a. Then we can consider a configuration of the form

Φ(x) = xiσi |x| f(|x|)a (1.3.4)

where i = 1, 2, 3 and f(r) is a dimensionless function such that

lim r0f(r) = 0, lim rf(r) = 1. (1.3.5)

At the spatial infinity, the vev of Φ is conjugate to (1.3.2), and therefore this configuration can be thought of as an excitation of the vacuum given by (1.3.2).

The unbroken U(1) within SU(2) is along Φ. A more general definition of the U(1) field strength FμνU(1), at least when r 0, is then the combination

FμνU(1) := 1 2atrFμνΦ. (1.3.6)

In the region r 0, let us try to bring the configuration (1.3.4) to (1.3.2) by a gauge transformation. This can be done smoothly except at the south pole, by using the gauge transformation

exp[iφ 2 (σ1 sin 𝜃 + σ2 cos 𝜃)],wherex |x| = (cos φ, sin φ cos 𝜃, sin φ sin 𝜃). (1.3.7)

This gives a gauge transformation around the south pole given by

i(σ1 sin 𝜃 + σ2 cos 𝜃) = exp[i𝜃σ3] (iσ2). (1.3.8)

As 𝜃 goes from 0 to 2π, we see that the U(1) field FμνU(1) has the magnetic charge m = 1, and therefore is a monopole. This was originally found by ’t Hooft and Polyakov. Note that its Dirac pairing with the particle of the field Φ is 2, which is twice the minimum allowed value.

Let us evaluate the energy contained in the field configuration. The kinetic energy is 1g2 times

d3x trBiBi + trDiΦDiΦ =d3x tr(Bi DiΦ)2 ± 2trBiDiΦ (1.3.9) ±2d3xtrBiDiΦ = ±2d3xitrBiΦ (1.3.10) = ±2S2dn trBΦ (1.3.11)

where the final integral is over the sphere at the spatial infinity, which according to (1.3.6) evaluates to ± 2(2a)(2πm), where m is the magnetic charge. Therefore we have the bound

(energy of the monopole) 4π g2(2a)|m| (1.3.12)

This is called the Bogomolnyi-Prasad-Sommerfield (BPS) bound. The inequality is saturated if and only if

Bi = ±DiΦ, (1.3.13)

which is called the BPS equation. This fixes the form of the function f(r) in (1.3.4).

1.3.2 Semiclassical features

Given such an explicit monopole solution, there is a way to construct other solutions related by the symmetry. First, the configuration (1.3.4) has a center at the origin of the coordinate system. We can shift the center of the monopole at an arbitrary point y of the spatial 3. These give three zero-modes.

Another zero mode is obtained by the gauge transformation:

eiαΦa (1.3.14)

Note that a gauge transformation which vanishes at infinity is a redundancy of the physical system, but a gauge transformation which does not vanish at infinity is considered to change the classical configuration. For general α, this transformation (1.3.14) changes the asymptotic behavior of Fij(x), but for α = π, the transformation (1.3.14) trivially acts on the fields in the adjoint representation. Therefore α is an angular variable 0 α < π.

The semiclassical quantization of the monopole involves the Fock space of non-zero modes, together with a wavefunction ψ(y,α) depending on the zero modes y and α. The wavefunction along y represents the spatial motion of the center of mass of the monopole. The wavefunction along α represents the electric charge of the monopole, which can be seen as follows.

By comparing (1.3.14) with (1.3.3) and (1.3.6), we see that the unbroken U(1) global gauge transformation by eiφ shifts α by

α α + φ. (1.3.15)

Recall that a state |ψ with electric charge n behaves under the U(1) global transformation by eiφ by

|ψ einφ|ψ. (1.3.16)

Now, as α is a variable with period π, ψ(α) can be expanded as a linear combination of ei2dα where d is an integer. Under (1.3.15) the wavefunction changes as in (1.3.16) with n = 2d, therefore we see that the monopole state with this zero-mode wave function has the electric charge 2d.

Summarizing, the combination of the electric charge and the magnetic charge (n,m) we obtain from the semi-classical quantization has the form (n,m) = (2d, 1) where d is an integer. This was found originally by Julia and Zee: once we quantize the ’t Hooft-Polyakov monopole, we not only have a purely-magnetic monopole but a whole tower of dyon states, with d = to + .

Finally let us consider the effect of the fermionic zero modes in the ’t Hooft-Polyakov monopole (1.3.4). First let us consider two Weyl fermions λ, λ̃ in the adjoint representation, with the Lagrangian

trλ̄γμDμλ + trλ̃̄γμDμλ̃ + c(trλ[Φ,λ̃] + trλ̄[Φ,λ̃̄]). (1.3.17)

We regard both the gauge potential in the covariant derivative D and the scalar field as backgrounds, and decompose λ, λ̃ into eigenstates of the angular momentum. The lower bound of the orbital angular momentum is given by the Dirac pairing, which is here. The spinor fields have spin 2. Therefore the state with lowest angular momenta has spin 2. When the coefficient c takes a value in a certain range, it is known that there is a pair of zero modes bα where α = 1, 2 the spinor index of the SO(3) spatial rotation. The semiclassical quantization promotes them into a pair of fermionic oscillators

{bα,bβ} = δαβ. (1.3.18)

This creates four states starting from one state |ψ from the semiclassical quantization of the bosonic part:

b1|ψ b1b2|ψ |ψ b2|ψ . (1.3.19)

This counts as one complex boson and one fermion.

Suppose we introduce another pair λ, λ̃ of the adjoint Weyl fermions. Then we will have another pair of fermionic oscillators bα. Together, they generate 24 = 16 states, consisting of one massive vector (with 3 states), four massive spinors (with 8 states) and five massive scalars.

Next, consider having 2N Weyl fermions ψia in the doublet representation where a = 1, 2 and i = 1,, 2N, with the Lagrangian

ψ̄iγμDμψi + c(ψiaΦ(ab)ψib + ψ̄iaΦ(ab)ψ̄ib). (1.3.20)

Note that the Lagrangian has an SO(2N) flavor symmetry acting on the index i.

The electric charge of the quanta of ψ, ψ̃ with respect to the unbroken U(1) is now 1. Then the Dirac pairing is 2. Tensoring with the intrinsic spin 2, we find that the minimal orbital angular momentum is 0. It is known that for a suitable choice of c, this fermion system has zero modes γi, i = 1,, 2N. After semiclassical quantization, it becomes a set of fermionic operators with the commutation relation

{γi,γj} = δij. (1.3.21)

This is the commutation relation of the gamma matrices of SO(2N). Monopole states are representations of γi’s, meaning that they transform as a spinor representation of the flavor symmetry SO(2N).

Fields in a doublet representation of the SU(2) gauge symmetry has an another effect. Namely, in the gauge zero mode (1.3.14), α = π gives the matrix

1 0 0 1 (1.3.22)

which acts nontrivially on the fields in the doublet representation. Then the periodicity of the gauge zero mode α is now 2π, and the wavefunction along the α direction can now be einα for arbitrary integer n. Therefore, the electric charge n can either be even or odd. The operators γi come from the modes of the fields in the doublet representation, and therefore it changes the electric charge by ± 1.

We can define the flavor spinor chirality Γ by

Γ = γ1γ2γ2N, (1.3.23)

by which the spinor of SO(2N) can be split into positive-chirality and negative-chirality spinors. The action of the operators γi changes the chirality of the flavor spinors. Combined with the behavior of the U(1) electric charge we saw in the previous paragraph, we conclude that the parity of the U(1) electric charges of the monopole states is correlated with the chirality of the flavor spinor representation.