’t Hooft-Polyakov monopoles

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 [25, 26], or the textbook [27]. The review by Coleman [28] is also very instructive.³

1.3.1 Classical features

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

S = \int d^{4} x \frac{1}{g^{2}} [\frac{1}{2} t r F_{μ ν} F_{μ ν} + t r D_{μ} Φ D_{μ} Φ] .

(1.3.1)

The ﬁeld $Φ$ is a traceless Hermitean $2 \times 2$ matrix.

Consider the vacuum where

Φ = (\begin{matrix} a & 0 \\ 0 & - a \end{matrix}) .

(1.3.2)

When $a \neq 0$ , the $S U (2)$ gauge symmetry is broken to $U (1)$ . Indeed, the vev (1.3.2) commutes with a gauge ﬁeld strength of the form

F_{μ ν} = (\begin{matrix} F_{μ ν}^{U (1)} & 0 \\ 0 & - F_{μ ν}^{U (1)} \end{matrix})

(1.3.3)

where $F_{μ ν}^{U (1)}$ 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 $\pm a$ . Then we can consider a conﬁguration of the form

Φ (x) = \frac{x_{i} σ^{i}}{| x |} f (| x |) a

(1.3.4)

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

lim_{r \to 0} f (r) = 0, lim_{r \to \infty} f (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 $S U (2)$ is along $Φ$ . A more general deﬁnition of the $U (1)$ ﬁeld strength $F_{μ ν}^{U (1)}$ , at least when $r ≫ 0$ , is then the combination

F_{μ ν}^{U (1)} : = \frac{1}{2 a} t r F_{μ ν} Φ .

(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 [i \frac{φ}{2} (- σ^{1} sin 𝜃 + σ^{2} cos 𝜃)], where \frac{\vec{x}}{| 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 (- σ^{1} sin 𝜃 + σ^{2} cos 𝜃) = exp [- i 𝜃 σ^{3}] \cdot (i σ^{2}) .

(1.3.8)

As $𝜃$ goes from $0$ to $2 π$ , we see that the $U (1)$ ﬁeld $F_{μ ν}^{U (1)}$ 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 $1 ∕ g^{2}$ times

\begin{align} \int d^{3} x [t r B_{i} B_{i} + t r D_{i} Φ D_{i} Φ] & = \int d^{3} x [t r {(B_{i} \mp D_{i} Φ)}^{2} \pm 2 t r B_{i} D_{i} Φ] & (1.3.9) \\ \geq \pm 2 \int d^{3} x t r B_{i} D_{i} Φ = \pm 2 \int d^{3} x \partial_{i} t r B_{i} Φ & (1.3.10) \\ = \pm 2 \int_{S^{2}} d \vec{n} \cdot t r \vec{B} Φ & (1.3.11) \end{align}

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

(energy of the monopole) \geq \frac{4 π}{g^{2}} (2 a) | m |

(1.3.12)

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

B_{i} = \pm D_{i} Φ,

(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 $\vec{y}$ of the spatial $ℝ^{3}$ . These give three zero-modes.

Another zero mode is obtained by the gauge transformation:

e^{i α Φ ∕ a}

(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 $F_{i j} (x)$ , but for $α = π$ , the transformation (1.3.14) trivially acts on the ﬁelds in the adjoint representation. Therefore $α$ is an angular variable $0 \leq α < π$ .

The semiclassical quantization of the monopole involves the Fock space of non-zero modes, together with a wavefunction $ψ (\vec{y}, α)$ depending on the zero modes $\vec{y}$ and $α$ . The wavefunction along $\vec{y}$ 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 $e^{i φ}$ shifts $α$ by

α \to α + φ .

(1.3.15)

Recall that a state $| ψ ⟩$ with electric charge $n$ behaves under the $U (1)$ global transformation by $e^{i φ}$ by

| ψ ⟩ \to e^{i n φ} | ψ ⟩ .

(1.3.16)

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

Summarizing, the combination of the electric charge and the magnetic charge $(n, m)$ we obtain from the semi-classical quantization has the form $(n, m) = (2 d, 1)$ 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 $d = - \infty$ to $+ \infty$ .

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 $λ$ , $\tilde{λ}$ in the adjoint representation, with the Lagrangian

t r \bar{λ} γ^{μ} D_{μ} λ + t r \bar{\tilde{λ}} γ^{μ} D_{μ} \tilde{λ} + c (t r λ [Φ, \tilde{λ}] + t r \bar{λ} [Φ, \bar{\tilde{λ}}]) .

(1.3.17)

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

{b_{α}, b_{β}^{†}} = δ_{α β} .

(1.3.18)

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

\begin{matrix} \leftrightarrow & b_{1}^{†} | ψ ⟩ & \leftrightarrow & b_{1}^{†} b_{2}^{†} | ψ ⟩ \\ | ψ ⟩ & \leftrightarrow & b_{2}^{†} | ψ ⟩ & \leftrightarrow \end{matrix} .

(1.3.19)

This counts as one complex boson and one fermion.

Suppose we introduce another pair $λ^{'}$ , ${\tilde{λ}}^{'}$ of the adjoint Weyl fermions. Then we will have another pair of fermionic oscillators $b_{α}^{'}$ . Together, they generate $2^{4} = 16$ states, consisting of one massive vector (with 3 states), four massive spinors (with 8 states) and ﬁve massive scalars.

Next, consider having $2 N$ Weyl fermions $ψ_{i}^{a}$ in the doublet representation where $a = 1, 2$ and $i = 1, \dots, 2 N$ , with the Lagrangian

{\bar{ψ}}_{i} γ^{μ} D_{μ} ψ_{i} + c^{'} (ψ_{i}^{a} Φ_{(a b)} ψ_{i}^{b} + {\bar{ψ}}_{i}^{a} Φ_{(a b)} {\bar{ψ}}_{i}^{b}) .

(1.3.20)

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

The electric charge of the quanta of $ψ$ , $\tilde{ψ}$ with respect to the unbroken $U (1)$ is now $1$ . Then the Dirac pairing is $ℏ ∕ 2$ . Tensoring with the intrinsic spin $ℏ ∕ 2$ , 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 $γ_{i}$ , $i = 1, \dots, 2 N$ . After semiclassical quantization, it becomes a set of fermionic operators with the commutation relation

{γ_{i}, γ_{j}} = δ_{i j} .

(1.3.21)

This is the commutation relation of the gamma matrices of $S O (2 N)$ . Monopole states are representations of $γ_{i}$ ’s, meaning that they transform as a spinor representation of the ﬂavor symmetry $S O (2 N)$ .

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

(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix})

(1.3.22)

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

We can deﬁne the ﬂavor spinor chirality $Γ$ by

Γ = γ_{1} γ_{2} \dots γ_{2 N},

(1.3.23)

by which the spinor of $S O (2 N)$ can be split into positive-chirality and negative-chirality spinors. The action of the operators $γ_{i}$ 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.

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