Why does adjoint representation matter in some field theories?

Question

Recently I am reading a paper about monopoles. In several cases, it seems that writing fields in adjoint representation of the gauge group makes a difference.

Once it leads to different group after symmetry breaking when using other representation. And I also noticed statement like this, "An important open question is whether an analogous Bogomolny monopole's mass bound can be obtained if the Higgs field is not in the adjoint representation."

Can anyone kindly shed light on this. Thanks!

Update: I reckon any field (either EM field in real space or Higgs field in internal isotopic space) be in a certain type of representation space of the symmetry group associated with the Lagrangian or action. This space also dictates some constraints on the fields, e.g., specific tensor or spinor structures (anything more???). And what representation space you use contains physics as well, that is to say, we have to check it by experiments. Perhaps this question addresses on a particular case. Either does the explicit and concrete 2nd answer.

Is this understanding correct?

Answer 1

I am not sure if I know the correct answer (as I am a student my self), but I will try (and if I am wrong, someone please correct me).

The first thing that took me some time to figure out is what they mean by adjoint representation. In Georgi's book he defines the adjoint representation of a generator as: \begin{equation} [T_i]_{jk} \equiv -if_{ijk} \end{equation} which is equivalent to the adjoint representation of a Lie algebra. However, when discussing monopole, they actually mean the adjoint representation of a Lie group . This means that $\phi$ takes values in the Lie algebra (the vector space formed by the generators) and can be expressed in terms of the generators in an arbitrary representation: \begin{equation} \phi = \phi^a t^a \end{equation} where $t^a$ denote the generators in an arbitrary representation (and there is an implicit sum over repeated indices).

Now, let us look at the simplest example, which is the bosonic part of the $\mathrm{SU(2)}$ gauge invariant Georgi-Glashow model: \begin{equation} \mathcal{L}=\frac{1}{8} \mathrm{Tr} (F_{\mu \nu} F^{\mu \nu}) - \frac{1}{4} \mathrm{Tr}(D_\mu \phi D^\mu \phi) - \frac{\lambda}{4}(1-\phi^a \phi^a )^2 \end{equation} We can write the kinetic and potential energy, $T$ and $V$, as: \begin{equation} T=\int \left( - \frac{1}{4} \mathrm{Tr} (F_{0i}F_{0i}) - \frac{1}{4} \mathrm{Tr}(D_0 \phi D_0 \phi) \right) \mathrm{d^3}x \end{equation} and: \begin{equation} V=\int \left( - \frac{1}{8} \mathrm{Tr} (F_{ij}F_{ij}) - \frac{1}{4} \mathrm{Tr}(D_i \phi D_i \phi) + \frac{\lambda}{4}(1-\phi^a \phi^a )^2 \right) \mathrm{d^3}x \end{equation} where we used $L= \int \mathcal{L} \; \mathrm{d^3}x = T-V$. In order to get finite energy solutions we have to impose boundary conditions such that the total energy of the model vanished at spatial infinity. It should be clear that one of the requirements to ensure that the energy vanishes is: \begin{equation} \phi^a \phi^a =1 \end{equation} This implies that the Higgs vacuum corresponds to an infinite amount of degenerate vacuum values lying on the surface of a unit two-sphere in field space, which we will denote by $S^2_1$. Furthermore, by imposing the aforementioned finite energy boundary condition, this gives rise to the following map: \begin{equation} \phi : S^2_\infty \to S^2_1 \end{equation} where $S^2_\infty$ denotes the two-sphere associated with spatial infinity (in 3 dimensions). This is in fact the definition of the winding number (or degree) between two two-dimensional spheres and is therefore classified by $\pi_2(S^2)=\mathbb{Z}$ (and it is in theory possible to construct topological solitons). Now, if $\phi$ was in the fundamental representation, then I don't think it is possible to construct these topological solitons.

Answer 2

Theories with fundamental quarks which experience spontaneous chiral symmetry breaking:

$$ SU_L(N_f) \times SU_R(N_f) \rightarrow SU_A(N_f)$$

($N_f$ is the number of flavors)

(This is the observed approximate symmetry breaking in nature where the pions are the approximate Goldstone bosons).

In contrast, theories with adjoint quarks experience the chiral symmetry breaking pattern:

$$ SU(N_f) \rightarrow SO(N_f)$$

(modulo discrete groups). Please see for example the following article Auzzi, Bolognesi, and Shifman.

The reasoning is that since the adjoint representation is real, it has only one copy of $SU(N_f)$ flavor symmetry and $SO(N_f) $ is am maximal subgroup of $SU(N_f)$, thus any symmetry breaking will start in this pattern.

The Goldstone Boson manifold will be:

$$\mathcal{M} = SU(N_f)/SO(N_f)$$

The topology of the Goldstone Boson manifold determines the existence of t’ Hooft-Polyakov monopoles, since an non trivial homotopy group $\pi_2(\mathcal{M} )$ is required for a stable monopole solution to exist. This happens in our case when $N_f =2$, in this:

$$\mathcal{M} = SU(2)/SO(2) = S^2$$

Thus $\pi_2(\mathcal{M} ) = \mathbb{Z}$ and monopoles exist.

In addition for any number of flavors there will be Skyrmions in $\mathcal{M}$ as elaborated in , Bolognesi, and Shifman's article.