Recently I’ve been learning about spontaneous symmetry breaking (SSB) for gauge theories. I’ve stumbled across some really helpful resources, such as Hamilton’s Mathematical Gauge Theory – With Applications to the Standard Model.

However I think I’m getting confused about the symmetries involved. In particular, I’m getting mixed up about when we want something to be $G$-invariant vs $\mathcal{G}$-invariant.

Suppose we have a gauge theory involving:

• A principal $G$-bundle $P \rightarrow M$ with structure group given by a compact Lie group $G$.
• A complex representation $\rho: G \rightarrow \text{GL} \left( W \right)$, where $W$ is a finite complex vector space with the standard Hermitian form.
• The associated vector bundle $E \rightarrow M$ with $E = P \times_{\rho} W$.
• Gauge group $\mathcal{G}$ of $P$.

In general, my question is this: when do we expect objects in our theory to be $G$-invariant vs $\mathcal{G}$-invariant?

Let me give some examples which have confused me. I know connection 1-forms describe our gauge fields, and sections of $E$ describe our Higgs fields.

• If we introduce a potential function $V: \Gamma \left( E \right) \rightarrow \mathbb{R}$ for our Higgs fields, should $V$ be $G$-invariant or $\mathcal{G}$-invariant?
• Should our Lagrangian be $G$-invariant or $\mathcal{G}$-invariant?
• Given a vacuum vector $w_0 \in W$, which minimizes $V$, we can define the unbroken subgroup to be the isotropy group / stabilizer $H$ of $w_0$. If $H \subset G$ is a proper subgroup, then we have a spontaneously broken symmetry. However I thought the full initial symmetry we expected of our system was $\mathcal{G}$, not $G$?

I thought $\mathcal{G}$ was the symmetry group for our theory, so we always should expect objects to be $\mathcal{G}$-invariant. However I often seen authors assume $G$-invariance instead.

Perhaps I’ve got something basic wrong here. Or maybe I’m confused because I am interested in SSB for arbitrary gauge theories, while texts like Hamilton often make assumptions I’m not accustomed to (due to their interest in physical applications). For example, Hamilton assumes $G$ is compact and $M$ is Minkowski space (so connected and simply connected) so $P$ is trivial. In this case, I know $\mathcal{G} \simeq \text{Map} \left( M, G \right)$. Perhaps this is why some objects can be said to be $G$-invariant rather than $\mathcal{G}$-invariant?

Recently I’ve been learning about spontaneous symmetry breaking (SSB) for gauge theories. I’ve stumbled across some really helpful resources, such as Hamilton’s Mathematical Gauge Theory – With Applications to the Standard Model.

However I think I’m getting confused about the symmetries involved. In particular, I’m getting mixed up about when we want something to be $G$-invariant vs $\mathcal{G}$-invariant.

Suppose we have a gauge theory involving:

• A principal $G$-bundle $P \rightarrow M$ with structure group given by a compact Lie group $G$.
• A complex representation $\rho: G \rightarrow \text{GL} \left( W \right)$, where $W$ is a finite complex vector space with the standard Hermitian form.
• The associated vector bundle $E \rightarrow M$ with $E = P \times_{\rho} W$.
• Gauge group $\mathcal{G}$ of $P$.

In general, my question is this: when do we expect objects in our theory to be $G$-invariant vs $\mathcal{G}$-invariant?

Let me give some examples which have confused me. I know connection 1-forms describe our gauge fields, and sections of $E$ describe our Higgs fields.

• If we introduce a potential function $V: \Gamma \left( E \right) \rightarrow \mathbb{R}$ for our Higgs fields, should $V$ be $G$-invariant or $\mathcal{G}$-invariant?
• Should our Lagrangian be $G$-invariant or $\mathcal{G}$-invariant?
• Given a vacuum vector $w_0 \in W$, which minimizes $V$, we can define the unbroken subgroup to be the isotropy group / stabilizer $H$ of $w_0$. If $H \subset G$ is a proper subgroup, then we have a spontaneously broken symmetry. However I thought the full initial symmetry we expected of our system was $\mathcal{G}$, not $G$?

I thought $\mathcal{G}$ was the symmetry group for our theory, so we always should expect objects to be $\mathcal{G}$-invariant. However I've often seen authors assume $G$-invariance instead.

Perhaps I’ve got something basic wrong here. Or maybe I’m confused because I am interested in SSB for arbitrary gauge theories, while texts like Hamilton often make assumptions I’m not accustomed to (due to their interest in physical applications). For example, Hamilton assumes $G$ is compact and $M$ is Minkowski space (so connected and simply connected) so $P$ is trivial. In this case, I know $\mathcal{G} \simeq \text{Map} \left( M, G \right)$. Perhaps this is why some objects can be said to be $G$-invariant rather than $\mathcal{G}$-invariant?