Vacuum Orbits and Unbroken Subgroups

Choose a vacuum state $\omega_0$ of a theory with exact ordinary global symmetry $G$. The unbroken subgroup is the stabilizer

$$H=\{g\in G\mid g\mathbin{\cdot}\omega_0=\omega_0\}.$$

Every state obtained by acting with $G$ lies on the orbit $G\mathbin{\cdot}\omega_0$. If $G$ is a finite-dimensional Lie group, its action is smooth, and $H$ is closed, that orbit is abstractly the homogeneous space $G/H$. Its tangent space at the reference vacuum is $\mathfrak g/\mathfrak h$, so $\dim G-\dim H$ counts continuous broken directions and supplies the number of local orbit coordinates.

This result organizes one symmetry orbit, not necessarily the complete vacuum set. Disconnected vacua do not contribute tangent dimensions; distinct orbits and transverse moduli need not fit into one quotient; and the number of orbit coordinates is not yet a theorem about propagating Goldstone particles. A classical set of degenerate minima is only a candidate for this structure until stable, phase-selected quantum vacua are established.

Required background. Symmetry Realization and Order Parameters supplies the state stabilizer and the distinction between a complete state diagnostic and one order-parameter expectation value. Groups, Actions, Quotients, and Covers supplies group actions, stabilizers, cosets, quotient groups, and faithful actions.

Helpful background. Homotopy, Degree, Winding, and Covering Spaces is useful when the global topology of an orbit matters. This page identifies that topology but does not classify defects from it.

From one selected vacuum to its orbit

Throughout, “vacuum” means a selected infinite-volume state of the kind constructed in Finite Volume, Thermodynamic Limits, and Pure Phases, not a finite-volume wavefunction or merely a minimum of a classical potential.

Let $\alpha_g$ be the active action of $G$ on physical operators. It induces a left action on states by

$$(g\mathbin{\cdot}\omega)(X) =\omega\!\left(\alpha_{g^{-1}}(X)\right).$$

The inverse makes the left-action law work:

$$g\mathbin{\cdot}(k\mathbin{\cdot}\omega) =(gk)\mathbin{\cdot}\omega.$$

Because inversion permutes $G$, the condition $g\mathbin{\cdot}\omega_0=\omega_0$ is equivalent to $\omega_0(\alpha_g(X))=\omega_0(X)$ for every physical $X$, exactly as on the first page of this chapter. This state-functional definition does not assume that every broken vacuum is a vector in one common infinite-volume Hilbert-space representation.

Assume first that $G$ acts faithfully on the full physical theory. If the full physical action has a closed normal kernel $K$, then $K\subseteq H$ for every vacuum and one should replace $G$ by $G/K$. The kernel of one chosen order-parameter multiplet is not enough for this quotient. The orbit is unchanged because

$$(G/K)/(H/K)\simeq G/H.$$

The unbroken subgroup depends on the reference vacuum, but only up to conjugacy. If $\omega_g=g\mathbin{\cdot}\omega_0$, then

$$H_{\omega_g}=gHg^{-1}.$$

Thus a chosen orientation fixes a particular embedded subgroup, while the orbit carries its conjugacy class as invariant information.

Why the orbit is G/H

Define

$$\begin{aligned} F:G/H&\longrightarrow G\mathbin{\cdot}\omega_0,\\ F(gH)&=g\mathbin{\cdot}\omega_0. \end{aligned}$$

The map is well defined because $h\mathbin{\cdot}\omega_0=\omega_0$ for every $h\in H$, so $gh$ and $g$ give the same state. It is surjective by the definition of the orbit. If two cosets give the same state, then

$$g_1\mathbin{\cdot}\omega_0 =g_2\mathbin{\cdot}\omega_0 \quad\Longrightarrow\quad g_2^{-1}g_1\in H,$$

and hence $g_1H=g_2H$; the map is injective. Here $G/H$ is the homogeneous space of left cosets $gH$. It is a quotient group only in the special case that $H$ is normal in $G$.

Closedness of $H$ is what makes $G/H$ a smooth Hausdorff manifold. In the common case of a continuous action on a Hausdorff vacuum-state space, a stabilizer is closed; otherwise this must be assumed. With the quotient-manifold structure, $G/H$ is the abstract orbit. If the vacua or order-parameter values also lie in a smooth ambient manifold and the action is smooth, the natural orbit map is generally an injective immersion, not automatically an embedding with the subspace topology. These distinctions are stated in Etingof 2020, §§ 4.1 and 4.4, pp. 28–31, official PDF.

Tangent directions and local coordinates

Let $\Phi_{\omega_0}:G\to G\mathbin{\cdot}\omega_0$ be the orbit map $g\mapsto g\mathbin{\cdot}\omega_0$. Its differential at the identity has kernel $\mathfrak h$, the Lie algebra of $H$. Therefore

$$\begin{aligned} T_{\omega_0}(G\mathbin{\cdot}\omega_0) &\simeq\mathfrak g/\mathfrak h,\\ \dim(G/H)&=\dim G-\dim H. \end{aligned}$$

The quotient vector space is canonical; a complementary subspace to $\mathfrak h$ inside $\mathfrak g$ need not be. After choosing representatives $X_a$ of a basis of $\mathfrak g/\mathfrak h$, local coordinates near the reference vacuum can be written

$$\begin{aligned} g(\pi)&=\exp\!\left(i\pi^aX_a\right),\\ \omega(\pi)&=g(\pi)\mathbin{\cdot}\omega_0. \end{aligned}$$

The $X_a$ use the site’s Hermitian-generator convention. These are local coordinates: the exponential need not cover all of $G/H$, different patches can be required, and right multiplication by $H$ does not change the represented vacuum. Constructing fields and an effective action from these coordinates belongs to Cosets and Nonlinear Realizations.

The dimension formula counts tangent directions only. It neither counts disconnected components nor proves that every direction produces an independent gapless particle. The hypotheses for a gapless pole and the qualifications on particle counting are treated on the two Goldstone pages.

State stabilizer versus order-parameter stabilizer

Suppose physical local operators $\mathcal O_i$ transform in a representation $R$ and

$$\begin{aligned} v_i&=\omega_0(\mathcal O_i),\\ H_v&=\{g\in G\mid R(g)v=v\}. \end{aligned}$$

The exact unbroken group is the stabilizer $H=H_{\omega_0}$ of the whole state. As established on the first page of this chapter,

$$H_{\omega_0}\subseteq H_v.$$

Consequently $G/H_v$ is the orbit of this expectation-value vector, which can be smaller than the state orbit $G/H_{\omega_0}$. The two agree only when the chosen diagnostic distinguishes every broken transformation relevant to the state.

At the infinitesimal level, a Hermitian generator $T$ belongs to the order-parameter stabilizer when

$$T v=0.$$

If $Tv\neq0$, then $iTv$ is a nonzero tangent direction in the order-parameter orbit and proves that $T$ is broken in the state. If $Tv=0$, no conclusion follows about the full state without additional observables.

One orbit is not the whole vacuum set

The complete set of vacua decomposes set-theoretically into $G$-orbits,

$$\mathcal M_{\mathrm{vac}} =\bigsqcup_{\lambda}\mathcal O_\lambda, \qquad \mathcal O_\lambda\simeq G/H_\lambda,$$

under the smoothness and closed-stabilizer conditions above. There can be several reasons for more than one orbit:

• two degenerate nonzero radii can give two distinct circles under the same $U(1)$ action;
• neutral moduli can label directions not generated by $G$;
• isolated vacua and continuous orbits can coexist; or
• stabilizer dimensions can jump, so the full vacuum set need not itself be one smooth manifold.

Even a single $G/H$ can be disconnected. The dimension formula sees only the identity-component tangent geometry. For broken $\mathbb Z_2$ with trivial stabilizer,

$$G/H\simeq\mathbb Z_2$$

is two points: it has two vacua but dimension zero. Global topology and disconnected components can support domain walls, strings, or other sectors, but their classification and dynamics require downstream treatments in Nonperturbative Dynamics and Many-Body QFT and Quantum Matter.

The complex scalar: circle, tangent, and discrete anisotropy

For the exact global $U(1)$ scalar with classical potential

$$\begin{aligned} V_0(\phi) &=m^2|\phi|^2+\frac{\lambda}{2}|\phi|^4,\\ m^2&<0, \qquad \lambda>0. \end{aligned}$$

the classical minimum set is

$$\begin{aligned} \mathcal M_{\mathrm{cl}} &=\{\phi_0e^{i\theta}\mid 0\leq\theta<2\pi\},\\ |\phi_0|^2&=-\frac{m^2}{\lambda}. \end{aligned}$$

Choose the reference minimum $\phi_0>0$. For a charge-one field its stabilizer under phase rotations is trivial, so

$$U(1)/\{1\}\simeq S^1.$$

The infinitesimal tangent is $i\phi_0$, whereas changing $|\phi_0|$ is radial and not generated by $U(1)$. Thus the angular coordinate lies on the orbit while the radial coordinate is transverse to it. This classical construction is presented in Tong 2019, § 2.2, pp. 58–61, official PDF.

The quantum statement is conditional. If the exact quantum theory admits stable selected vacua

$$\omega_\theta(\phi) =\frac{v_R}{\sqrt2}e^{i\theta},$$

then they form the corresponding physical $U(1)$ orbit; the renormalized radius need not equal the classical $|\phi_0|$. The state-based and infinite-volume qualifications are developed in Weinberg 1995, §§ 19.1–19.2, pp. 163–169. A temporary source $J$ selects one $\omega_\theta$ and is removed only after the infinite-volume limit, using the ordered prescription on Finite Volume, Thermodynamic Limits, and Pure Phases.

Now keep a permanent anisotropy $h\phi^N+h^*(\phi^*)^N$ with $h\neq0$ and $N\geq2$, in a regime where the deformed theory is stable or within a declared cutoff effective theory. The exact phase-rotation group is only $\mathbb Z_N$. If a selected phase breaks it completely, its orbit is

$$\mathbb Z_N/\{1\},$$

a set of $N$ disconnected vacua with zero tangent dimension. A shallow approximate $U(1)$ valley is not an exact $U(1)$ vacuum orbit; its lifting and the resulting pseudo-Goldstone scale belong to Explicit Breaking and Pseudo-Goldstone Modes.

A nontrivial stabilizer: O(N) to O(N−1)

Let a real $N$-component order parameter have reference value $v_0=v e_N$ with $v\neq0$ and $N\geq2$. The group $O(N)$ acts transitively on vectors of length $|v|$. The transformations fixing $e_N$ form $O(N-1)$, so

$$\begin{aligned} O(N)/O(N-1)&\simeq S^{N-1},\\ \dim S^{N-1}&=N-1. \end{aligned}$$

The tangent vectors are precisely the variations orthogonal to $e_N$; the radial direction is again transverse. Choosing another point on the sphere conjugates the displayed stabilizer but does not change its orbit type. The general QFT notation $G\to H$, the quotient $G/H$, and its continuous broken directions are summarized in Tong 2019, § 2.2, p. 65, official PDF.

Common pitfalls

“The vacuum manifold is always $G/H$.” One orbit is $G/H$ under the stated hypotheses. The full vacuum set may contain several orbits, transverse moduli, isolated points, or singular orbit types.

“Every classical minimum is a quantum vacuum.” Quantum corrections can lift or reshape a classical degeneracy, and an exact convex effective-potential treatment has additional phase-selection subtleties. Stable selected infinite-volume states are required.

“The stabilizer of one order parameter is the unbroken group.” It is an upper bound on the state stabilizer. Equality needs a complete enough diagnostic family.

“The quotient dimension counts vacua.” It counts continuous tangent directions. A finite broken group can have many vacua and dimension zero.

“The quotient dimension is the Goldstone-particle count.” That equality needs additional dynamical and kinematic hypotheses and can fail outside the standard relativistic internal-symmetry setting.

“Every orbit is automatically an embedded submanifold.” A general smooth Lie-group orbit is naturally immersed. Embeddedness requires additional conditions.

“A gauge orbit is a vacuum manifold of physical states.” Gauge-related configurations are redundant descriptions, not distinct physical vacua. The gauge-invariant Higgs discussion appears later in this chapter.

Check the vacuum geometry

These questions are for self-study and are not graded.

1. For $O(4)$ with a nonzero vector order parameter, identify $H$, the orbit, and its dimension.
2. If exact $\mathbb Z_6$ is completely broken, how many vacua lie on the orbit, and how many continuous tangent directions does it have?
3. If one expectation value has stabilizer $H_v$ larger than the state stabilizer $H_\omega$, what can be said about the dimensions of the two quotients?

Check

1. The stabilizer is $O(3)$, the orbit is $O(4)/O(3)\simeq S^3$, and its dimension is $3$.
2. The orbit $\mathbb Z_6/\{1\}$ contains six points and has zero continuous tangent dimension.
3. Since $H_\omega\subseteq H_v$, the state orbit $G/H_\omega$ has at least as many continuous tangent directions as the order-parameter orbit $G/H_v$. Equal dimensions show only that their Lie algebras have equal dimensions; the stabilizers may still differ by disconnected or finite pieces. Equality of the actual stabilizers requires a complete diagnostic.

What follows

The stabilizer fixes the equivalence relation on symmetry transformations, $G/H$ organizes one selected-vacuum orbit, and $\mathfrak g/\mathfrak h$ supplies its local continuous coordinates. Those statements are kinematic; dynamics and global structure require separate steps.

• Goldstone’s Theorem: Hypotheses and Pole Argument asks when a broken direction forces a gapless pole.
• Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions separates orbit dimension from mode counting and states the infrared exceptions.
• Cosets and Nonlinear Realizations turns local orbit coordinates into fields and invariant effective interactions.
• Elitzur’s Theorem and the Gauge-Invariant Higgs Mechanism explains why gauge redundancy does not produce a physical $G/H$ vacuum orbit.

References

• Etingof, Pavel. Lie Groups and Lie Algebras I. 18.745 lecture notes. Cambridge, MA: Massachusetts Institute of Technology OpenCourseWare, 2020. Official course page. Official PDF.
• Tong, David. The Standard Model: 2 Broken Symmetries. Part III lecture notes. Cambridge: University of Cambridge, Department of Applied Mathematics and Theoretical Physics, 2019. Official PDF accessed August 2, 2026. Official course page. Official PDF.
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1995. DOI.