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Enforce the unit-vector constraint using a Lagrange multiplier $\Lambda (x)$:Integration by parts in a compactly supported variation gives $-\Delta {\varphi }^a+\Lambda {\varphi }^a=0$. Contracting with ${\varphi }^a$ gives $\Lambda ={\varphi }^b\Delta {\varphi }^b$, henceDifferentiating ${\varphi }^a{\varphi }^a=1$ twice also gives $\varphi \cdot \Delta \varphi =-|\nabla \varphi {|}^2$, so an equivalent equation is $\Delta \varphi +|\nabla \varphi {|}^2\varphi =0$. These are the harmonic map equations for the O3 nonlinear sigma model.

For arbitrary maps, the printed finite-energy compactification assertion is too strong. The following counterexample shows that finite sigma-model energy need not give a limit at infinity. Outside a large disk, setInterpolate smoothly to a constant inside the disk. There is no angular dependence, and the exterior energy isNevertheless the field keeps rotating and has no limit as $r\to \infty$. Thus energy integrability alone does not prove a continuous map on the one-point compactification.

The usual sigma-model lump sector adds the boundary condition $\varphi (x)\to {\varphi }_{\infty }\in S^2$ uniformly as $|x|\to \infty$. Defining the value at the added point to be ${\varphi }_{\infty }$ then gives a continuous map $S^2\to S^2$: neighborhoods of the point at infinity are precisely complements of compact subsets of the plane, and uniform convergence supplies continuity there. For the topological degree calculation using differential forms below, take a smooth compactified map, as in the regular lump sector. This boundary/regularity information is additional to finite energy for unrestricted fields.

Write $a_i={\partial }_i\varphi$ and $b_i={ϵ}_{ij}\varphi \times {\partial }_j\varphi$. Since $\varphi \cdot {\partial }_i\varphi =0$ and $|\varphi |=1$,Also the given double-index charge reduces toFor $s=\pm 1$, expanding the nonnegative square givesTherefore the Bogomolny degree bound for the O3 sigma model isThe charge integral is absolutely convergent already for finite-energy fields, because $|\varphi \cdot ({\partial }_1\varphi \times {\partial }_2\varphi )|\le \frac{1}{2}(|{\partial }_1\varphi {|}^2+|{\partial }_2\varphi {|}^2)$. The inequality itself consequently does not need the compactification hypothesis. Equality requires ${\partial }_i\varphi +s{ϵ}_{ij}\varphi \times {\partial }_j\varphi =0$ for $s$ chosen to have the sign of $Q$.

For the topological interpretation, the standard oriented area form on the target sphere isIts pullback is ${\varphi }^{\ast }\omega =\varphi \cdot ({\partial }_1\varphi \times {\partial }_2\varphi )d{x}^1\wedge d{x}^2$. The topological degree is defined by ${\varphi }_{\ast }[S^2]=\mathrm{deg}(\varphi )[S^2]$ in $H_2(S^2;\mathbb{Z})\cong \mathbb{Z}$. Pairing with $\omega$ givesEquivalently, the integer is the signed number of preimages of a regular value. It is invariant under smooth homotopy: if $\Phi :S^2\times [0,1]\to S^2$ is that homotopy, then Stokes theorem and $d\omega =0$ make the difference between the two charge integrals equal to ${\int }_{S^2\times [0,1]}d({\Phi }^{\ast }\omega )=0$. This identifies the charge as the degree charge of an O3 sigma-model lump, rather than merely an arbitrary energy integral.

A nonlinear sigma model describes a field taking values in a Riemannian target rather than an unconstrained linear space. Its static configurations are harmonic maps, critical points of the Dirichlet energy. The standard example for sigma-model lumps is the O3 nonlinear sigma model in two spatial dimensions: a unit vector $\mathbf{n}(x_1,x_2)\in S^2$ hasThe same target is the complex projective line, giving the CP1 nonlinear sigma model description. A lump is a localized finite-energy static solution, whose existence and minimum energy are governed by the degree of its field map.

Impose a fixed angle-independent value ${\mathbf{n}}_{\infty }$ at spatial infinity and require the field to extend smoothly there. The one-point compactification of the plane is a sphere, so the configuration becomes a map $S^2\to S^2$. Its topological degree is the integerThis is the degree charge of an O3 sigma-model lump: the integrand is the pullback of the oriented unit-sphere area form, normalized by its area $4\pi$. Equivalently, the degree counts inverse images of a regular target point with orientation signs. It is unchanged under smooth homotopy with the prescribed boundary value. Consequently fields of different degree belong to distinct topological sectors.

Topology also determines an energy lower bound. The derivatives are tangent to the sphere, so $|\mathbf{n}\times {\partial }_2\mathbf{n}|=|{\partial }_2\mathbf{n}|$. Since
${\partial }_1\mathbf{n}\cdot (\mathbf{n}\times {\partial }_2\mathbf{n})=-\mathbf{n}\cdot ({\partial }_1\mathbf{n}\times {\partial }_2\mathbf{n})$, square completion givesChoosing the sign of $N$ proves the Bogomolny degree bound for the O3 sigma modelFor positive degree, equality holds precisely when ${\partial }_1\mathbf{n}+\mathbf{n}\times {\partial }_2\mathbf{n}=0$; for negative degree the opposite sign applies. These are the Bogomolny equations for the lump. A saturating field minimizes energy in its degree sector and hence solves the static field equation. Explicitly, that equation is $\Delta \mathbf{n}+|\nabla \mathbf{n}{|}^2\mathbf{n}=0$, obtained by varying under ${\mathbf{n}}^2=1$.

The first-order equations become especially simple in stereographic projection. Use $z=x_1+i{x}_2$ andThe round target metric is $4dwd\bar{w}/(1+|w{|}^2{)}^2$. In the one-half energy normalization used here, the stereographic energy of the O3 sigma model and degree areThus $E-4\pi N=8\int |w_{\bar{z}}{|}^2/(1+|w{|}^2{)}^2$. Positive-degree lumps are holomorphic maps, $w_{\bar{z}}=0$, and negative-degree lumps are antiholomorphic maps, $w_z=0$.

A holomorphic sphere map is meromorphic in a stereographic coordinate and hence a rational map $w=p(z)/q(z)$ with coprime polynomials. To see why the rational form suffices, its finitely many coordinate poles can be removed by subtracting their principal parts, leaving a holomorphic function on the compact sphere and therefore a constant. The degree of a rational map of the Riemann sphere isafter canceling common factors. A generic target value $c$ has inverse images solving $p-cq=0$, with the root at infinity counted in homogeneous coordinates when necessary; the number is $N$, and holomorphic local degrees are positive. Poles of $w$ do not make the physical field singular: the reciprocal target coordinate $1/w$ is regular there. Antiholomorphic rational maps have the opposite topological degree and the same minimum energy.

For example, the unit sigma-model lump and size modulus isIts degree is one and radial integration gives $E=4\pi$. The complex center $z_0$ supplies two position parameters, while the magnitude and phase of $\rho$ give size and internal orientation. More generally, fixing $w(\infty )=0$ allows a monic degree-$N$ denominator and a numerator of degree at most $N-1$, with no common roots. Their $2N$ complex coefficients give the based rational-map moduli of sigma-model lumps.

Two-dimensional Derrick scaling leaves this pure-gradient energy unchanged, so degree fixes the minimum energy but not a preferred size. As $\rho \to 0$, the unit-lump energy concentrates and the field limit is singular at its center. A degree cannot change through a smooth boundary-preserving deformation, but concentration can leave that smooth configuration space. Thus topological degree protects the smooth sector and determines the Bogomolny energy, while not by itself stabilizing the lump size. There is no contradiction with the scalar obstruction in Question 1: a curved-target harmonic map obeys a nonlinear constrained field equation, rather than having harmonic unconstrained components.

The kink panel uses the normalization of Question 1. The lump panel has $z_0=0$ and $|\rho |=1$ in the unit sigma-model lump and size modulus; changing its size redistributes the density while keeping its total energy $4\pi$.

Let $T=\frac{1}{2}\int |\nabla \varphi {|}^2{d}^{n}x$ and $W=\int U(\varphi )d^{n}x$. For a sufficiently regular finite-energy field configuration with vacuum asymptotics, the scaled field ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$ stays in the same boundary sector. Changing variables gives the Derrick scaling formulaA static solution must be stationary under this variation, so the Derrick virial identity isFor $n>2$, both terms are nonnegative, so both vanish and the field is constant at a vacuum. Hence there are no nonconstant smooth finite-energy static solitons of this scalar type for $n>2$. For $n=1$ the identity is $T=W$, allowing kinks. For $n=2$ it requires $W=0$: a nonzero potential contribution excludes a static soliton, but the scaling identity alone does not exclude a purely derivative energy.

There is a useful refinement specific to the unconstrained flat target ${\mathbb{R}}^l$. If $U$ is differentiable and nonnegative, $W=0$ implies $U(\varphi (x))=0$ and ${\nabla }_{\varphi }U=0$ on the image. The static Euler-Lagrange field equations then give $\Delta \varphi =0$. Each first derivative is an entire harmonic function in L2 space. Its mean-value bound over a disk of radius $R$ is at most its total squared norm divided by $\pi R^2$; letting $R\to \infty$ shows that derivative is zero. Thus even the exceptional two-dimensional case has only constant smooth finite-energy solutions into this flat target. This two-dimensional flat-target Derrick obstruction uses the field equations in addition to the scale identity.

The Yang-Mills theory with a Higgs field escapes the original scaling because the gauge curvature is quadratic in spatial scaling. In three dimensions use $A_i^{\lambda }(x)=\lambda A_i(\lambda x)$ and ${\Phi }^{\lambda }(x)=\Phi (\lambda x)$. Then $F_{ij}^{\lambda }={\lambda }^2{F}_{ij}(\lambda x)$ and $D_i^{\lambda }{\Phi }^{\lambda }=\lambda D_i\Phi (\lambda x)$. Writing the magnetic, Higgs-gradient and Higgs-potential energies as $V_B,V_D,V_U$, the Derrick scaling of Yang-Mills-Higgs energy givesat stationarity. These terms can balance. In particular, the zero-potential Bogomolny-Prasad-Sommerfield monopole has $V_B=V_D\ne 0$.

For a nonlinear sigma model in two dimensions, the harmonic map energy is purely quadratic in first derivatives and is scale invariant: $V_{\lambda }=V$. Its target is a curved manifold, not the unconstrained Euclidean target of the refinement above. Its equation is the nonlinear harmonic-map equation, not componentwise $\Delta \varphi =0$. The Derrick theorem therefore gives no obstruction. Nonconstant finite-energy maps into $S^2$, as in the O3 nonlinear sigma model, provide examples with nonzero degree.

For the complex theory, regard $\varphi$ and $\bar{\varphi }$ as independent variables in the variation, and write $s=|\varphi {|}^2$, $U=U(s)$. Variation in $\bar{\varphi }$ givesThusThe factor 2 follows from the printed $1/2$ normalization of the complex kinetic term; equivalently vary its two real components.

A non-topological soliton is a localized finite-energy persistent field solution with no protecting winding or other nontrivial topological sector. In this complex theory a Q-ball has a stationary spatial profile and a rotating phase,with real $f$. It is not static as a field even though its energy density is time-independent. The global phase symmetry gives the Noether chargeThe profile equation is $-\Delta f+2U^{\prime }(f^2)f-{\omega }^{2}f=0$. Its energy is $T+W+{\omega }^{2}I/2$. To vary the size at fixed charge one must also vary the frequency, because scaling $f$ at fixed $\omega$ would change $Q$. Eliminating $\omega =Q/I$ gives charge-constrained scalar-field scaling:The new positive-power term can balance the other terms for $n>1$, including $n>2$, so the static Derrick theorem does not rule out these solutions. Equivalently, at fixed frequency the profile extremizes $T+\int [U(f^2)-{\omega }^2{f}^2/2]$, whose effective potential need not be nonnegative. This explains both the time dependence and the charge constraint behind the evasion; a complex field by itself is not enough.

For an explicit example take the sextic unique-vacuum Q-ball potentialIt is nonnegative, since $U(s)=s[(s-1/2{)}^2+3/4]$, and its only vacuum is zero. Its small-field mass squared is $2U^{\prime }(0)=2$, while ${\min }_{s>0}2U(s)/s=3/2$. It supports localized Q-balls, for example radial ones in three dimensions with $3/2<{\omega }^2<2$. In that range the profile potential is positive quadratic near zero but negative somewhere away from zero, permitting a localized nontrivial profile. This example uses a conserved phase charge and no vacuum winding.

Harmonic maps give nonconstant stationary wave maps. For $n=2$, inverse stereographic projection givesIt satisfies $|Q|=1$ and $\Delta Q=-|\nabla Q{|}^{2}Q$, so $\varphi (t,x)=Q(x)$ is a smooth global wave map. Its wave map energy is $4\pi$, since $|\nabla Q{|}^2=8/(1+|x{|}^2{)}^2$. Translations and rescalings give further examples; these harmonic maps approach a constant at infinity.

A time-independent wave map is a harmonic map of its spatial domain into the target. For target $S^2$, its equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. In two dimensions inverse stereographic projection gives a smooth nonconstant example of finite wave map energy $4\pi$.

For smooth fields into ${\mathbb{R}}^l$ with nonnegative differentiable potential and finite energy, Derrick scaling in two spatial dimensions forces the potential integral to vanish. Thus the field takes values where $U=0$ and $\nabla U=0$, so its static field equation makes every component harmonic. Each first derivative is an entire harmonic function in L2 space; the mean-value inequality on arbitrarily large balls forces it to vanish. Hence the field is constant. The scale identity alone does not exclude a pure-gradient model; this additional argument uses the flat unconstrained target. Curved-target harmonic maps, for example the O3 nonlinear sigma model, can instead have finite-energy nonconstant solutions.

A wave map is a harmonic map with Lorentzian domain: a critical point of the action obtained by contracting the pullback target metric with the domain Lorentzian metric. For target sphere $S^2$ in Euclidean space and $\square =-{\partial }_t^2+\Delta$, its extrinsic equation is $\square \varphi =\varphi (|{\varphi }_t{|}^2-|\nabla \varphi {|}^2)$, subject to $|\varphi |=1$. The derivative contractions are null forms for wave equations. Wave map Cauchy data must include a tangent initial velocity.