Gauge-theory soliton

A gauge-theory soliton is a finite-energy localized classical field configuration stabilized by topology or a balance among differently scaling energy terms.

Derrick's scaling argument tests a static field configuration by rescaling space. A soliton can be stationary only if energy terms with opposite scaling powers balance at the original scale.

The Abelian Higgs model couples a complex scalar field to a $U(1)$ gauge field and permits magnetic vortex solitons in two spatial dimensions.

An Abelian Higgs vortex has quantized magnetic flux and a Higgs field whose phase winds at spatial infinity. Its vortex number is $N=(2\pi {)}^{-1}\int F$ under the convention $D=d-iA$.

At critical coupling on a surface with conformal metric $g=\Omega dzd\bar{z}$, the vortex equations are

For $u=\log |\varphi {|}^2$ and zeros of multiplicities $N_r$ at $z_r$, the Taubes equation is

Integrating the Taubes equation on a compact surface of area $A$ gives $A\ge 4\pi N$. A nontrivial vortex with nonzero Higgs field requires the strict inequality $A>4\pi N$.

A Yang-Mills instanton is a finite-action Euclidean gauge field with self-dual or anti-self-dual curvature. On ${\mathbb{R}}^4$, its asymptotic pure gauge defines a map $S_{\infty }^3\to SU(2)$ whose degree is the instanton number.

For an $SU(2)$ connection under a standard trace convention,Its sign depends on orientation and on whether the instanton is self-dual or anti-self-dual.