Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 313 (Solitons, Instantons, and Geometry)

Write the scalar field energy as $E=T+V$, whereUnder the Derrick scaling ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$, a change of variables givesA finite-energy solution of the Euler-Lagrange equation is stationary under this admissible variation. The Derrick virial identity is therefore

The static field equation is ${\varphi }^{\prime \prime }=U^{\prime }(\varphi )$, so integration givesThe polynomial before $C$ is nonnegative and vanishes, so requiring the minimum to be zero fixes $C=0$. Hence the vacuum manifold iswhich has three elements. A finite-energy scalar-field kink can join only adjacent vacua: a solution cannot cross the intermediate vacuum at finite $x$ because its first integral would have $\varphi ={\varphi }^{\prime }=0$ there and the Picard-Lindelof theorem would make it constant. There are therefore four oriented topological sectors,comprising two increasing kinks and their two antikinks. Symmetry under $\varphi \mapsto -\varphi$ and spatial reflection generates all four from one profile.

For the $0\to 1$ sector, completing the square gives the Bogomolny boundEquality holds for the Bogomolny equationWith $y={\varphi }^2$, this becomes the logistic differential equation $y^{\prime }=2y(1-y)$. Translation invariance supplies an arbitrary center $x_0$, and the explicit kink in a phi-six model isIt tends to $0$ and $1$ at the two spatial ends and saturates the bound. Its mass is consequently

Finite energy requires $|\varphi |\to 1$ and $D\varphi \to 0$ on the circle at spatial infinity. Writing $\varphi \sim e^{i\chi }$ there gives $A\sim d\chi$. The vortex number is the winding numberBy Stokes theorem, the magnetic flux is quantized:

For the rotationally symmetric Abelian Higgs vortex ansatz, $A=f(r)d\theta$ and $\varphi =h(r)e^{ik\theta }$ giveAfter the angular integration, the Abelian Higgs model energy becomesCompleting the square in the two pairs of terms givesRegularity at the origin and approach to the vacuum at infinity requireMore precisely, $h(r)=O(r^k)$ and $f(r)=O(r^2)$ near the origin. The boundary term is $k$, soThe bound is saturated exactly when both squares vanish, giving the radial Bogomolny vortex equationsThe asymptotic value $f(\infty )=k$ also makes the flux $2\pi k$, so the ansatz has vortex number $N=k$.

On an oriented pseudo-Riemannian vector space $({\mathbb{R}}^n,\eta ,\mathrm{vol})$, the Hodge star operator is the unique linear mapsatisfyingfor all $p$-forms $\alpha ,\beta$. If $s$ is the number of negative metric directions, thenon $p$-forms under this convention. For the two-forms in the question,whereasThe induced inner product is symmetric, so these expressions are negatives of one another. Hence a self-dual differential form and an anti-self-dual differential form are orthogonal and

Write $w=x^1+i{x}^2$ and $z=x^3+i{x}^4$. The metric is conformal to the standard Euclidean metric, and the Hodge star on middle-degree differential forms is conformally invariant. Taking $e^{ij}=d{x}^i\wedge d{x}^j$, the three real forms arebecause ${\omega }_1+i{\omega }_2=dw\wedge dz$. For the orientation specified by
$dw\wedge dz\wedge d\bar{w}\wedge d\bar{z}$, one hasThe corresponding relations for the complementary basis forms immediately giveThus these forms give a real basis of the self-dual two-forms.

Let $D_{\alpha }={\partial }_{\alpha }+A_{\alpha }$ be the gauge covariant derivative. In these complex coordinates the Anti-self-dual Yang-Mills equations areIntroduce the spectral parameter $\lambda$ and the linear operatorsTheir commutator isTherefore the Lax pair for the anti-self-dual Yang-Mills equationsis compatible for every $\lambda$ exactly when the ASDYM equations hold.

In particular, $F_{wz}=0$ says that the connection restricted to each $(w,z)$ surface is a flat connection. On a simply connected coordinate patch, the compatible equationshave an invertible solution $g$. Applying the associated gauge transformation setsThis conclusion is local; global topology can obstruct a single such gauge over the whole space.