Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 (Solitons, Instantons and Geometry)

Treat $\varphi$ and $\bar{\varphi }$ as independent fields. The Lagrangian density isThe Euler-Lagrange equation for $\bar{\varphi }$ isorThe equation from variation of $\varphi$ is its complex conjugate.

Put $\mu =|E|=-E>0$ and seek $\varphi =e^{-iEt}f(x)$ with real $f$. The equation becomesFor $b=\sqrt{2\mu }$, the identityshows thatsatisfies the stationary equation. Hence $e^{-iEt}{f}_E(x)$ is the bright soliton of the focusing nonlinear Schrödinger equation.

Let $y=x-X(t)$ and substituteSeparating the coefficient of $f_E^{\prime }$ and the constant and linear coefficients multiplying $f_E$ givesThis is the Galilean boost of the stationary soliton.

Write $b=\sqrt{2|E|}$. Its conserved squared norm isUsinggivesThereforeThe moving soliton behaves like a classical particle of inertial mass $N(E)$, with negative internal binding energy $-N(E{)}^3/24$.

Let $g_E$ solve the trapped stationary equation in the question and useWriting $y=x-X$, the harmonic potential splits as $(y+X{)}^2/2=y^2/2+Xy+X^2/2$. Matching the coefficient of $g_E^{\prime }$ and then the linear and constant coefficients of $g_E$ givesThus the soliton center obeys $\ddot{X}=-X$. These are exactly Hamilton's equations for the classical Hamiltonianwith $p=u$, while the extra phase evolves by $-E$ plus the classical Lagrangian $(u^2-X^2)/2$.

Use the four-parameter ansatz from part (ii), now allowing every parameter to vary slowly. Oddness of $y{f}_E(y{)}^2$ and the stationary-profile integrals give the collective-coordinate effective LagrangianwhereVariation of $\theta$ makes $N(E)$, and hence $E$, constant. Variation of $u$, $X$, and $E$ then givesThus the center follows the classical Hamiltonianto leading adiabatic order. The first neglected profile correction is of order ${ϵ}^3$, because the term linear in $y$ integrates to zero.

The curvature of $D=d+A$ isor $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]$. The covariant exterior derivative and the Jacobi identity give the Bianchi identity

Since the Hodge star operator satisfies ${\ast }^2=1$ on Euclidean two-forms,Using the opposite choice of sign as well givesEquality holds precisely for a self-dual or anti-self-dual Yang-Mills instanton, with the sign selected by that of $k$.

Now assume ${\partial }_0{A}_{\mu }=0$ and write $\Phi =A_0$. ThenThe spatial Bianchi identity and the self-duality equations becomewhere the upper four-dimensional sign gives the first displayed reduced sign under the orientation used here. Consequently

Gauge invariance of the inner product givesFor $w=1-|\Phi {|}^2$,Since $w\to 0$ at infinity, the weak maximum principle for elliptic operators excludes a negative interior minimum. Hence

With $d^{4}x=d{x}^0{d}^{3}x$, direct decomposition givesBecause $D_i{B}_i=0$,so the Pontryagin density reduces to the surface charge $4\pi N$ per unit $x^0$. The conventional three-dimensional energyobeys the Bogomolny bound $E_3\ge 4\pi |N|$, saturated by the Bogomolny-Prasad-Sommerfield monopole equation. The four-dimensional action density per unit $x^0$ is $2E_3$; signs relating $k$ and $N$ depend on the self-duality and orientation convention.

For the hedgehog ansatz for a monopole in the question, direct differentiation with $[T_a,T_b]=-{ϵ}_{abc}{T}_c$ givesOn a sphere of radius $r$,ThereforeThe boundary conditions $f\to 1$ and $r^2\alpha \to 1$ make the integrand equal to $r^{-2}+o(r^{-2})$. Thus

Write $F_{12}={\partial }_1{A}_2-{\partial }_2{A}_1$. Up to a total derivative, the covariant-gradient identity and completion of the magnetic and potential terms giveFor positive flux, equality in the Bogomolny bound therefore gives the Bogomolny vortex equationsThe opposite simultaneous signs describe negative winding.

For the orientation $d{x}^1\wedge d{x}^2>0$, the Hodge star operator isThus

A radial vortex of winding $N>0$ haswith $f(0)=a(0)=0$, $f(1)=1$, $a(1)=N$, and equations

For $e^{2\rho }=8/(1-|z{|}^2{)}^2$, put $u=\log |\Phi |$. Away from zeros, the first vortex equation givesSincethe field $\psi =u-\log (1-z\bar{z})+\log 2$ satisfies the Liouville equationFor holomorphic $g:D\to D$, direct use of the Cauchy-Riemann equations verifiesChoosing $g(z)=z^{N+1}$ gives the radial Witten hyperbolic vortexFor $N=1$ these reduce toSince $Bd{\mu }_g=dA=a_1^{\prime }(r)dr\wedge d\theta$,This directly verifies the Abelian Higgs vortex flux relation for unit winding.