Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 56 (Solitons and Instantons)

The nonexistence conclusion concerns nonconstant fields: a constant scalar-field vacuum with $U=0$ is a zero-energy critical point of an energy functional in every dimension. Assume the potential is smooth, so the classical Euler-Lagrange equation is well defined. Write the two nonnegative energy terms asFor the admissible Derrick scaling ${\varphi }_{\lambda }(x)=\varphi (\lambda x)$,Stationarity at $\lambda =1$ gives the Derrick virial identityFor $D>2$, both terms have nonpositive sign, forcing $T=V=0$ and hence a constant vacuum. For $D=2$, scaling alone only proves $V=0$; the gradient term is scale invariant, so it must not simply be declared zero.

The two-dimensional flat-target Derrick obstruction supplies that remaining step. Since $U\ge 0$ and its integral is zero, continuity gives $U(\varphi (x))=0$ everywhere; smooth nonnegative $U$ has $U^{\prime }(\varphi (x))=0$ at those values. The field equation $\Delta \varphi =U^{\prime }(\varphi )$ therefore becomes $\Delta \varphi =0$. Each ${\partial }_i\varphi$ is an entire harmonic function in $L^2({\mathbb{R}}^2)$. Its mean value property and the Cauchy-Schwarz inequality give, for every radius $R$,Let $R\to \infty$ to get ${\partial }_i\varphi =0$. Thus $D=2$ also has only constant stationary field configurations of finite energy. The unconstrained real scalar target is essential in this argument; the spherical target in Question 2 has different field equations.

In $D=1$ the field equation is ${\varphi }^{\prime \prime }=U^{\prime }(\varphi )$. Multiplication by ${\varphi }^{\prime }$ gives the first integralFinite energy makes $\frac{1}{2}({\varphi }^{\prime }{)}^2+U(\varphi )$ an integrable nonnegative function, so along some sequence $x_n\to +\infty$ it tends to zero. Evaluating the constant first integral on that sequence gives $K=0$. ChooseConsequently $({\varphi }^{\prime }{)}^2=2U(\varphi )$ andFor a nonconstant entire solution the sign is fixed. Indeed a finite-$x$ zero of ${\varphi }^{\prime }$ would have $U(\varphi )=U^{\prime }(\varphi )=0$; uniqueness of the smooth second-order equation with initial data $(\varphi ,{\varphi }^{\prime })=(v,0)$ would make the solution the constant vacuum. Thus ${\varphi }^{\prime }$ never vanishes in a nonconstant solution. Conversely a smooth solution of the displayed first-order equation satisfies ${\varphi }^{\prime \prime }=U^{\prime }(\varphi )$ wherever it is nonconstant.

Under the usual soliton boundary conditions $\varphi (x)\to v_{\pm }$ as $x\to \pm \infty$, with $v_{\pm }$ in the vacuum manifold $\{U=0\}$, the two endpoint components label the topological sector. For isolated scalar-field vacua this is an ordered pair of vacuum labels, not necessarily one universally normalized integer. More explicitly, for any smooth $F$, the topological currentis identically conserved, taking ${ϵ}^{01}=1$ and boundary conditions fixed during evolution. The choices $F(u)=u$ and $F(u)=W(u)$ give the field-difference charge and the kink energy charge. The square completion for a one-dimensional kink givesThe first-order critical solution saturates this Bogomolny bound. A nonzero difference between endpoint vacuum components prevents continuous deformation to a vacuum while those boundary conditions remain fixed. Finite energy alone need not give finite endpoint values for arbitrary potentials, so these topological labels use the stated vacuum boundary conditions.

To evade the scalar-only obstruction, introduce a gauge field and a charged, generally multicomponent Higgs field. For a Yang-Mills-Higgs energy write $E=T_H+V+E_B$, where $T_H$ contains $|D_i\Phi {|}^2$ and $E_B$ contains $|F_{ij}{|}^2$. Rescale ${\Phi }_{\lambda }(x)=\Phi (\lambda x)$ and $A_{i,\lambda }(x)=\lambda A_i(\lambda x)$. Then $D_i\Phi$ scales by $\lambda$, while $F_{ij}$ scales by ${\lambda }^2$, soThe magnetic term supplies the missing opposing scaling power. In $D=2$ the relation is $E_B=V$, permitting Abelian Higgs vortices; in $D=3$ it is $E_B=T_H+3V$, permitting 't Hooft-Polyakov monopoles. In the monopole Bogomolny limit $V=0$, $B_i=\pm D_i\Phi$ achieves $E_B=T_H$. Also, asymptotic ordinary derivatives need not vanish when a Higgs phase winds: the relevant vanishing quantity is the gauge covariant derivative. This allows nontrivial boundary winding with finite energy. Merely appending a gauge potential to a neutral one-component real field would not by itself furnish that Higgs mechanism.

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.

Use Euclidean coordinates $(x^1,x^2,x^3,x^4)$ and orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$. For $D_{\mu }={\partial }_{\mu }+A_{\mu }$, the gauge curvature is $F_{\mu \nu }=[D_{\mu },D_{\nu }]$. The ASDYM equations are $F=-\ast F$, equivalentlyPut $w=x^1+i{x}^2$, $z=x^3+i{x}^4$, with ${\partial }_w=({\partial }_1-i{\partial }_2)/2$ and similarly for $z$. The bars label the complex conjugate coordinates on the Euclidean real slice; after complexification they can be treated as independent coordinates. A Lax pair for the anti-self-dual Yang-Mills equations isAt infinity rescale the operators and use the other affine coordinate. Their commutator isCompatibility for every $\lambda$ is therefore precisely $F_{wz}=F_{\bar{w}\bar{z}}=0$ and $F_{w\bar{w}}+F_{z\bar{z}}=0$. To verify the real component signs,The conjugate equation supplies the remaining real components. This is compatibility of an overdetermined system, not an additional field equation on $\Psi$ for only one value of the spectral parameter.

Impose invariance under translations in $x^1,x^2,x^3$, writing $t=x^4$. In a symmetry-adapted gauge the potentials depend only on $t$. A further $t$-dependent Yang-Mills gauge transformation sets $A_4=0$ locally, by solving the usual parallel transport ordinary differential equation; it does not reintroduce spatial dependence. ThenThe three real equations above become ${\dot{A}}_1=[A_2,A_3]$, ${\dot{A}}_2=[A_3,A_1]$ and ${\dot{A}}_3=[A_1,A_2]$, namelyThese are the Nahm equations. The orientation chosen above gives the requested plus sign; reversing orientation exchanges self-duality of gauge curvature and anti-self-duality of gauge curvature. Translation invariance here is a local dimensional reduction and is not a claim of finite four-dimensional action over all three translation directions.

To prove conservation, abbreviate $P=A_1+i{A}_2$, $Q=A_1-i{A}_2$ and $R=A_3$, so $A(\lambda )=P+2R\lambda -Q{\lambda }^2$ and $B(\lambda )=-iR+iQ\lambda$. Expanding the commutator, including the cancellation of the cubic term, givesThe Nahm equations giveThus the polynomial Lax representation of the Nahm equations isTake a finite-dimensional matrix representation of the Lie algebra and its complexification. For every positive integer $p$, differentiating the product and using cyclicity of the matrix trace givesThis proves the trace invariants of a Lax equation directly. Since $\mathrm{Tr}(A(\lambda {)}^p)$ is a polynomial of degree at most $2p$ in the affine coordinate $\lambda$, its derivative vanishes identically only if the derivative of every coefficient vanishes. Therefore every trace-polynomial coefficient is independent of $t$. At projective infinity these polynomials are sections of $\mathcal{O}(2p)$; they need not be constant functions of $\lambda$ on the whole projective sphere.

Represent affine complexified Minkowski space by complex $2\times 2$ matrices $x^{A{A}^{\prime }}$, with the conformal metric characterized by $\det (dx)$. Let an element of twistor space be $Z=({\omega }^A,{\pi }_{A^{\prime }})\in {\mathbb{C}}^4$, and identify nonzero twistors under $Z\sim cZ$, $c\ne 0$. The compact projective twistor space is ${\mathbb{CP}}^3$. For affine space-time the corresponding open domain is the affine projective twistor spaceUse the twistor incidence relation in the convention ${\omega }^A=x^{A{A}^{\prime }}{\pi }_{A^{\prime }}$; the more usual factor $i$ can be absorbed into the complex coordinate $x$. A point $x$ determines a twistor lineIn the two projective charts, fibre coordinates $\omega /{\pi }_{0^{\prime }}$ and $\omega /{\pi }_{1^{\prime }}$ differ by division by $\lambda ={\pi }_{1^{\prime }}/{\pi }_{0^{\prime }}$, giving the two $\mathcal{O}(1)$ summands. Two distinct space-time points have intersecting lines precisely when $(x-y)\pi =0$ for some nonzero $\pi$, equivalently $\det (x-y)=0$; this recovers conformal null separation.

The twistor correspondence is the double fibrationFor a fixed twistor, the incidence solutions form an affine alpha-plane, a two-dimensional totally null plane. Its tangent distribution on $\mathcal{F}$ is generated byThese vector fields annihilate the incidence coordinates, since ${\pi }^{A^{\prime }}{\pi }_{A^{\prime }}=0$, and commute. This distribution is the geometric source of the linear system in the Penrose-Ward correspondence.

For the inverse construction, take a rank-$r$ holomorphic vector bundle $E$ on the twistor domain corresponding to a space-time open set $U$. The crucial hypothesis isThese are line-trivial bundles in the Ward correspondence. It is not enough to specify an arbitrary holomorphic vector bundle, nor just zero First Chern class on its lines: $\mathcal{O}(1)\oplus \mathcal{O}(-1)$ has degree zero but is nontrivial. A line with a nontrivial splitting is a jumping line and does not support the regular linewise construction there. Thus a globally nonsingular field on all affine complexified space-time requires triviality on all its relevant lines; a local construction uses only the lines over $U$.

Choose holomorphic vector bundle bundle frames on two twistor patches, and let $P$ be the invertible transition matrix on their overlap. Pull it back to $\mathcal{F}$. Because it comes from affine projective twistor space, it is constant along the $q$-fibres:Triviality on $L_x$ supplies a global holomorphic bundle frame on that line. Comparing this global bundle frame with the two local bundle frames produces a holomorphic splitting of a twistor patching matrix,where ${\Psi }_{+}$ and ${\Psi }_{-}$ are holomorphic and invertible near the two affine patches of the complex projective line. The splitting can be chosen holomorphically in $x$ locally: the restrictions have $r$ independent global sections and $H^1({\mathbb{CP}}^1,{\mathcal{O}}^{\oplus r})=0$, so these bundle frames extend through a neighborhood of the given line. This explains where regularity and linewise triviality enter the construction.

Differentiate the factorization using $V_{A}P=0$. Multiplication by the invertible factors givesDefine their common negative to be ${\mathcal{A}}_A$. It is regular on both projective patches and homogeneous of degree one in $\pi$, because $V_A$ has that degree while the bundle frames have degree zero. ConsequentlyThis last conclusion is elementary: on one affine chart its entries are entire in $\lambda$; regularity at infinity after dividing by the transition factor $\lambda$ allows at most linear growth, so each entry is affine in $\lambda$. Equivalently, a global section of $\mathcal{O}(1)$ is a linear homogeneous polynomial. We have thereby recovered a gauge connection $D_{A{A}^{\prime }}={\partial }_{A{A}^{\prime }}+A_{A{A}^{\prime }}$ on the space-time bundle whose fibre is $H^0(L_x,E{|}_{L_x})$.

The splitting factors now obeyTheir invertibility implies the compatibility conditionsIn the spinor decomposition of gauge curvature,the two spinors are symmetric. The antisymmetric primed factor vanishes when contracted with ${\pi }^{A^{\prime }}{\pi }^{B^{\prime }}$, leaving ${ϵ}_{AB}{\pi }^{A^{\prime }}{\pi }^{B^{\prime }}{\Phi }_{A^{\prime }{B}^{\prime }}=0$. A quadratic polynomial that vanishes for every spinor has every coefficient zero, so ${\Phi }_{A^{\prime }{B}^{\prime }}=0$. With the orientation convention used here, this is exactly anti-self-duality of gauge curvature, hence the ASDYM equations. This establishes the field equation from the holomorphic data, rather than assuming it as an extra input.

The construction naturally gives a gauge class. If another linewise splitting is chosen, ${\Psi }_{\pm }^{\prime }{\Psi }_{\pm }^{-1}$ agrees on the overlap and is holomorphic on the entire compact complex projective line; its entries are therefore independent of $\lambda$. Write this common matrix as $g(x{)}^{-1}$. Thenthe ordinary Yang-Mills gauge transformation. Changing the original holomorphic vector bundle bundle frames or replacing $E$ by an isomorphic bundle produces the same space-time gauge class. A determinant trivialization or another holomorphic structure-group reduction gives the corresponding restriction from $GL(r,\mathbb{C})$ to $SL(r,\mathbb{C})$ or another complex gauge group.

For perspective, the forward direction starts from an ASD gauge connection and uses its flat gauge covariant derivative along each alpha-plane to descend parallel sections to affine projective twistor space. The reconstructed linear system is exactly that parallel-section system. Going back through it recovers the original line-trivial bundle, while changing space-time bundle frames amounts to Yang-Mills gauge transformation. Thus, locally and with the stated regularity and triviality conditions, holomorphic line-trivial bundles determine solutions of the ASDYM equations uniquely up to gauge.

No unitary reality condition is imposed on complexified Minkowski space in this question. Obtaining real Euclidean gauge fields requires the compatible antiholomorphic twistor involution and a bundle reality structure; obtaining finite-action instantons requires further global and boundary conditions. Those are additional restrictions on the complex correspondence, not automatic consequences of having any holomorphic vector bundle on projective twistor space.