Spectral Flow of Vortex Shape Modes over the BPS 2-Vortex Moduli Space

The flow of shape eigenmodes of the small fluctuation operator around BPS 2-vortex solutions is calculated, as a function of the intervortex separation $2d$. For the rotationally-invariant 2-vortex, with $d = 0$, there are three discrete modes; the lowest is non-degenerate and the upper two are degenerate. As $d$ increases, the degeneracy splits, with one eigenvalue increasing and entering the continuous spectrum, and the other decreasing and asymptotically coalescing with the lowest eigenvalue, where they jointly become the eigenvalue of the 1-vortex radial shape mode. The behaviour of the eigenvalues near $d=0$ is clarified using a perturbative analysis, and also in light of the 2-vortex moduli space geometry.


Introduction
Collective coordinate dynamics has been successfully implemented in [1] to describe the low-energy effective theory governing collisions between ϕ 4 -kinks.The collective coordinates associated to each kink consist of the kink center a and the amplitude A of the unique shape mode of kink fluctuations.The reduced system in the centre of mass frame, assuming reflection symmetry, has a two-dimensional Lagrangian with a kinetic energy involving a metric depending on A and a potential energy also depending on A.
It seems feasible to develop a similar effective theory for BPS n-vortex dynamics in the Abelian Higgs model at critical coupling.For a 1-vortex, the collective coordinates would be the vortex center and the amplitude of the unique discrete fluctuation mode -the radial shape mode.Finding a collective coordinate treatment in the case of 2-vortices would be far more interesting.Here, the discrete eigenvalues and eigenfunctions of the second-order fluctuation operator vary with the separation of the 1-vortex constituents.Internal shape modes of vortices of various kinds have been studied, e.g., in refs.[2,3,4,5], but so far, almost exclusively for coincident-vortex solutions.
The moduli space of BPS n-vortices is the space of sets of n unordered points in the plane -the points where the Higgs field vanishes.This is a non-singular complex manifold [6,7,8], even where vortices coincide, but one needs to be careful about the choice of coordinates.In particular, for 2-vortices with fixed centre of mass at the origin, if the constituent vortices have a separation 2d, then it is d 2 rather than d that is a good radial coordinate on the moduli space.
For each n > 0 there is a unique n-vortex, rotationally-invariant around the origin, for which the n constituent vortex locations coincide.The spectral problem of the second-order vortex fluctuation operator in this case resembles that of the planar Hydrogen atom, but the potential well is bounded from below and reaches the continuous-spectrum threshold exponentially fast as the radius increases, which implies that there exist at most a finite number of discrete shape modes.In [2], these facts guided the search for discrete modes.It was found, for example, that for n = 1 there is only one discrete shape mode whereas for n = 2 there are three.
The generic BPS 2-vortex is formed from two 1-vortices separated by an arbitrary finite distance.The spectrum of vortex fluctuations is then akin to the quantum spectrum of a particle moving in the field created by two centers of force, where the centers are freely movable.In this paper we investigate how the shape modes and their eigenfrequencies vary with the vortex separation.The number of discrete modes remains finite, and in fact decreases from three to two as the separation increases.
Our results should lead to a generalisation of the notion of geodesic flow on the 2-vortex moduli space as an approximation to 2-vortex dynamics.Allowing for the possibility of excited shape modes will require a more complicated collective coordinate model for the low-energy dynamics of vortices, including a potential on the moduli space.If the vortex motion is slow, and the evolution of the shape mode amplitudes is treated adiabatically, then a Berry connection will probably also be needed.

BPS vortices in the Abelian Higgs model: SUSY structure of the fluctuation operator
We start from the action of the relativistic Abelian Higgs model, describing the minimal coupling between a U (1) gauge field and a charged scalar field in a phase where the gauge symmetry is broken spontaneously.We focus on the BPS critical value of the coupling (the strength of the Higgs potential) where the Higgs and gauge field masses are equal [9,10].In terms of non-dimensional coordinates, couplings and fields, the action functional for this system is The ingredients here are a complex scalar (Higgs) field ϕ Our Minkowski-space metric tensor is g µν = diag(1, −1, −1), with µ, ν = 0, 1, 2, and we use the Einstein summation convention.In the temporal gauge A 0 = 0, the energy of static field configurations becomes We interchangably use Cartesian and polar coordinates ⃗ x = (x 1 , x 2 ) = (x, y) = (r cos θ, r sin θ) with d 2 x = dx dy.The energy (2), treated non-relativistically, models the free energy of a superconducting material arising in the Ginzburg-Landau theory of superconductivity -see formula (17) in [11] where the order parameter |ϕ| 2 = ϕ ϕ corresponds to the density of Cooper pairs.Critical points of V [ϕ, A] that satisfy the boundary conditions on the circle at infinity S 1 ∞ (i.e. as r → ∞) have finite energy.Indeed, it can be checked that the configuration space of static fields, is the union of Z topologically disconnected sectors.Here, n is the vorticity or winding number of the map ϕ| ∞ : S 1 ∞ −→ S 1 from the circle at infinity to the vacuum orbit |ϕ| 2 = 1, parametrised by the phase of ϕ.It follows from the vanishing of the covariant derivative of ϕ at infinity, that n is also the normalised magnetic flux, Φ ≡ 1 2π R 2 d 2 x F 12 = n.In the BPS regime, V [ϕ, A] can be written in the form, see [9], which implies that BPS vortices are solutions of the first-order PDEs and are absolute minima of the energy for each n, with V [ϕ, A] = π|n|.For n positive, the upper signs in (6) need to be chosen.Then, given n, there exist BPS vortex solutions characterized by n arbitrary locations (points) in the plane, which are the zeros of the scalar field counted with multiplicity, and simultaneously the locations of maximal magnetic field.n-vortex solutions therefore have 2n real moduli.
For the sake of clarity in later formulas we shall denote the scalar field profile ϕ(⃗ x) of an n-vortex solution as 2 (⃗ x) and the gauge (vector The main theme of this paper is the construction of field fluctuations around 2-vortices {ψ (2) k (⃗ x)}, and analysis of how they depend on the vortex separation.
To find the linear fluctuation modes, we consider the evolution of small perturbations φ j (⃗ x) and a k (⃗ x) around BPS vortex fields ψ with ϵ small, and they still belong to the n-vortex topological sector.To discard pure gauge fluctuations, we impose the background gauge as the gauge fixing condition on the perturbations.Substituting (7) into the field equations and linearizing, the eigenfrequencies ω and eigenmodes are found to be solutions of the spectral problem H + ξ λ (⃗ x) = ω 2 λ ξ λ (⃗ x), see [12,13,14].Here, λ is a label in either the discrete or continuous spectrum, and H + is the second-order vortex fluctuation operator where |ψ| 2 = ψ 2 1 +ψ 2 2 , and {ψ, V k } satisfy eqs.(6).The fluctuation vectors ξ(⃗ x) belong in general to a rigged Hilbert space.There are square integrable eigenfunctions ] is bounded, as well as continuous spectrum eigenfunctions.
Weinberg [15] proved that there are 2n linearly-independent normalizable zero modes (having eigenvalues ω 2 = 0) for any BPS n-vortex solution.These can be characterized as lying in the kernel of the operator Analysis of these zero modes was further developed in [16] and [17], motivated by the study of vortex scattering at low energies within the approach of geodesic dynamics in the n-vortex moduli space, see e.g.[18].A crucial point for the calculations in this paper is that are SUSY (supersymmetry) partners, and therefore isospectral in the strictly positive part of the spectrum.Moreover, is a simpler operator than H + , and its spectrum is easier to investigate.For H − , it was proved in [2,3] that it is sufficient to find eigenfunctions of the form The corresponding eigenfunction of the SUSY partner operator H + , sharing the positive eigenvalue ω 2 λ and the same normalization, is 3 The spectrum of the BPS 2-vortex fluctuation operator In this Section we obtain numerically the discrete, positive eigenvalues and eigenfunctions of the secondorder fluctuation operator H + evaluated at BPS 2-vortex solutions {ψ (2) k (⃗ x)}.Recall that these 2-vortices can be interpreted as two separated 1-vortices.If the 2-vortex mass center is located at the origin, the 1-vortex locations (zeros of ψ (2) ) can be assumed to be at (d, 0) and (−d, 0), with equivalent solutions being obtained by translation and rotation.The first task is to construct the 2-vortex solution with these zeros.
We start from the more general, rotationally-invariant n-vortex solution, having the polar coordinate form and with the radial gauge A (n) r = 0 imposed.The functions f n (r) and β n (r) need to satisfy the first-order coupled equations and the asymptotic conditions f n (r) → 1 and β n (r) → 1 as r → ∞.The requirement of regularity at the origin fixes the behaviour for small r to be f n (r) ∼ d n r n and β n (r) ∼ 1 4n r 2 , where d n is a constant depending on n.Using eqs.(17), the radial profiles f n (r) and β n (r) for n = 1 and n = 2 (and higher n) are easily generated.A 2-vortex whose zeros are well separated is then approximated by superposing two rotationally-invariant 1-vortices, translated to have the desired zeros.A 2-vortex with coincident zeros is the rotationally-invariant solution with radial profiles f 2 (r) and β 2 (r).
To construct 2-vortex solutions whose zeros are separated by an intermediate distance 2d, we take advantage of the Bogomolny energy bound, which for 2-vortices is V [ψ (2) , V (2) ] = 2π.Our strategy is to numerically construct an n = 2 configuration with zeros at (d, 0) and (−d, 0), that accurately saturates this bound.Indeed, the deviation of the energy from this bound is an estimate of the solution's precision.To generate initial data, we use a generalized n = 1 configuration centered at (d, 0), where r = (x − d) 2 + y 2 and θ = arctan y x−d are polar coordinates around the centre.Here, α is a free parameter whose value is chosen later.The desired n = 2 configuration can now be constructed using the standard superposition Substituting this into (2) gives the energy depending on α.Our numerical mesh ranges over the spatial rectangle [−30, 30] × [−15, 15], with I max = 3201 points in the x-component and J max = 1601 in the y-component.The mesh points are ⃗ p ij = (x min + i δx, y min + j δy) where x min = −30 and y min = −15, δx = δy = 0.01875 are the spatial steps, i = 0, . . ., I max − 1 and j = 0, . . ., J max − 1.A second-order finite difference scheme is employed to approximate the spatial derivatives arising in the functional energy (20).Now, for each d, the value of α is chosen to minimize (20).The analytical relation α = d ϵ /(d ϵ + γ) with ϵ = 3.4 and γ = 1.4 is a good approximation.Finally, from this configuration, numerical gradient flow is employed to refine the solution.The resulting energies saturate the Bogomolny bound with a relative error less than 0.03%.These numerical 2-vortex solutions are precise enough to address the spectral problem of this paper.For later convenience, we denote their scalar field at the mesh points by ψ (2) The next task involves the spatial discretization of the spectral problem ( 14) for n = 2, which will lead us to the positive discrete spectrum of the 2-vortex fluctuation operator H − , and hence H + .Here, we can work with a less fine mesh than the previous one.We introduce new mesh points ⃗ q ij = (x min + i∆x, y min + j∆y) where ∆x and ∆y are the new spatial steps in each direction, i = 0, . . ., i max − 1 and j = 0, . . ., j max −1.The fluctuation field values at the mesh points, (a 1 ) ij = a 1 (⃗ q ij ), are arranged in a single column (a 1 ) s ≡ ((a 1 ) 00 (a 1 ) 01 . . .(a 1 ) 0(jmax−1) (a 1 ) 10 . . .(a 1 ) (imax−1)(jmax−1) ) t where s = j max • i + j , as also are the values of the background potential well, With this arrangement, the eigenvalue problem, discretized up to second order, is Dirichlet boundary conditions have been assumed for the eigenfunctions, which implies that the value (a 1 ) s = 0 is imposed for s < 0 and for s ≥ j max • i max .In addition, if s mod j max = 0 then (a 1 ) [s/jmax]jmax − 1 = 0 where [z] denotes the integer part of z, and likewise, (a 1 ) [s/jmax]jmax = 0 for s mod j max = j max − 1.The procedure approximates the fluctuation operator by a finite matrix, which can be analysed to obtain the full spectrum of discrete eigenvalues and eigenfunctions.It has been checked that the choice i max = 201 and j max = 101 with ∆x = ∆y = 0.3, giving a fluctuation operator approximated by a 20301 × 20301 matrix, provides precise enough results.In Figure 1, the potential wells U (⃗ q s ) of the discrete Schrödinger-type equation ( 22) are plotted for intervortex distance parameters d = 0, d = 2 and d = 5.
The main result of this paper is displayed in Figure 2.This shows the three discrete positive eigenvalues ω 2 λ , plotted as functions of d.To better understand the result we recall the spectrum of the fluctuation operator in the case of rotationally-invariant 1-and 2-vortices.Using the notation introduced in [2, 3], for the 1-vortex there is only one shape eigenmode with angular momentum number k = 0, eigenvalue ω 2 10 ≈ 0.777476 and eigenfunction v 10 (r).For the 2-vortex there is one eigenmode with angular momentum number k = 0, eigenvalue ω 2 20 ≈ 0.53859 and eigenfunction v 20 (r), and also a doubly-degenerate pair of eigenmodes with angular momentum number k = 1, eigenvalue ω 2 21 ≈ 0.97303 and eigenfunctions v 21 (r) cos θ and v 21 (r) sin θ.The behaviour of the radial profiles v 10 (r), v 20 (r) and v 21 (r) is shown in  .These modes correspond to the symmetric and antisymmetric combinations of the localized, radial shape modes associated to the individual 1-vortices.
To emphasize this picture, in Figure 3 the eigenmode a 1 of H − with eigenvalue ω 2 1 (d) is plotted for selected values of d.We observe how starting at d = 0 from the k = 0 solution a 1 (r) = v 20 (r), the eigenmode remains symmetric.Asymptotically, it may be understood as the symmetric linear combination of the 1-vortex modes, a 1 ≃ v 10 (r + ) + v 10 (r − ) with r ± = (x ± d) 2 + y 2 .In Figure 4 we plot the eigenmode with eigenvalue ω 2 2 (d), starting from the k = 1 mode a 1 = v 21 (r) cos θ that is antisymmetric under x → −x.The mode profile retains its antisymmetry as d grows.In particular, for large d, the mode becomes the antisymmetric linear combination of the 1-vortex modes, a 1 ≃ v 10 (r + ) − v 10 (r − ).
The eigenmodes of the 2-vortex fluctuation operator H + can be obtained from these eigenmodes of H − by using the intertwining formula (15).Since these are 4-component vectors, it is difficult to illustrate their precise form, or the way that they excite a 2-vortex solution.However, we can plot the potential energy density of an excited 2-vortex.Figure 5 shows snapshots of the oscillating 2-vortex solution at separation parameter d = 1.5, excited by the mode of lowest positive frequency, ω 1 .The constituent 1-vortices shrink and stretch in phase.Figure 6 shows the oscillations of the same 2-vortex excited by the discrete mode with the higher frequency, ω 2 .In this case, the 1-vortices oscillate in counterphase, i.e. while one vortex shrinks the other stretches.

Perturbation theory near d = 0
Here, we present some analytical calculations based on perturbation theory, that explain the spectral structure of the 2-vortex modes near the rotationally-invariant 2-vortex at d = 0. Specifically, we shall consider the eigenvalue problem (14) where the potential well is determined by the scalar part of a 2-vortex solution for small d, which we denote by ψ (2) (⃗ x) is the rotationally-invariant 2-vortex scalar field and δψ (2) (⃗ x) is derived from the zero-frequency mode ξ 0 that splits the locations of the two overlapping 1-vortices, whose form is where h 20 (r) can be obtained numerically1 (see Section 4.1 in ref. [3]), and note that h 20 (r) is a decreasing function so h ′ 20 (r) < 0. Therefore, the expansion (23) simplifies to A relation between the perturbation parameter ϵ and the small distance parameter d can be derived from the zeros of (25).Near the origin,  Notice that ( 25) is a first-order expansion in ϵ, so we are restricted here to a first-order treatment of the spectral problem ( 14).A higher-order expansion is a non-trivial task beyond the scope of this paper.
In addition to (23), the eigenmodes and eigenvalues must be expanded as where a 1 (⃗ x) and ω 2 are a joint eigenmode and eigenvalue of the fluctuation operator H − associated to the rotationally-invariant 2-vortex, and ϵ is related to d as in (26).At first order, eq.( 14) reduces to From the Fredholm alternative it is known that the projection of the right-hand side of (28) on to the homogeneous solution (the eigenmode a 1 (⃗ x) at d = 0) must be zero in order to obtain bounded solutions.This implies that We can now evaluate the integrals in (29) to estimate the three eigenvalues arising for small d.
1.For the perturbed k = 0 mode at d = 0, because the angular integral vanishes.Therefore, the lowest non-degenerate eigenvalue has no quadratic dependence on d, so the leading dependence is quartic (at least), 2. For the k = 1 mode proportional to cos θ at d = 0, 3. For the k = 1 mode proportional to sin θ at d = 0, Combining the last two results, we find the splitting of the eigenvalues ω 2 3 and ω 2 2 that are degenerate at d = 0, as shown in Figure 7 (left).The similar dependence on d 2 , apart from the sign, is striking, and will be clarified in the next Section.5 Insight from the 2-vortex moduli space The work of Taubes on BPS n-vortex solutions [6] and the later work of Samols on the Riemannian geometry of the n-vortex moduli space [7], reviewed in [8], make clear that it is best to set z = x + iy, and to use the complex variable Z = X + iY to denote the location of a vortex (a zero of the scalar field), instead of the Cartesian 2-vector (X, Y ).An n-vortex solution is characterised by its n unordered zeros; good complex coordinates on the moduli space are therefore the n elementary symmetric polynomials in these zeros.If the zeros are at {Z 1 , Z 2 , . . ., Z n }, these coordinates are (up to sign) the coefficients of the polynomial Taubes showed that an n-vortex with these zeros exists and is unique up to gauge transformations, and in a convenient gauge has a scalar field ϕ that is a product of a real function with the polynomial P (z).For a 2-vortex with zeros at Z 1 and Z 2 , so good, 2-vortex moduli space coordinates are the centre of mass 1 2 (Z 1 +Z 2 ) and the product −Z 1 Z 2 (this sign choice is convenient).We are interested in vortices with centre of mass at the origin.In particular, if the vortices have Cartesian locations (d, 0) and (−d, 0), as above, then Z 1 = d and Z 2 = −d, and the good coordinate for these centred vortices is c = −Z 1 Z 2 = d 2 .When c = 0, the vortices coincide at the origin, and the 2-vortex is rotationally invariant.Samols showed that the metric on the moduli space of centred 2-vortices has the form ds 2 = F (|c|) dc dc, with F smooth and positive, including at c = 0.
There is a geodesic motion through moduli space, where c moves smoothly along the real axis from positive to negative values and the velocity of c remains negative throughout.What this means is that the vortices scatter through a right angle.If the vortices have Cartesian locations (0, d) and (0, −d), then Z 1 = id and Z 2 = −id, so c = −Z 1 Z 2 = −d 2 .When c is positive, the vortex locations are on the x-axis; when c is negative, they are on the y-axis.
The eigenvalues and eigenmodes of the fluctuation operator around a 2-vortex are expected to flow smoothly over the moduli space.In particular, for centred vortices moving on the x-axis, the flow is smooth as a function of c = d 2 .c is a better coordinate than d, and the flow remains smooth if the range of c is extended to negative values, corresponding to a right-angle scattering of the vortices.Furthermore, there is a symmetry between the vortex configuration with modulus c, and with modulus −c; they differ by a rotation through a right angle.So the eigenvalues of the discrete modes are the same at c and −c, and the eigenmodes should be related by a right-angle rotation.(2) for small c, the dependence of the lowest eigenvalue ω 2  1 on c appears to be quadratic, as expected for a smooth symmetric function with a minimum.This quadratic dependence was verified through the perturbative analysis of Section 4. It means that the dependence of ω 2 1 on d, shown in Figure 2, is quartic, something that would be rather curious if one did not take into account the moduli space geometry; (3) the two eigenvalues ω 2 2 and ω 2 3 cross over smoothly and linearly at c = 0.This happens because for c > 0, the eigenmode for ω 2 2 is antisymmetric in x and symmetric in y, whereas for ω 2 3 it is the opposite way.As the vortices scatter through a right angle, the symmetry axes are exchanged, and the eigenvalues exchange their order.The linear dependence of these eigenvalues on c = d 2 , and the smooth crossover, is verified by the formula (33) obtained through the perturbative analysis.In terms of d, the eigenvalues have quadratic dependence near d = 0, and crucially, the coefficients of the quadratic terms have opposite signs.The numerical results, shown in Figure 7, do not exactly match these expectations.This could be because the numerical analysis is tricky for d very small, or it could be because the locations of the vortex zeros have moved slightly during the relaxation of the vortex configuration from the initial ansatz (19) to the optimised solution.
Recall that eigenvalue crossing is not generic as one parameter varies; instead the eigenvalues tend to repel and avoid crossing.However, here the crossing eigenvalues have eigenmodes with opposite symmetries, so there is no repulsion.

Outlook
We have obtained some detailed understanding of the discrete shape modes of BPS 2-vortex solutions, and how they vary with the separation of the 1-vortex constituents.It would be interesting to study the modification to the low-energy scattering of vortices when these modes are excited, either classically or quantum mechanically.The geodesic motion through the 2-vortex moduli space [7] will be supplemented by oscillation of the discrete modes and some potential energy function on the moduli space.An adiabatic treatment should be possible if the energy in the modes of oscillation is comparable with the kinetic energy of the translational motion of the vortices, and both are small.The simplest case would be when just one shape mode is excited.As the shape modes have different symmetries, the transfer of energy from one mode to another is likely to be suppressed.A complication may occur at the critical separation where one mode enters the continuum, should that mode be excited.We have seen that if vortices approach from a large distance, and their radial shape modes are excited in counterphase, then after they approach and scatter through a right angle, then it is this mode that enters the continuum.
A related program would be a search for fermionic bound states to vortices and an examination of their properties.This involves replacing Weinberg's first-order differential operator by the Dirac operator, and investigating the spectral problem −iγ j ∂ ∂x j − iV j (x 1 , x 2 ) + 2|ψ(x 1 , x 2 )| Ψ(x 1 , x 2 ; ω) = ωΨ(x 1 , x 2 ; ω) , where γ 0 , γ 1 , γ 2 are 2 × 2 matrices that generate the Clifford Algebra of R Finally, we mention that a similar analysis may be performed for the BPS vortices in the gauged massive non-linear sigma model, discussed in refs.[19,20,21,22] for example.Although such models are nonrenormalizable, they may arise as low-energy effective theories within non-Abelian gauge theory or even string theory.

Figure 2 :
Figure 2: Eigenvalues of the 2-vortex fluctuation operator plotted against the intervortex separation parameter d (left).The red boxed area in the left figure has been enlarged on the right.

Figure 5 :
Figure 5: Snapshots of the oscillating 2-vortex at d = 1.5, excited by the mode of frequency ω 1 .

Figure 6 :
Figure 6: Snapshots of the oscillating 2-vortex at d = 1.5 excited by the mode of frequency ω 2 .

Figure 8 :
Figure 8: Dependence of the 2-vortex fluctuation operator eigenvalues on the parameter c = d 2 (left).The eigenvalue crossover region in the left figure has been enlarged on the right.