Slow Schrödinger dynamics of gauged vortices

N.M. Romão^{∗}

Max-Planck-Institut für Mathematik Vivatsgasse 7, D–53111 Bonn, Germany

J.M. Speight^{†} Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, England

Abstract

Multivortex dynamics in Manton’s Schrödinger–Chern–Simons variant of the Landau– Ginzburg model of thin superconductors is studied within a moduli space approximation. It is shown that the reduced flow on M_{N}, the N vortex moduli space, is hamiltonian with respect to ω_{L^{2}}, the L^{2} Kähler form on M_{N}. A purely hamiltonian discussion of the conserved momenta associated with the euclidean symmetry of the model is given, and it is shown that the euclidean action on (M_{N}, ω_{L^{2}}) is not hamiltonian. It is argued that the N = 3 flow is integrable in the sense of Liouville. Asymptotic formulae for ω_{L^{2}} and the reduced Hamiltonian for large intervortex separation are conjectured. Using these, a qualitative analysis of internal 3-vortex dynamics is given and a spectral stability analysis of certain rotating vortex polygons is performed. Comparison is made with the dynamics of classical fluid point vortices and geostrophic vortices.

1 Introduction

The Landau–Ginzburg theory of an idealized planar superconductor consists of a complex scalar field ϕ representing the electron pair condensate, and a U(1) gauge potential A_{i} (i = 1, 2), interacting via the potential energy functional

V = ∫ (\frac{1}{2}B^{2} + \frac{1}{2} ∑_{i} D_{i}ϕ\overline{D_{i}ϕ} + \frac{μ^{2}}{8} (1 − |ϕ|^{2})^{2}) d^{2}x.  (1.1)

Here D_{i}ϕ = ∂_{i}ϕ − iA_{i}ϕ is the gauge covariant derivative, B = ∂_{1}A_{2} − ∂_{2}A_{1} is the magnetic field and μ is a coupling constant. The model admits topologically stable, spatially localized solutions (minimals of V) called vortices, the planar analogues of magnetic flux tubes in 3dimensional superconductors. These have finite V, and so have a well-defined winding number at infinity, N ∈ ℤ, which is interpreted physically as the net vortex number. The case of critical coupling, μ = 1, is special because here, given any unordered choice of N complex numbers (not necessarily distinct) {z_{1}, …, z_{N}}, there exists a minimal of V with N vortices located at z = x_{1} + ix_{2} = z_{r}, r = 1, …, N, unique up to gauge equivalence [6]. More precisely the Higgs field ϕ of the solution has zeroes precisely at the prescribed points z_{r}. Physically, one says that static critically coupled vortices exert no net forces on one another. It follows that the moduli space of static N-vortex solutions is M_{N} ≡ ℂ^{N}. Note that the vortex positions {z_{r}} are not good global coordinates on M_{N} because we must identify solutions which differ by permutation of {z_{r}}. Good global coordinates are provided by the coefficients w_{r} of the unique monic polynomial in z whose roots are {z_{r}} (counted with multiplicity), that is

∗e-mail: nromao@mpim-bonn.mpg.de †e-mail: j.m.speight@leeds.ac.uk

z^{N} + w_{1}z^{N−1} + ⋯ + w_{N} := (z − z_{1}) ⋯ (z − z_{N}).  (1.2)

To introduce dynamics to the theory, one must also define a kinetic energy functional, T, and many choices are possible. Manton has advocated the use of a Schrödinger–Chern–Simons functional linear in first time derivatives, explicitly,

T = γ ∫ (\frac{i}{2}(\overline{ϕ}D_{0}ϕ − ϕ\overline{D_{0}ϕ}) + BA_{0} + E_{2}A_{1} − E_{1}A_{2} − A_{0}) d^{2}x  (1.3)

where A_{0} is the temporal gauge field, E_{i} = ∂_{i}A_{0} − ∂_{0}A_{i} is the electric field and γ is another coupling constant [8]. The resulting dynamics (governed by Lagrangian L = T − V) is first order in time and non-dissipative, and is hoped to give a good description of vortex dynamics in thin superconductors at very low temperatures.

In order to make progress on the problem of N-vortex dynamics in this model, one could resort to numerical solution of the field equations. Manton [8] has, however, pursued a different strategy, namely that of adiabatic approximation. One assumes that at each time the field configuration lies in M_{N}, the moduli space of static solutions of the μ = 1 system, but that the field’s position in M_{N} may vary slowly with time. This slow variation is assumed to be governed by the reduced Lagrangian, that is, T − V restricted to M_{N}. The result is a finite dimensional dynamical system, in fact, a hamiltonian flow on M_{N}, as we shall see, which hopefully captures the important features of the full, infinite dimensional field dynamics. Of course if μ = 1 exactly, the restricted potential is constant, and there is no dynamics — the vortices remain static. On the other hand, if μ differs greatly from 1 then approximating multivortex profiles by critical N-vortices is ill-justified. So the approximation is valid only in the near critical regime. This strategy has proved to be extremely successful in the study of vortex dynamics in the relativistic version of the Landau–Ginzburg model [15, 16], and has been justified by rigorous error estimates [19].

The present paper gives an analysis of vortex dynamics within the adiabatic approximation, following on from Manton’s original paper [8]. Some of the results are anticipated by the study of the equivalent model on the two-sphere [14]. In section 2, for example, we prove that the reduced flow on M_{N} is hamiltonian with respect to the L^{2} Kähler form, a result which can be understood from the results on S^{2}, at least formally, once the sphere’s radius is taken to infinity. It is worthwhile giving independent proofs of the relevant results for two reasons, however. First, the planar model has far more direct physical significance, both as a model of extremely thin superconductors, and of the 3-dimensional case where translation symmetry is imposed. The spherical model is useful to study the quantum mechanics of the system, but its direct advantage is mathematical convenience (M_{N} ≡ ℂℙ^{N} is then compact, hence the methods of geometric quantization are amenable to more explicit treatment) rather than physical applicability. Second, the formulation of the model on S^{2} entails considerable technical subtleties absent in the planar model. When dealing with the planar model, therefore, many of the arguments simplify considerably.

Neither [8] nor [14] contains a quantitative discussion of classical vortex dynamics, and for good reason — neither the Kähler form nor the Hamiltonian on M_{N} are explicitly known. In the second half of this paper we will restrict attention to the regime of well-separated vortices, where explicit formulae can be inferred, and hence more progress is possible. The results turn out to be similar to the dynamics of so-called geostrophic vortices [12], of interest in meteorology, and, to a lesser extent, classical fluid point vortices [1]. We give a complete description of the internal 3-vortex dynamics and analyze the spectral stability properties of rigidly rotating regular polygons.

Lange and Schroers have studied the slow Schrödinger dynamics of ungauged Landau– Ginzburg vortices within an adiabatic approximation [7]. For them, there is no concept of critical coupling related to a self-duality structure, so no moduli space of static N-vortex solutions is available. Instead, they define M_{2} to be the unstable manifold of the coincident double vortex, which they construct via numerical gradient flow. Their emphasis is very different from the present work, therefore. They use intensive numerical techniques to study two-vortex dynamics thoroughly, concentrating particularly on the case of overlapping vortices. Indeed, one of their main motivations was to provide the first (necessarily numerical) implementation of the unstable manifold method for solitons in spatial dimension exceeding one. In contrast, we focus on the opposite physical regime (large separation), for N ≥ 3, where heavy numerical analysis is not required.

2 The reduced system

Let q^{α}, α = 1, …, 2N, be some choice of local coordinates on M_{N} (for example the real and imaginary parts of the vortex positions z_{r} ∈ ℂ). Then, since T depends only linearly on time derivatives, the Lagrangian restricted to M_{N} must take the form

L|_{M_{N}} = A_{α}(q)q̇^{α} − V (q)  (2.1)

where A_{α} are some functions of the coordinates {q^{β}} which may naturally be interpreted as the components of a U(1) connexion form A = A_{α}dq^{α} on M_{N}, since changing A by an exact form A ↦ A+dΦ changes L|_{M_{N}} by an irrelevant total time derivative [8]. The Euler–Lagrange equations resulting from L|_{M_{N}} are

B_{αβ}q̇^{α} = −\frac{∂V}{∂q^{β}} ,  (2.2)

where B = dA is the curvature form of A. Clearly, B is closed. Further, assuming that (2.2) is a well-defined flow on M_{N}, B must also be a nondegenerate bilinear form. So we may reinterpret B as a symplectic form on M_{N}. Given any function F ∈ C^{∞}(M_{N}), let X_{F} denote its symplectic gradient, that is the unique vector field satisfying

ι_{X_{F}} B = −dF  (2.3)

where ι denotes interior multiplication, ι_{X}B(Y) := B(X, Y). System (2.2) is then hamiltonian flow along X_{V}.

No explicit formula for B or A exists. However, Manton has shown how A can be related to the analytic properties of the Higgs field ϕ near its zeroes. Let us assume that the vortex positions z_{r} are all distinct, and use these as local complex coordinates on M_{N}. Then it is known that in a neighbourhood of z_{r}

log |ϕ(z)|^{2} = log |z − z_{r}|^{2} + a_{r} + \frac{1}{2}\overline{b}_{r}(z − z_{r}) + \frac{1}{2}b_{r}(\overline{z} − \overline{z}_{r}) + ⋯  (2.4)

where a_{r} and b_{r} are some unknown functions of the vortex positions z_{s}, a_{r} real, b_{r} complex. We have used here the conventions of [10]. Samols [15], adapting earlier work of Strachan [18], showed that the L^{2} metric on M_{N} could be written entirely in terms of the unknown coefficients b_{r}, namely

g_{L^{2}} = π ∑_{r,s=1}^{N} (δ_{rs} + 2\frac{∂b_{s}}{∂z_{r}} ) dz_{r}d\overline{z}_{s}.  (2.5)

This is a hermitian metric on T M_{N} which can be shown to be Kähler [15]. Using similar techniques, Manton has obtained [8] a similar formula for the connexion form A; in our notation,

A = iπγ ∑_{r=1}^{N} [(\overline{b}_{r} + \frac{1}{2}\overline{z}_{r})dz_{r} − (b_{r} + \frac{1}{2}z_{r})d\overline{z}_{r}] .  (2.6)

We will now establish the following proposition:

Proposition 1 The connexion A has curvature form B = −2γω_{L^{2}} where ω_{L^{2}} is the Kähler form corresponding to g_{L^{2}}.

Proof: We will prove the formula on M_{N}\Δ_{N}, where Δ_{N} is the measure zero set on which vortices coincide, and appeal to smoothness. Let us define the (0, 1)-form

b := ∑_{r} b_{r}d\overline{z}_{r}.  (2.7)

Hermiticity of g_{L^{2}} and (2.5) imply

\frac{∂b_{s}}{∂z_{r}} ≡ \frac{∂\overline{b}_{r}}{∂\overline{z}_{s}} ⇒ ¯∂\overline{b} = −∂b.  (2.8)

Note that the connexion A and ω_{L^{2}} may be written, using (2.7) and (2.8)

A = iπγ [\frac{1}{2} ∑_{r}(\overline{z}_{r}dz_{r} − z_{r}d\overline{z}_{r}) + \overline{b} − b] ,

ω_{L^{2}} = \frac{iπ}{2} (∑_{r} dz_{r} ∧ d\overline{z}_{r} + ∂b − ¯∂\overline{b}) .  (2.9)

It follows that

2ω_{L^{2}} + \frac{1}{γ} dA = iπ(∂b − ¯∂\overline{b} + d\overline{b} − db) = iπ(∂\overline{b} − ¯∂b).  (2.10)

Now ∂ ¯∂b = −¯∂∂b = ¯∂^{2}\overline{b} = 0 by (2.8), so ¯∂b is an antiholomorphic (0, 2)-form. Hence its component functions are antiholomorphic functions on M_{N}\Δ_{N}. But it is known [15] that the coefficients b_{r}, and their derivatives, decay exponentially fast at large vortex separation. Hence these antiholomorphic functions must in fact vanish. So ¯∂b = 0 = ∂\overline{b}, and the proposition follows. □

For future reference, we note that ¯∂b = 0 implies that

\frac{∂b_{r}}{∂\overline{z}_{s}} = \frac{∂b_{s}}{∂\overline{z}_{r}}  (2.11)

for all r, s.

So the adiabatic approximation to N-vortex dynamics is hamiltonian flow on (M_{N}, ω_{L^{2}}) with Hamiltonian

H = −\frac{1}{2γ} V |_{M_{N}} .  (2.12)

Henceforth X_{F} will denote the symplectic gradient of F with respect to ω_{L^{2}}, rather than B.

3 Conservation laws

We may now give a hamiltonian discussion of the conservation laws discovered by Manton and Nasir [9]. The natural action of the euclidean group E(2) ≅ U(1) ⋉ ℂ on the physical plane ℂ induces a E(2)-action on M_{N} by (e^{iθ}, c) : {z_{r}} ↦ {e^{iθ}z_{r} + c} (strictly speaking, this only defines the action on M_{N}\Δ_{N}, but one can use (1.2) to deduce a well defined action on the global coordinates w_{r}). This action is manifestly holomorphic. It is also isometric, since it leaves L^{2}-norms invariant. Hence the E(2)-action is symplectic. We would like to construct a moment map μ : M_{N} → 𝔢(2)^{∗} for this action and identify its components with respect to a natural basis for 𝔢(2)^{∗} as the conserved momenta of the system, just as it has been done for the symplectic SO(3)-action in the S^{2} model [14]. Unfortunately, no such moment map exists.

Certainly, given any X ∈ 𝔢(2) ≅ T_{(1,0)}E(2), the induced vector field X^{♯} ∈ Γ(T M_{N}) is hamiltonian, because all symplectic vector fields on M_{N} are hamiltonian (Y symplectic implies ι_{Y} ω_{L^{2}} is closed; it is also exact since H^{1}(M_{N}) = 0, whence Y is the symplectic gradient of some smooth function). Using the terminology of [11], the E(2)-action on M_{N} is weakly hamiltonian. So functions P_{j} : M_{N} → ℝ , j = 0, 1, 2 satisfying −dP_{j} = ι_{X_{j}}ω_{L^{2}} must exist, where

X_{0} = \frac{∂^{♯}}{∂θ} = i ∑_{r} (z_{r} \frac{∂}{∂z_{r}} − \overline{z}_{r} \frac{∂}{∂\overline{z}_{r}} ) ,  (3.1)

X_{1} = \frac{∂^{♯}}{∂c_{1}} = ∑_{r} (\frac{∂}{∂z_{r}} + \frac{∂}{∂\overline{z}_{r}} ) ,  (3.2)

X_{2} = \frac{∂^{♯}}{∂c_{2}} = i ∑_{r} (\frac{∂}{∂z_{r}} − \frac{∂}{∂\overline{z}_{r}} )  (3.3)

generate rotations and translations in M_{N}. In fact, these functions are unique up to additive constants. The problem is that the P_{j} cannot be assembled into the components of an equivariant map M_{N} → 𝔢(2)^{∗}. In other words, we have:

Proposition 2 The symplectic E(2)-action on M_{N} is not hamiltonian.

Proof: Assume, to the contrary, that a moment map μ exists. Each M_{N} contains a symplectic submanifold (ℂ, \frac{iπN}{2} dZ ∧ d\overline{Z}), namely the E(2)-orbit of any configuration of N coincident vortices (here Z denotes the common vortex position). The restriction μ̂ of μ to this orbit defines a moment map for the standard action of E(2) on ℂ. Since μ̂ is equivariant, and the SO(2) action fixes 0,

μ̂(0) = μ̂((e^{iθ}, 0) · 0) = Ad_{(e^{iθ},0)}^{∗}μ̂(0)  (3.4)

where Ad^{∗} denotes the coadjoint action of E(2) on 𝔢(2)^{∗}. Hence μ̂(0) = αdθ for some real constant α. But this completely determines μ̂ by transitivity of the translation action:

μ̂(Z) = μ̂((1, Z) · 0) = Ad_{(1,Z)}^{∗}αdθ = αdθ.  (3.5)

Thus, any equivariant map ℂ → 𝔢(2)^{∗} is constant, and cannot generate a nontrivial group action. □

Nonetheless, generating functions P_{j} for the X_{j} do exist, and one expects, given Manton and Nasir’s Lagrangian analysis [9], they may be written locally (on M_{N}\Δ_{N}) as

P_{0} = \frac{π}{2} ∑_{r}(|z_{r}|^{2} + b_{r}\overline{z}_{r} + \overline{b}_{r}z_{r})  (3.6)
P_{1} = i\frac{π}{2} ∑_{r}(z_{r} − \overline{z}_{r})  (3.7)
P_{2} = \frac{π}{2} ∑_{r}(z_{r} + \overline{z}_{r}).  (3.8)

Note that P_{1}, P_{2} are easily globalized since P_{1} + iP_{2} = −iπw_{1} in the global coordinates. A smooth globalization of P_{0} must exist, but no obvious formula suggests itself.

It is a routine exercise to verify that dP_{j}(Y) = ω_{L^{2}}(Y, X_{j}) for all Y ∈ Γ(T M_{N}) as required, if one makes use of the identities

∑_{r} b_{r} ≡ 0  (3.9)
∑_{r}(b_{r}\overline{z}_{r} − \overline{b}_{r}z_{r}) ≡ 0.  (3.10)

Identity (3.9) is proved in [15], while (3.10) is proved for the spherical model in [14]. For the sake of completeness, we will derive it directly for the planar model.

Let ϕ ∈ M_{N} be the Higgs field of the static solution with zeroes at {z_{r}}, and let ϕ^{θ} ∈ M_{N} be the field obtained from this by rotating all the vortex positions z_{r} ↦ e^{iθ}z_{r}. Then clearly ϕ^{θ}(e^{iθ}z) ≡ ϕ(z), and hence (2.4) implies that

b_{r}({e^{iθ}z_{r}}) = e^{iθ}b_{r}({z_{r}})  (3.11)

for all r. It follows that

X_{0}[b_{r}] = \frac{d}{dθ}|_{θ=0} b_{r} = ib_{r}, and X_{0}[\overline{b}_{r}] = −i\overline{b}_{r},  (3.12)

since X_{0} is real. But using our local expression for X_{0}, (3.1), we see that

X_{0}[b_{s}] = i ∑_{r} (z_{r} \frac{∂b_{s}}{∂z_{r}} − \overline{z}_{r} \frac{∂b_{s}}{∂\overline{z}_{r}} ) = i\frac{∂}{∂\overline{z}_{s}} ∑_{r}(z_{r}\overline{b}_{r} − \overline{z}_{r}b_{r}) + ib_{s}  (3.13)

where we have used (2.8) and (2.11). Comparing (3.13) and its complex conjugate with (3.12) one sees that ∑_{r}(b_{r}\overline{z}_{r}−\overline{b}_{r}z_{r}) is constant. That the constant vanishes follows from consideration of the limit where all N vortices coincide. In fact, constancy of the sum, rather than vanishing, is sufficient to prove that X_{P_{0}} = X_{0}.

Our Hamiltonian H = −\frac{1}{2γ} V |_{M_{N}} is manifestly invariant under the E(2)-action, so

{P_{j}, H} = ω_{L^{2}}(X_{j}, X_{H}) = dH(X_{j}) = X_{j}[H] = 0  (3.14)

and the momenta P_{j} are conserved by the flow of H.

The Lie subalgebra generated by the P_{j} in the Poisson algebra C^{∞}(M_{N}) is determined by the relations

{P_{j}, P_{0}} = ϵ_{jk}P_{k},  j = 1, 2  (3.15)
{P_{1}, P_{2}} = Nπ,  (3.16)

where ϵ_{jk} denotes the antisymmetric tensor with ϵ_{12} = +1. So we can see that it is a central extension 𝔢(2) ⊕ ℝ of the euclidean algebra. The nonvanishing of the Poisson bracket (3.16) is consistent with a field theory calculation of Hassaïne et al. [5]. In our context, given that [X_{1}, X_{2}] = 0, it provides another proof of Proposition 2. In fact, it implies that the E(2)-action on M_{N} corresponds to a nonzero 2-cocycle in the Lie algebra cohomology group H^{2}(𝔢(2); ℝ) ≅ ℝ; this group parametrizes obstructions of weakly hamiltonian E(2)-actions to be hamiltonian [11].

It is worthwhile to look at the content of Proposition 2 in the context of the results on the spherical model [14]. In the latter, spatial isometries are described by the group SO(3), which is simple. Therefore, H^{2}(𝔰𝔬(3); ℝ) = 0 and the SO(3)-action on M_{N} ≡ ℂℙ^{N} is necessarily hamiltonian. Dualizing the corresponding moment map, we obtain an isomorphism between 𝔰𝔬(3) and the Lie algebra of conserved angular momenta. The large radius limit R → ∞ that relates the spherical model to the planar model determines a contraction of Lie algebras

𝔰𝔬(3) → 𝔢(2)  (3.17)

with parameter \frac{1}{R}. This is the well-known contraction of 𝔰𝔬(3) to the graded Lie algebra associated to the filtration [4]

0 ⊂ 𝔰𝔬(2) ⊂ 𝔰𝔬(3),  (3.18)

where the choice of the 𝔰𝔬(2) subalgebra corresponds to fixing an axis through the centre of the sphere. We can realize this contraction as a deformation of Lie algebras of vector fields, using the isometry actions on each M_{N}. At the level of the Poisson algebras, however, no homomorphism of Lie algebras from 𝔢(2) to C^{∞}(M_{N}) is available by Proposition 2, so it is not surprising that the closure of the generators P_{j} does not yield 𝔢(2) in the planar model.

4 Well-separated vortices

Although the moduli space approximation has simplified the dynamical N-vortex problem in principle, we are still faced with the problem that neither the Kähler form ω_{L^{2}} nor the Hamiltonian H = −\frac{1}{2γ} V are explicitly known. There is one physical regime in which explicit progress is possible, namely when all vortices are well separated from one another. In this situation, an asymptotic formula for the coefficients b_{r}, and hence ω_{L^{2}} is available [10]. By translation invariance, the formulae involve only z_{rs} := z_{r} − z_{s}. They are

b_{r}(z_{1}, …, z_{N}) = \frac{q^{2}}{2π^{2}} ∑_{s≠r} K_{1}(|z_{rs}|)\frac{z_{rs}}{|z_{rs}|}  (4.1)

ω_{L^{2}} = i\frac{π}{2} [∑_{r} dz_{r} ∧ d\overline{z}_{r} − \frac{q^{2}}{4π^{2}} ∑_{r} ∑_{s≠r} K_{0}(|z_{rs}|)dz_{rs} ∧ d\overline{z}_{rs}] ,  (4.2)

where K_{n} denotes the modified Bessel’s function of the second kind, and q is an unknown real constant which may be interpreted as the scalar monopole charge or magnetic dipole moment of a single vortex at critical coupling. Based on a string theoretic duality argument, Tong has conjectured that q = −2π8^{\frac{1}{4}} , and this value is consistent with numerics [20, 17]. The formulae are believed to be correct up to linear order in exponentially small quantities (note K_{n}(ρ) is exponentially small for large ρ). They have not been proved rigorously, but they have been derived via two independent physical arguments, one using matched asymptotic expansions and the other appealing to a point source formalism, and there is a good fit to known numerics. Similarly derived formulae for monopole moduli spaces have subsequently been proved rigorously, so one has good grounds for confidence.

It is not so straightforward to obtain an asymptotic formula for H. It is known [8] that

V |_{M_{N}} = Nπ + \frac{μ^{2} − 1}{8} ∫ (1 − |ϕ|^{2})^{2}d^{2}x,  (4.3)

but no obvious method of estimating the integral for well-separated vortices suggests itself. We shall instead proceed by an indirect physical argument. For μ close to 1, one expects that V |_{M_{N}} is close to V_{N}^{μ}, the Landau–Ginzburg energy of N well-separated coupling μ vortices held at positions {z_{r}}, and for this an asymptotic formula has been found [17]:

V_{N}^{μ} = \frac{1}{2} ∑_{r} ∑_{s≠r} Ũ(|z_{rs}|)  (4.4)

where Ũ is the asymptotic 2-vortex interaction potential,

Ũ(ρ) := \frac{1}{2π} [m(μ)^{2}K_{0}(ρ) − q(μ)^{2}K_{0}(μρ)],  (4.5)

and we have discarded an irrelevant constant (N times the vortex mass). The factor of \frac{1}{2} is included in (4.4) since the sum is over all ordered distinct vortex pairs. Note that the scalar monopole charge q(μ) and magnetic dipole moment m(μ) vary with the coupling, and are not Figure 1: The quantity ∫ (1−|ϕ|^{2})^{2}d^{2}x for a vortex pair, as a function of the vortex separation ρ. Solid line and crosses: Shah’s numerical data, taken from [16]; dashed line: our conjectured asymptotic formula, as in equation (4.8).

equal except when μ = 1. Note also that, at μ = 1, Ũ ≡ 0, as one expects. Now let μ = 1+δμ, δμ small, and assume that, at sufficiently large vortex separation, V |_{M_{N}} = V_{N}^{μ} to leading order in δμ. Again discarding additive constants, one is led to conjecture that

V |_{M_{N}} = \frac{1}{2} ∑_{r} ∑_{s≠r} U(|z_{rs}|)

U(ρ) := \frac{q^{2}}{4π} (μ^{2} − 1) [ρK_{1}(ρ) − νK_{0}(ρ)] ,  (4.6)

where ν = 2(q′(1) − m′(1))/q ≈ 2.7060, using the tabulated data for m(μ) and q(μ) in [17]. One way to test this conjecture is to deduce from it an asymptotic formula for the integral

f_{Shah}(ρ) = ∫ (1 − |ϕ|^{2})^{2}d^{2}x  (4.7)

for the case of critically coupled N = 2 static vortices with intervortex separation ρ (note that E(2) symmetry implies the integral may only depend on ρ in this case). The function f_{Shah} has been evaluated numerically by Shah [16]. Formula (4.6) implies that

f_{Shah}(ρ) − f_{Shah}(∞) = \frac{2q^{2}}{π} [ρK_{1}(ρ) − νK_{0}(ρ)]  (4.8)

asymptotically at large ρ. A comparison of Shah’s numerical data with our conjectured asymptotic form is given in figure 1. The fit for ρ > 4 is very good.

The N-vortex equations of motion in the asymptotic regime are obtained by extracting the leading asymptotic term in the symplectic gradient of H. Since H is already exponentially small, an exponential correction to ω_{L^{2}} makes no contribution at this order, so we can actually set ω_{L^{2}} = πω_{0}, where

ω_{0} = \frac{i}{2} ∑_{r} dz_{r} ∧ d\overline{z}_{r}  (4.9)

is the canonical symplectic form on ℂ^{N}. We find, to leading order,

ż_{r} = i ∑_{s≠r} F (|z_{rs}|)z_{rs}, F (ρ) := −\frac{U′(ρ)}{2πγρ}.  (4.10)

Note that system (4.10) is similar to the equations of motion for a system of identical fluid point vortices, or geostrophic vortices, of vorticity 1/γ, which would result from replacing U by

U_{fluid}(ρ) = log ρ, U_{geo}(ρ) = K_{0}(ρ)  (4.11)

respectively [1, 12]. The system with U_{fluid} is particularly well studied. We shall see that our system behaves rather more like the geostrophic vortex system, which has not been so heavily studied, though there are still significant differences.

Two-vortex dynamics in the moduli space approximation is almost trivial: the vortices orbit their centre of mass at constant speed and separation. This is a special case of the rotating polygon solutions whose stability properties we will analyze in section 6. We turn now to a discussion of 3-vortex dynamics.

5 The dynamics of three vortices

Novikov has given a thorough treatment of the internal dynamics of identical vortex triples interacting via U_{fluid} in [13]. His method can be adapted readily to deal with our system in its asymptotic form, (4.10). The basic idea is to identify trajectories in the internal phase space of the system with level curves of the quantities conserved by flow (4.10). Since Novikov’s method relies on exploiting the conservation laws enjoyed by the system, it is more satisfactory to apply it directly to the momenta conserved by the full flow, in asymptotic form, rather than the momenta conserved by the asymptotic flow (4.10), and this is how we shall proceed. Of course, either approach yields the same results.

To begin with, let N be general. Let us define the centroid of an N-vortex configuration by

Z := \frac{1}{iπN} (P_{1} + iP_{2}) = \frac{1}{N} ∑_{r} z_{r}  (5.1)

and note that

∑_{r} |z_{r}|^{2} = N|Z|^{2} + \frac{1}{2N} ∑_{r} ∑_{s≠r} |z_{rs}|^{2}.  (5.2)

Since Z and P_{0} are conserved, it follows that

Q := \frac{2N}{π} P_{0} − N^{2}|Z|^{2}
= ∑_{r} (N(b_{r}\overline{z}_{r} + \overline{b}_{r}z_{r}) − \frac{1}{2} ∑_{s≠r} |z_{rs}|^{2})  (5.3)

is conserved also. The corresponding hamiltonian vector field can be readily computed as

X_{Q} = \frac{2N}{π} X_{0} − \frac{1}{π^{2}} (2P_{1}X_{1} + 2P_{2}X_{2})
= \frac{2Ni}{π} ∑_{r} ((z_{r} − Z)\frac{∂}{∂z_{r}} − (\overline{z}_{r} − \overline{Z})\frac{∂}{∂\overline{z}_{r}} ) ,  (5.4)

and this describes rigid rotations of vortex configurations about their centroids. Using (2.8), (2.11) and (3.10), one finds

{P_{j}, Q} = 0,  j = 1, 2.  (5.5)

For a moment, let us consider N = 3. Equation (5.5) implies that the set of three conserved quantities {P_{1}, Q, H} is in involution. If the corresponding hamiltonian vector fields are linearly independent at all points of an open dense subset M_{3} \ S ⊂ M_{3}, the dynamics of three vortices is Liouville integrable [3] on the six-dimensional phase space M_{3}. This manifold is then foliated by invariant tori [2], some of them possibly degenerate. Clearly, X_{1} is independent of both X_{Q} and X_{H} everywhere in M_{N}. Hence, the question of Liouville integrability reduces to the condition of linear independence of X_{Q} and X_{H}. Our intuition about the dynamics of N = 2 vortices [8] leads us to claim that this condition is generic on M_{3}. Actually, we expect X_{Q} and X_{H} to be linearly dependent exactly at the points of a subset S ⊂ M_{3} consisting of configurations with special symmetry: z_{r} either at the vertices of an equilateral triangle, at the ends and midpoint of a line segment, or on the locus of coincidence Δ_{3}; notice that all these conditions are algebraic in the coordinates z_{r} (and their complex conjugates), so that S is indeed closed with dense complement. We shall see in the following that this picture is consistent with what we can learn about the dynamics of 3-vortices in the asymptotic regime of large separation. The configurations in S that correspond to well-separated 3-vortices are special cases of the vortex polygons that we shall discuss in section 6.

When all |z_{rs}| are large, the angular momentum becomes

P_{0} = \frac{π}{2} [∑_{r} |z_{r}|^{2} + \frac{q^{2}}{2π^{2}} ∑_{r} ∑_{s≠r} |z_{rs}|K_{1}(|z_{rs}|)] ,  (5.6)

whereas

Q = \frac{1}{2} ∑_{r} ∑_{s≠r} [|z_{rs}|^{2} + \frac{Nq^{2}}{π^{2}} |z_{rs}|K_{1}(|z_{rs}|)] .  (5.7)

Note that, unlike Z and P_{0}, Q and H depend only on the relative separations of the vortices |z_{rs}|, to this order.

If we now set N = 3 and define x = (|z_{23}|, |z_{31}|, |z_{12}|), we see that x(t) is confined to a level curve of (Ĥ, Q̂) in ℝ^{3}, where

Ĥ = U(x_{1}) + U(x_{2}) + U(x_{3}),
Q̂ = g(x_{1}) + g(x_{2}) + g(x_{3}), g(x) := x^{2} + \frac{Nq^{2}}{π^{2}} xK_{1}(x).  (5.8)

Our strategy is, then, to construct such level curves. This almost completely determines the internal dynamics of the triple (meaning the dynamics up to rigid rotations), up to reparametrization of time. The only thing not determined is a discrete variable, the orientation of the triple. Two orientations are possible: traversing the triangle with vertices at z_{r} clockwise can order the vertices 123 or 132.

To construct a level curve of (Ĥ, Q̂), we may choose some initial data x(0) = x_{0} and then solve the first order system

ẋ(s) = \frac{∇Ĥ × ∇Q̂}{|∇Ĥ × ∇Q̂|}  (5.9)

to yield a level curve in arc-length parametrization (|ẋ(s)| ≡ 1). Since we work only to linear order in exponentially small quantities, and Ĥ is small, we may keep only the leading term in Q̂, that is, take

Q̂ = |x|^{2}  (5.10)

in (5.9). So level sets of Q̂ may be approximated by concentric spheres of radius Q̂^{\frac{1}{2}} in ℝ^{3}, and the level curves of (Ĥ, Q̂) are simply the intersection curves between level sets of Ĥ and these spheres. Alternatively, given a radius Q̂^{\frac{1}{2}} , we may seek level curves of HQ̂ : S^{2} → ℝ,

HQ̂(y) = U(Q̂^{\frac{1}{2}} y_{1}) + U(Q̂^{\frac{1}{2}} y_{2}) + U(Q̂^{\frac{1}{2}} y_{3}),  (5.11)

where x = Q̂^{\frac{1}{2}} y and S^{2} is the unit 2-sphere in ℝ^{3}. It is straightforward to generate contour plots of HQ̂ for given Q̂ within any convenient coordinate system on S^{2}, and hence deduce the corresponding internal trajectories. Note that Q̂ determines the root-mean-square of the triangle side lengths, x_{1}, x_{2}, x_{3}. It turns out that the level curves are not very sensitive to Q̂, provided we choose it to be fairly large (which it should be, in the asymptotic regime), so let us imagine that Q̂ has been fixed.

Not every y ∈ S^{2} represents a valid vortex triple. Each y_{i} must be non-negative, and since they represent the side lengths of a euclidean triangle, none may exceed the sum of the other two. So y(t) is confined to a triangular region in S^{2}, bounded by great-circular arcs (the intersection of the sphere with the colinearity planes y_{1} = y_{2} + y_{3}, y_{2} = y_{3} + y_{1}, y_{3} = y_{1} + y_{2}), and centred on y_{0} = (1, 1, 1)/3^{\frac{1}{2}} . This central point represents an equilateral triple. The orthogonal projection of this region onto the tangent plane T_{y_{0}}S^{2} is depicted in figure 2. The boundaries project to congruent elliptic arcs. Any point on the boundary represents a colinear vortex triple, and the boundaries intersect at coincidence points (where a vortex pair coalesces). Since our analysis is restricted to the well-separated regime, level curves which approach these corners closely are of suspect validity.

Figure 2 depicts level curves for the choice Q̂ = 441 and several values of Ĥ. Our discussion is qualitatively similar to Novikov’s [13], so will be kept brief. The minimum possible value of HQ̂ is

HQ̂ ; min = HQ̂(y_{0}) = 3 U(\sqrt{Q̂/3})  (5.12)

attained by the equilateral triple. The level “curve” for this value consists of the single point y_{0}, so the internal configuration of this triple remains constant. In fact the triangle rotates about its centre at constant speed, as will be discussed in section 6. Increasing HQ̂, we obtain closed curves centred on y_{0}. In this low energy regime, the shape of the triangle varies periodically in time as the triangle spins, but the triangle’s orientation remains constant. The Figure 2: Level curves of ĤQ̂ for Q̂ = 441. Here H_{1} = ĤQ̂ ; min and H_{2} = ĤQ̂ ; crit. The dashed curves represent the colinearity boundaries, which intersect at coincidence points.

periods of shape change and rotation are generically incommensurate. Clearly, these solutions are all stable.

Increasing HQ̂ further, the level curve eventually becomes tangent to the colinearity boundaries, at the critical value

HQ̂ ; crit = HQ̂((1, 1, 2)/6^{\frac{1}{2}} ) = 2 U(\sqrt{Q̂/6}) + U(2\sqrt{Q̂/6}).  (5.13)

This level curve should not be interpreted as a closed orbit. Rather, we have three fixed points, (1, 1, 2)/\sqrt{6}, (1, 2, 1)/\sqrt{6}, (2, 1, 1)/\sqrt{6}, joined by 3 heteroclinic orbits. The fixed points represent colinear triples with the middle vortex equidistant from the outer vortices, a special case of the filled rotating polygons which will be discussed in section 6. This is stationary (but not static) and clearly unstable: small perturbations lead to divergent dynamical regimes.

For HQ̂ > HQ̂ ; crit, the level curve splits into 3 disjoint pieces. The actual trajectory stays on one segment of this curve – it reverses direction each time it hits a colinearity boundary, with a corresponding change in the triangle’s orientation. At these points, the triangle collapses and turns itself inside out, so for larger energies, the triple’s internal motion consists of a periodic “flip-flopping”. As the energy grows unbounded, the trajectory approaches a coincidence point, and hence escapes the presumed region of validity of our approximation.

It is interesting to note that every trajectory intersects the isosceles lines (the straight lines containing y_{0} and the coincidence points), so every 3-vortex motion passes, at least once, through an isosceles configuration.

6 Rigidly rotating polygons

If N vortices are placed at the vertices of a regular N-gon, they will rotate at constant speed about their centroid, which remains fixed [8]. This behaviour is familiar from both the fluid and geostrophic vortex systems. Given the asymptotic equations of motion (4.10), it is straightforward to derive frequency-radius relations and spectral stability properties of these solutions, valid for large polygon side length. Note that a change in the coupling constants μ, γ may always be absorbed into a rescaling and/or reversal of time, so the stability properties will be independent of our choice of parameters.

It is convenient to transform to a co-rotating frame by defining new coordinates y_{r}(t) ∈ ℂ,

z_{r}(t) =: y_{r}(t)e^{iΩt}  (6.1)

for some fixed choice of frequency Ω ∈ ℝ, so that frequency Ω solutions are static in the new frame. System (4.10) in these coordinates is

ẏ_{r} = i ∑_{s≠r} F (|y_{rs}|)y_{rs} − iΩy_{r},  (6.2)

where y_{rs} := y_{r} − y_{s}. Note that (6.2) is hamiltonian flow on (ℂ^{N}, ω_{0}) with respect to the modified Hamiltonian

H_{Ω} = ∑_{r} ∑_{s≠r} U(|y_{rs}|) − Ω ∑_{r} |y_{r}|^{2}.  (6.3)

Let λ := e^{2πi/N}. Then y_{r} = λ^{r}σ, σ ∈ (0, ∞), is a static solution of (6.2) provided

Ω = ∑_{j=1}^{N−1} F (|1 − λ^{j}|σ).  (6.4)

This is the frequency-radius relation for a rotating N-gon.

We turn now to stability properties. Let y ∈ ℂ^{N} be a fixed point of the co-rotating flow. Then we may define the linearization of the flow about y,

Λ_{y} : T_{y}ℂ^{N} → T_{y}ℂ^{N},  Λ_{y} : X ↦ ∇_{X}X_{H_{Ω}},  (6.5)

where ∇ is any affine connexion on ℂ^{N}, and X_{H_{Ω}} is the symplectic gradient of H_{Ω}. The fixed point y is spectrally stable if Λ_{y} has no eigenvalues with positive real part. Now, since X_{H_{Ω}} is symplectic, it follows that Λ_{y} ∈ 𝔞𝔲𝔱(T_{y}ℂ^{N}, ω_{0}), that is,

ω_{0}(Λ_{y}X, Y) + ω_{0}(X, Λ_{y}Y) ≡ 0  (6.6)

for all X, Y ∈ T_{y}ℂ^{N}. It follows that, if η is an eigenvalue of Λ_{y}, so are −η, \overline{η} and −\overline{η}. In order that y be spectrally stable, therefore, spec Λ_{y} must be purely imaginary.

In the case of the rotating N-gon, y = ỹ = (λσ, λ^{2}σ, …, σ), one can find spec Λỹ explicitly due to the cyclic symmetry of the configuration [12]. Let π : ℤ → ℤ_{N} be the group homomorphism with π(1) = 1. Then a N × N real matrix A is right circulant if there exists Â ∈ ℝ^{N} such that

A_{ij} = Â_{π(j−i+1)}  (6.7)

for all i, j. In order to exploit this symmetry, it is convenient to define a twisted complex basis e_{(i)} ∈ ℂ^{N}, i = 1, …, N, such that

e_{(i)j} = λ^{i}δ_{ij}  (6.8)

there being no summation implied. If we write down the matrix representing Λỹ with respect to the associated basis for ℝ^{N}, that is, {Re e_{(1)}, …, Re e_{(n)}, Im e_{(1)}, …, Im e_{(n)}}, we find that

Λỹ = [ \begin{matrix}A & B \\ C & A \end{matrix} ]  (6.9)

where each of the real N × N blocks is right circulant. The explicit formulae for Â, B̂, Ĉ ∈ ℝ^{N} are rather complicated, and are presented in the appendix. Of more interest are the following symmetries possessed by these coefficients:

\begin{matrix}Â_{1} = 0, Â_{N+2−s} & = & −Â_{s} \\ B̂_{N+2−s} & = & B̂_{s} \\ Ĉ_{N+2−s} & = & Ĉ_{s} \end{matrix} \} s = 2, …, N.  (6.10)

Right circulancy is a powerful symmetry because the eigenvectors of a right circulant matrix are independent of its entries. For each j ∈ {1, …, N}, let

x_{j} = (1, λ^{j}, λ^{2j}, …, λ^{(N−1)j}) ∈ ℂ^{N}.  (6.11)

Then x_{j} is simultaneously an eigenvector of A, B and C, with eigenvalues

α_{j} = Â · x_{j}, β_{j} = B̂ · x_{j}, γ_{j} = Ĉ · x_{j}.  (6.12)

From (6.10), (6.11) and (6.12), one sees that α_{j} is purely imaginary, while β_{j} and γ_{j} are real. Note that \overline{x}_{j} = x_{N−j} for all 1 ≤ j ≤ N − 1, and that \overline{x}_{N} = x_{N}. It follows that

α_{N−j} = \overline{α}_{j} = −α_{j}, β_{N−j} = \overline{β}_{j} = β_{j}, γ_{N−j} = \overline{γ}_{j} = γ_{j},  (6.13)

for 1 ≤ j ≤ N − 1, and that α_{N} = \overline{α}_{N} = −α_{N}, so α_{N} = 0. Short calculations using the detailed forms of B̂, Ĉ (see appendix) establish that β_{N} = 0 and γ_{1} = −β_{1}.

Now, relative to the basis {x_{1}^{+}, x_{1}^{−}, …, x_{N}^{+}, x_{N}^{−}}, where x_{j}^{±} := (x_{j}, ±x_{j}) ∈ ℂ^{2N}, Λỹ is block diagonal, with 2 × 2 blocks

Λ_{j} = [ \begin{matrix}α_{j} + \frac{1}{2}(β_{j} + γ_{j}) & \frac{1}{2}(β_{j} + γ_{j}) \\ −\frac{1}{2}(β_{j} − γ_{j}) & α_{j} − \frac{1}{2}(β_{j} + γ_{j}) \end{matrix} ] .  (6.14)

It follows that the eigenvalues of Λỹ are

η_{j}^{±} = α_{j} ± \sqrt{β_{j}γ_{j}}.  (6.15)

Owing to the symmetry properties of α_{j}, β_{j}, γ_{j}, (6.13), we see that

η_{N−j}^{±} ≡ −η_{j}^{±},  (6.16)

N  2 3 4 5 6 7 ≥ 8
gauged 1 1 1 1 0 0 0
fluid  1 1 1 1 1 1 0
geostrophic 1 1 1 1 0 0 0

Table 1: Comparison of spectral stability properties for rotating vortex N-gons of various types, 0 =unstable, 1 =stable. In the gauged and geostrophic cases, the entries refer to stability at large polygon radius σ. There may be windows of anomalous (in)stability at small σ, but these lie outside the regime of validity of the calculations. Entries for geostrophic and fluid vortices are taken from [12].

so that eigenvalues generically come in quartets. Note also that η_{N}^{±} = 0 (since α_{N} = β_{N} = 0) and that η_{1}^{+} = −η_{N−1}^{−} = iΩ. These eigenvalues originate from the E(2) symmetry enjoyed by system (6.2), as will be shown shortly.

Since α_{j} ∈ iℝ for all j, ỹ is spectrally stable provided that β_{j}γ_{j} ≤ 0 for all j. But β_{1} ≡ −γ_{1}, and β_{N−j}γ_{N−j} ≡ β_{j}γ_{j} for 2 ≤ j ≤ N − 1, so there are in fact only k − 1 stability criteria, where N = 2k or N = 2k + 1, namely

β_{j}γ_{j} ≤ 0,  j = 2, …, k.  (6.17)

For N = 2, 3 stability is automatic, for N = 4, 5 there is one criterion, for N = 6, 7 two criteria, and so on.

In the first nontrivial case, N = 4, λ = i and the single stability criterion is

(β_{2}γ_{2})(σ) = −8\sqrt{2}σ^{2}F′(\sqrt{2}σ)F′(2σ) ≤ 0.  (6.18)

Since F has no critical points in the well-separated (large σ) regime, the rotating square is spectrally stable for all σ sufficiently large. As N increases, the expressions for β_{j}γ_{j} become increasingly complicated so that it is not feasible to check their sign by hand. It is straightforward to check the criteria graphically by plotting the sign of β_{j}γ_{j} against σ, however. The results are summarized and compared with the previously studied cases of fluid and geostrophic vortices in table 1.

We may give a similar analysis of the case of N − 1 vortices on the vertices of a regular polygon orbiting another vortex at the polygon’s centre. Let λ = e^{2πi/(N−1)}. Then ŷ = (0, λσ, λ^{2}σ, …, σ) ∈ ℂ^{N} is a fixed point of the flow (6.2), provided that

Ω = F (σ) + ∑_{j=1}^{N−2} F (|1 − λ^{j}|σ)(1 − λ^{j}).  (6.19)

Once again the linearization Λŷ ∈ 𝔞𝔲𝔱(Tŷℂ^{N}, ω_{0}), so ŷ is spectrally stable if and only if spec Λŷ ⊂ iℝ. The presence of the centre vortex destroys the right circulant symmetry of Λ, however, so we must resort to a numerical algorithm to generate spec Λŷ. By plotting max{|Re η| : η ∈ spec Λŷ} against σ, we can determine which of the (N − 1)-gons are spectrally stable in the well separated regime. The results, which were generated using Matlab’s eigenvalue finder, are summarized in table 2.

N − 1  2 3 4 5 6 7 8 9 ≥ 10
gauged 0 1 1 1 1 1 0 0 0
fluid  0 1 1 1 1 1 1 1 0
geostrophic 0 0 1 1 1 1 0 0 0

Table 2: Comparison of spectral stability properties for vortex (N − 1)-gons rotating about a single central vortex for various vortex types, key and comments as for table 1.

These results have been derived in the approximation of large σ. In the full adiabatic approximation, they will receive corrections due to the subleading terms in ω_{L^{2}} and H, and we are assuming that these corrections will be very small when σ is large. What reason do we have to believe that the seemingly delicate property of spectral stability, spec Λ ⊂ iℝ, will not be destroyed by these perturbations? The spectra found exactly (for Λỹ), or numerically (for Λŷ), had no accidental degeneracies for σ sufficiently large, in the stable cases. All the eigenvalues were simple, with the exception of η = 0, which we will shortly return to. No simple imaginary eigenvalue can be perturbed off the imaginary axis because, due to the reflexion symmetries enjoyed by spec Λ, it would have to split in two, which is impossible by conservation of multiplicity. Further, the 0 eigenvalue, which turns out to have multiplicity 2, is fixed at 0 by symmetry considerations, as we shall now see.

We are free to transform to the co-rotating frame in the full adiabatic flow, by redefining our Hamiltonian

H ↦ H_{Ω} = H − ΩP_{0}.  (6.20)

Then ỹ and ŷ will still be fixed points of X_{H_{Ω}}, provided σ is chosen suitably. In each case (y = ỹ or ŷ) we may study the spectrum of the linearization Λ_{y} : X ↦ ∇_{X}X_{H_{Ω}} ∈ 𝔞𝔲𝔱(T_{y}M_{N}, ω_{L^{2}}), which will be a slightly perturbed version of the asymptotic spectrum already considered. Recall that X_{0}, X_{1}, X_{2} denote the vector fields generating rotations and translations (3.1)– (3.3). Now for any fixed point y ∈ M_{N},

Λ_{y}X_{j} = ∇_{X_{j}}X_{H_{Ω}} = [X_{j}, X_{H_{Ω}}] = −Ω[X_{j}, X_{0}]  (6.21)

since [X_{j}, X_{H}] = X_{{P_{j},H}} = 0 by the E(2) symmetry. Hence

Λ_{y}X_{0} = 0, Λ_{y}X_{1} = −ΩX_{2}, Λ_{y}X_{2} = ΩX_{1},  (6.22)

whence it follows that {0, iΩ, −iΩ} ⊂ spec Λ_{y}. Hence, the double 0 eigenvalue is fixed under any perturbation maintaining E(2) symmetry, so we conclude that the simple stability analysis given above is structurally stable at large σ.

7 Conclusion

In this paper we have confirmed the expectation raised by [14] that the adiabatic approximation to N-vortex dynamics in Manton’s model of a planar superconductor amounts to a natural hamiltonian flow on (M_{N}, ω_{L^{2}}). A hamiltonian account of the conserved momenta descending from the symplectic but nonhamiltonian action of E(2) on M_{N} has been given, and based on it we argued that the dynamics of three vortices is integrable in the sense of Liouville. We have derived an asymptotic formula for the 2-vortex interaction potential close to critical coupling, namely

U(ρ) = \frac{q^{2}}{4π} (μ^{2} − 1)[ρK_{1}(ρ) − νK_{0}(ρ)],  q = −2π8^{\frac{1}{4}} , ν ≈ 2.7060,  (7.1)

and used this to analyze internal 3-vortex dynamics at large separation. We have studied the spectral stability properties of rigidly rotating vortex polygons, both with and without a central vortex, and compared our results with the more familiar cases of fluid and geostrophic vortices.

Now that we have some quantitative dynamical predictions from the adiabatic approximation, it would be interesting to test its validity numerically. There are really two separate issues. First, does the hamiltonian flow give a good account of low-energy vortex dynamics near critical coupling? Second, do our asymptotic formulae for U and ω_{L^{2}} give a good approximation to this flow at large separation? Both questions may be addressed by numerical analysis of 2-vortex dynamics. According to the adiabatic approximation, two vortices orbit one another at constant radius and angular frequency indefinitely. Is this behaviour seen in numerical simulations of the full field theory? One would expect the orbiting pair to gently radiate energy and hence slowly drift either together or apart (depending on the choice of μ, γ), but hopefully this happens on a much longer time scale than the period of their orbit. If their separation ρ is large, then their angular frequency should be close to

Ω(ρ) = F (ρ) = \frac{q^{2}}{8πγ} (μ^{2} − 1) (1 + \frac{ν}{ρ}) K_{0}(ρ).  (7.2)

This is very different from the inverse square law found for ungauged Landau–Ginzburg vortices [7]. It would be interesting to test this formula numerically in terms of both its ρ and μ dependence.

Acknowledgements

The authors wish to thank Nick Manton for useful discussions and correspondence.

Appendix

The simplest way to construct the matrix representing Λỹ with respect to the twisted basis e_{(i)} defined in equation (6.8) is to incorporate the twisting into the co-rotating coordinate system, by defining x_{r}(t) such that y_{r}(t) = λ^{r}x_{r}(t). The equations of motion (6.2) become

ẋ_{r} = i ∑_{s≠r} F (|x_{r} − λ^{s−r}x_{s}|)(x_{r} − λ^{s−r}x_{s}) − iΩx_{r}.  (A1)

Then x̃ = (σ, σ, …, σ) is a fixed point of (A1) provided (6.4) holds. The linearization of (A1) about x̃ is

δẋ_{r} = iR(σ)δx_{t} − iS(σ)δ\overline{x}_{r} − i ∑_{s≠r} λ^{s−r}Q(|1 − λ^{s−r}|σ)δx_{s} + i ∑_{s≠r} P (|1 − λ^{s−r}|σ)δ\overline{x}_{s}, (A2)

where

P (σ) := \frac{1}{2}σF′(σ),
Q(σ) := F (σ) + P (σ),
R(σ) := ∑_{j=1}^{N−1}[λ^{j}F (|1 − λ^{j}|σ) + P (|1 − λ^{j}|σ)],

S(σ) := ∑_{j=1}^{N−1} λ^{j}P (|1 − λ^{j}|σ).  (A3)

Note that P, Q, R, S are all real. Decomposing (A2) into real and imaginary parts, with δx_{r} = δx_{r}^{1} + iδx_{r}^{2}, one finds that

[ δẋ_{1}^{1} ; ⋮ ; δẋ_{N}^{1} ; δẋ_{1}^{2} ; ⋮ ; δẋ_{N}^{2} ] = [ \begin{matrix}A & B \\ C & A \end{matrix} ] [ δx_{1}^{1} ; ⋮ ; δx_{N}^{1} ; δx_{1}^{2} ; ⋮ ; δx_{N}^{2} ]  (A4)

where A, B, C are the N × N right circulant matrices generated by Â, B̂, Ĉ ∈ ℝ^{N}:

\begin{matrix}Â_{1} = 0 & Â_{s} = Im (λ^{s−1})Q(|1 − λ^{s−1}|σ) & s = 2, …, N \\ B̂_{1} = −R(σ) − S(σ) & B̂_{s} = Re (λ^{s−1})Q(|1 − λ^{s−1}|σ) + P (|1 − λ^{s−1}|σ) & s = 2, …, N \\ Ĉ_{1} = R(σ) − S(σ) & Ĉ_{s} = −Re (λ^{s−1})Q(|1 − λ^{s−1}|σ) + P (|1 − λ^{s−1}|σ) & s = 2, …, N \end{matrix}

(A5)

The matrix in equation (A4) is Λỹ in this basis. The symmetries of the components of Â, B̂, Ĉ claimed in (6.10) follow immediately, once we note that λ^{N} = 1 and λ^{−1} = \overline{λ} so that λ^{(N+2−s)−1} = \overline{λ}^{s−1} when 2 ≤ s ≤ N. Note that

β_{N} = ∑_{s=1}^{N} B̂_{s} = −R(σ) − S(σ) + ∑_{s=2}^{N} [Re (λ^{s−1})Q(|1 − λ^{s−1}|σ) + P (|1 − λ^{s−1}|σ)]

= ∑_{j=1}^{N−1} [(Re (λ^{j}) − λ^{j})(F (|1 − λ^{j}|σ) + P (|1 − λ^{j}|σ)] = 0,  (A6)

as previously claimed. Note also that

β_{1} + γ_{1} = B̂_{1} + Ĉ_{1} + ∑_{j=1}^{N−1}(B̂_{j+1} + Ĉ_{j+1})λ^{j}

= −2S(σ) + 2 ∑_{j=1}^{N−1} λ^{j}P (|1 − λ^{j}|σ) = 0  (A7)

so that γ_{1} = −β_{1}, as was claimed.

References

[1] H. Aref, “Point vortex motions with a center of symmetry” Phys. Fluids 25 (1982) 2183– 2187.

[2] V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edition, New York, Springer-Verlag, 1989.

[3] M. Audin, Spinning Tops, Cambridge, Cambridge University Press, 1996.

[4] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge, Cambridge University Press, 1984.

[5] M. Hassaïne, P.A. Horváthy and J.-C. Yera, “Non-relativistic Maxwell–Chern–Simons vortices” Ann. Phys. 263 (1998) 276–294, hep-th/9706188.

[6] A. Jaffe and C. Taubes, Vortices and Monopoles, Boston, Birkhäuser, 1980.

[7] O. Lange and B.J. Schroers, “Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices” Nonlinearity 15 (2002) 1471–1488, nlin.PS/0201047.

[8] N.S. Manton, “First-order vortex dynamics” Ann. Phys. 256 (1997) 114–131,

hep-th/9701027.

[9] N.S. Manton and S.M. Nasir, “Conservation laws in a first-order dynamical system of vortices” Nonlinearity 12 (1998) 851–865, hep-th/9809071.

[10] N.S. Manton and J.M. Speight, “Asymptotic interactions of critically coupled vortices” Commun. Math. Phys. 236 (2003) 535–555, hep-th/0205307.

[11] D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford, Clarendon Press, 1998.

[12] G.K. Morikawa and E.V. Swenson, “Interacting motion of rectilinear geostrophic vortices” Phys. Fluids 14 (1971) 1058–1073.

[13] E.A. Novikov, “Dynamics and statistics of a system of vortices” Sov. Phys. JETP 41 (1975) 937–943.

[14] N.M. Romão, “Quantum Chern–Simons vortices on a sphere” J. Math. Phys. 42 (2001)

3445–3469, hep-th/0010277.

[15] T.M. Samols, “Vortex scattering” Commun. Math. Phys. 145 (1992) 149–179,

[16] P.A. Shah, “Vortex scattering at near-critical coupling” Nucl. Phys. B294 (1994) 259–

276, hep-th/9402075.

[17] J.M. Speight, “Static intervortex forces” Phys. Rev. D55 (1997) 3830–3835,

hep-th/9603155.

[18] I.A.B. Strachan, “Low-velocity scattering of vortices in a modified abelian Higgs model” J. Math. Phys. 33 (1992) 102–110.

[19] D. Stuart, “Dynamics of abelian Higgs vortices in the near Bogomolny regime” Commun. Math. Phys. 159 (1994) 51–91.

[20] D. Tong, “NS5-branes, T-duality and worldsheet instantons” JHEP 07 (2002) 013–036,

    hep-th/0204186.