# Magnetic zero-modes, vortices and Cartan geometry

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## Abstract

We exhibit a close relation between vortex configurations on the 2-sphere and magnetic zero-modes of the Dirac operator on \(\mathbb {R}^3\) which obey an additional nonlinear equation. We show that both are best understood in terms of the geometry induced on the 3-sphere via pull-back of the round geometry with bundle maps of the Hopf fibration. We use this viewpoint to deduce a manifestly smooth formula for square-integrable magnetic zero-modes in terms of two homogeneous polynomials in two complex variables.

## Keywords

Magnetic zero-modes Dirac operator Vortex equations Cartan geometry## Mathematics Subject Classification

53Z05 58Z05## 1 Introduction

The goal of this paper is to explain and exploit a link between magnetic zero-modes of the Dirac operator on Euclidean 3-space and vortices on the 2-sphere via a particular family of geometries on the 3-sphere.

The 3-sphere geometries are obtained from the standard round geometry via pull-back with a family of maps \(S^3\rightarrow S^3\) which are bundle maps of the Hopf fibration and cover holomorphic maps \(S^2\rightarrow S^2\). The bundle maps are given in terms of two complex polynomials, and one consequence of our analysis is a manifestly smooth and square-integrable expression both for the magnetic zero-modes and the vortex configurations in terms of these polynomials. Another is an interpretation of vortices on \(S^2\) in terms of Cartan geometry. In the remainder of this introduction we sketch the context for our results.

The problem of determining magnetic zero-modes of the Dirac operator in Euclidean 3-space was first posed and addressed in an influential paper by Loss and Yau [1] in 1986. Motivated by questions about the stability of atoms, the authors were interested in finding spinors \(\varPsi \) and magnetic gauge potentials *A* on \(\mathbb {R}^3\) such that \(\varPsi \) is a zero-mode of the (static) Dirac operator minimally coupled to *A*, and both the associated magnetic field and the spinor are square-integrable. In this paper, we call pairs of spinors \(\varPsi \) and magnetic gauge potentials *A* satisfying this condition magnetic zero-modes.

Loss and Yau gave explicit expressions for one family of magnetic zero-modes, which we call linear in the following, and derived a formula which determines a gauge field for a given spinor field such that the pair form a magnetic zero-mode. This formula is singular where the spinor field vanishes, but it was, nonetheless, used by Adam, Muratori and Nash (AMN) in a series of papers [2, 3, 4] to obtain magnetic zero-modes which satisfy an additional nonlinear equation, and which we call vortex zero-modes in this paper. AMN observed that their solutions can be expressed in terms of solutions of the Liouville equation on \(S^2\) and addressed the singularities in the resulting formulae. In [2], they also pointed out that the coupled Dirac and nonlinear equation can be obtained as the dimensional reduction in a perturbed Seiberg–Witten equation on \(\mathbb {R}^4\) with a crucial sign flip (the resulting equation is often called the Freund equation).

In 2000, Erdös and Solovej pointed out that the geometry underlying the existence of magnetic zero-modes is the conformal equivalence of \(\mathbb {R}^3\) and \(S^3{\setminus } \{ \text {point}\}\) and the Hopf fibration of \(S^3\) over \(S^2\) [5]. This was used in [6, 7] to show that the linear magnetic zero-modes found by Loss and Yau can be obtained directly by pulling eigenmodes (of any energy) of the Dirac operator on \(S^3\) back to \(\mathbb {R}^3\).

One motivation for this paper was to find a similarly geometrical but also explicit understanding of the vortex zero-modes, i.e. to use the geometrical insight of Erdös and Solovej for a better understanding and improvement in the formulae derived by AMN. A second motivation was to explore links to a vortex equation for a scalar Higgs field and an abelian gauge field on \(S^2\), recently proposed by Popov. The existence of such links is suggested by the appearance of the same data in Popov vortex solutions and the AMN expressions for magnetic zero-modes; it is the reason why we call the latter vortex zero-modes.

The Popov vortex equations were obtained in [8] as the reduction by *SU*(1, 1) symmetry of the self-duality equations for *SU*(1, 1) Yang–Mills theory on the product of the 2-sphere with hyperbolic 2-space. In [9], Manton pointed out that the Popov vortex equations can be solved in terms of rational maps \(S^2\rightarrow S^2\). His solution turns out to be a particularly simple illustration of an interesting subsequent observation by Baptista [10] that Bogmol’nyi vortex equations on a Kähler surface can be interpreted as degenerate Hermitian metrics.

In the terminology of Baptista’s paper, Manton showed that Popov vortices encode the geometry of the pull-back of the round metric on \(S^2\) with a rational map. If the rational map has degree *n*, the metric necessarily has conical singularities at the \(2n-2\) ramification points, which are also the zeros of the vortex Higgs field.

Here, we lift this picture from \(S^2\) to \(S^3\). This is geometrically natural for Popov vortices, since they live on a *U*(1) bundle over \(S^2\) whose total space is the Lens space \(S^3/\mathbb {Z}_{2n-2}\). Manton’s rational map characterising the vortex lifts to a bundle map \(S^3\rightarrow S^3\), and the pull-back of the round metric on \(S^3\) with this bundle map defines a metric which, generalising Baptista’s viewpoint, encodes a vortex configuration on \(S^3\). We then show that such a vortex configuration defines a magnetic zero-mode of the Dirac operator on \(S^3\). Using conformal equivalence we obtain the advertised smooth and manifestly square-integrable expression for vortex zero-modes on \(\mathbb {R}^3\) and, at the same time, establish the expected link to Popov vortices.

The fact that \(S^3\) is the Lie group *SU*(2) allows one to encode the round geometry of \(S^3\) in the Maurer–Cartan form \(h^{-1}\hbox {d}h\). In Cartan geometry, the same form also encodes the geometry of the round geometry of \(S^2\) by combining the orthonormal frame field with the spin connection 1-form of \(S^2\). Since all the geometries we discuss in this paper are pulled back from the round geometries of \(S^2\) and \(S^3\), it is not surprising that many of our results can be stated succinctly in terms of the pull-back of the Maurer–Cartan form via the bundle map \(S^3\rightarrow S^3\). In fact, the flatness condition of the *su*(2) gauge potentials defined by these pull-backs turns out to be equivalent to our vortex equations on \(S^3\) and to the Popov vortex equations on \(S^2\). This adds a further, non-abelian interpretation of the vortex zero-modes. It also provides an intriguing link with the self-duality equations for *SU*(1, 1) gauge fields from which the Popov equations arose.

In this paper we are interested in the geometry linking magnetic zero-modes and vortices, but also in manifestly smooth expressions for both. The paper contains a number of explicit calculations and formulae, and we therefore need to lay out conventions and coordinates in some detail at the beginning. To help the reader keep sight of the bigger picture, we have also produced a summary diagram of the geometries and the maps between them in Fig. 3. Although the picture is part of our final summary section, the reader may find it helpful to refer to it now or while reading the paper.

The paper is organised as follows. In Sect. 2 we collect our conventions for parameterising *SU*(2) both as a Lie group and a round 3-sphere, give the stereographic and gnomonic projection from \(S^3\) and \(\mathbb {R}^3\) in these coordinates, and explain how both enter a simple formula for relating orthonormal frames on \(S^3\) and \(\mathbb {R}^3\) and for mapping zero-modes of magnetic Dirac operators on \(S^3\) to zero-modes of magnetic Dirac operators on \(\mathbb {R}^3\). While the conformal covariance of the kernel of the Dirac operator is, of course, well known, we are not aware of a treatment which emphasises the role of the gnomonic projection in the way we do. We illustrate our discussion by constructing the linear magnetic zero-modes on \(\mathbb {R}^3\) from general eigenmodes on \(S^3\) in our conventions.

Section 3 contains our definition of vortex configurations on \(S^3\) and some of our main results: the equivalence between vortex configurations on \(S^3\) and flat *su*(2) gauge potentials, an expression for both in terms of bundle maps \(S^3\rightarrow S^3\), and the construction of magnetic zero-modes on both \(S^3\) and \(\mathbb {R}^3\) from vortex configurations. The allowed bundle maps can be expressed in terms of two polynomials, thus leading to the promised formulae for magnetic zero-modes. The section ends with a brief discussion of how linear and vortex zero-modes can be combined to form new magnetic zero-modes.

In Sect. 4 we review the definition of Popov vortices and show that our vortex configurations on \(S^3\) are equivariant descriptions of them. We explain the relation between our bundle map \(S^3\rightarrow S^3\) and the rational map \(S^2\rightarrow S^2\) used by Manton for solving the Popov equations and interpret the pull-back to \(S^2\) of the flat *su*(2) gauge potential on \(S^3\) in the language of Cartan geometry.

Finally, Sect. 5 contains a summary in the form of a diagram in Fig. 3 and an outlook onto open questions.

## 2 Magnetic Dirac operators on \(S^3\) and on \(\mathbb {R}^3\)

### 2.1 Conventions for *SU*(2) and the Hopf map

*su*(2) generators

*SU*(2) matrix

*h*in terms of a pair of complex numbers \((z_1,z_2)\) via

*H*for the hyperplane bundle over \(S^2\), one has the following commutative diagram [11, 12]:

### 2.2 Stereographic projection and frames

*H*rather than \(\mathrm {St}^{-1}\), even though the two maps strictly speaking take values on 3-spheres of different radii.

*G*and

*H*, which is illustrated in Fig. 1. For fixed \(\mathbf {x}\), \(G(\mathbf {x})\) and \(H(\mathbf {x})\) are rotations about the same axis, and it follows from elementary geometry that

*G*rotates by twice the rotation angle of

*H*. As an aside we note that describing spherical geometry in terms of \(\mathbb {R}^3\) via pull-back with

*H*and

*G*is analogous to describing hyperbolic geometry in terms of, respectively, the Poincaré and the Beltrami–Klein models.

*G*and

*H*can be used to pull back the left-invariant 1-forms \(\sigma _j\), \(j=1,2,3\), on the 3-sphere. The relation of the resulting frames on \(\mathbb {R}^3\) to each other and to the standard frame (2.22) is interesting, and important for the remainder of this paper. We therefore collect the relevant results here. Defining the scale function

*H*gives a frame which is related to the standard frame (2.22) by rotation with

*G*(acting in its adjoint representation), a reflection in the origin and rescaling by \(\varOmega \). For later use we note

## Lemma 2.1

*SU*(2) are related via

## Proof

### 2.3 Magnetic Dirac operators and their zero-modes

As we saw in the previous section, the frame (2.19) pulled back to \(\mathbb {R}^3\) via *H* and the flat frame (2.22) are related by a rotation with *G*, a reflection and rescaling by (2.29). This implies a simple relation between the zero-modes of the Dirac operators on \(S^3\) and \(\mathbb {R}^3\).

## Lemma 2.2

*U*(1) gauge field

*A*, then

## Proof

*A*in the frame (2.22) is

This lemma can be used to construct magnetic zero-modes on \(\mathbb {R}^3\) from magnetic zero-modes on \(S^3\). For the family constructed explicitly by Loss and Yau in [1], which we call linear in this paper, this was observed in [6] and elaborated in [7], where this family was obtained from eigenmodes of the Dirac operator on \(S^3\). The corresponding argument for the family of vortex zero-modes is one of the main results of our Sect. 3.2.

*k*runs over the values so that the factorials are well defined. These functions are orthonormal and satisfy

## 3 Vortex equations and magnetic zero-modes

### 3.1 Vortex equations on \(S^3\)

We are now ready to introduce the 3-dimensional geometries which will lead us to the smooth vortex zero-modes promised in the Introduction and provide the link with vortices on the 2-sphere. First, we define vortex configurations on the 3-sphere.

## Definition 3.1

*n*be a positive integer,

*A*be a 1-form on \(S^3\), and \(\varPhi : S^3\rightarrow \mathbb {C}\) be a complex-valued function. We say that the pair \((\varPhi ,A)\) is a vortex configuration on \(S^3\) with vortex number \(2n-2\) if the following conditions hold:

- 1.Normalisation :$$\begin{aligned} A(X_3)=n-1, \end{aligned}$$(3.1)
- 2.Equivariance:$$\begin{aligned} {{\mathcal {L}}}_{X_3}A =0, \quad i{{\mathcal {L}}}_{X_3}\varPhi =(n-1)\varPhi , \end{aligned}$$(3.2)
- 3.Vortex equations:where \(F_A=\hbox {d}A\).$$\begin{aligned} (\hbox {d}\varPhi +i A\varPhi )\wedge \sigma = 0, \quad F_A = \frac{i}{2} (|\varPhi |^2-1)\bar{\sigma } \wedge \sigma , \end{aligned}$$(3.3)

*iA*may be viewed as the connection 1-form on the total space \(S^3/\mathbb {Z}_{2n-2}\) (Lens space) of a

*U*(1) bundle over \(S^2\) of degree \(2n-2\). Comparing with (2.17) and referring to [11] for details, the equivariance requirement (3.2) means that \(\varPhi \) is the equivariant form of a section of the associated line bundle (the (\(2n-2)\)th power of the hyperplane bundle). In fact, we will show in Sect. 4.1, Lemma 4.1, that the vortices on \(S^2\) which are equivariantly described by our vortex configurations are Popov vortices.

The following theorem shows that any vortex configuration can be expressed in terms of the pull-back of the Maurer–Cartan form \(h^{-1}\hbox {d}h\) on *SU*(2) via a bundle map \(U:S^3 \rightarrow S^3\) of the Hopf fibration covering a rational map \(S^2\rightarrow S^2\). Since the Maurer–Cartan form encodes the frame (2.19) of the round 3-sphere, its pull-back encodes the pull-back of the round metric with *U*. In that sense, this result is a 3-dimensional version of Baptista’s interpretation of vortices as deformed 2-dimensional geometry.

## Theorem 3.2

*su*(2) connection on \(S^3\) satisfying the normalisation condition

*su*(2) connection on \(S^3\) satisfying (3.6) and of the form (3.7) can be trivialised as \(\mathcal {A}=U^{-1}\hbox {d}U\), where \(U:S^3\rightarrow S^3 \) has degree \(n^2\) and is a bundle map of the Hopf fibration, covering a rational map \(R:S^2\rightarrow S^2\) of degree

*n*. Up to a

*U*(1) gauge transformation (3.4), one can choose the bundle map

*U*to have the form

*n*with no common zeros

*U*via

Our condition on \(a_0,b_0,a_n,b_n\) will turn out to be convenient in the discussion of Popov vortices in Sect. 4.2 and facilitates comparison with the treatment in [9].

## Proof

*SU*(2) gauge potential \(\mathcal {A}\) on \(S^3\) can always be globally trivialised in terms of a function \(U:S^3 \rightarrow SU(2)\) as \(\mathcal {A}=U^{-1}\hbox {d}U\). We now show that the vortex form (3.7) and the normalisation (3.6) force the trivialising map to be a bundle map covering a rational map of degree

*n*. The normalisation (3.6) requires

*U*in terms of two functions \(P_1,P_2\) which do not vanish simultaneously as in (3.8) (but without assuming that \(P_1,P_2\) are polynomials) this map is simply the quotient \(P_2/P_1\). In terms of our stereographic coordinate

*z*for \(S^2\) and the section

*s*in (2.16) we define

*n*-fold cover of a circle which links with each of the

*n*circles in the pre-image of another point exactly once. It follows that the map

*U*has degree \(n^2\) and the map

*R*covered by

*U*has degree

*n*.

*U*in terms of two functions \(P_1,P_2\) but still not assuming that \(P_1,P_2\) are polynomials, the condition (3.16) implies

*R*has to be a holomorphic map \(S^2\rightarrow S^2\) of degree

*n*, which means it must be a rational map, as claimed.

*n*we require at least one of \(a_n,b_n\) to be nonzero (so that the maximum of the degrees of \(p_1\) and \(p_2\) is

*n*) and at least one of \(a_0, b_0\) to be nonzero (so that we cannot reduce the degree by cancellation). We can then arrange for all of \(a_0, b_0,a_n,b_n\) to be nonzero by left-multiplying

*U*with a constant

*SU*(2) matrix if necessary; this does not affect \(\mathcal {A}\) and therefore leaves the vortex configuration unchanged.

*U*to be the trivialisation in terms of the polynomials \(P_1,P_2\) in (3.8), we can define a new trivialisation

*R*. The non-abelian gauge potential \({\tilde{\mathcal {A}}}= {\tilde{U}}^{-1} d{\tilde{U}}\) differs from \(\mathcal {A}= U^{-1} \hbox {d}U\) by the gauge transformation (3.4), as claimed.

*n*, we obtain the claimed formula for the vortex field \(\varPhi \) from

*A*is a straightforward calculation, which makes use of

In order to make contact with discussions in the literature related to the potential *A* we note an expression for *A* in terms of polar coordinates, for later use.

## Lemma 3.3

*A*in (3.10) can be expressed via the formula

## Proof

### 3.2 Magnetic zero-modes from vortices

We are now ready to explain how one can construct magnetic zero-modes of the Dirac operator on the 3-sphere and on Euclidean 3-space from vortex configurations on the 3-sphere. We define spinorial vortex zero-modes as follows.

## Definition 3.4

## Theorem 3.5

## Proof

We can pull back the vortex zero-modes of the Dirac equation on \(S^3\) to \(\mathbb {R}^3\) using Lemma 2.2, but we also need to understand how the nonlinear equation behaves under this pull-back. It turns out that the resulting equations take their simplest form in vector notation for gauge potentials and their magnetic fields, i.e. when expanding a 1-form on \(\mathbb {R}^3\) as \(A=\mathbf {A}\cdot \hbox {d}\mathbf {x}\) and defining the magnetic field vector field via \(dA=\frac{1}{2} \epsilon _{jkl} B_j \hbox {d}x_k \wedge \hbox {d}x_l\) or \(\mathbf {B}=\nabla \times \mathbf {A}\).

## Corollary 3.6

Any pair of homogeneous polynomials \(P_1,P_2:\mathbb {C}^2 \rightarrow \mathbb {C}\) of the same degree and without common zeros uniquely determines a smooth and square-integrable magnetic zero-mode of the Dirac operator in Euclidean 3-space which satisfies the coupled equations (3.46).

## Proof

*SU*(2) matrix yields a vortex configuration \((\varPhi ,A)\) on \(S^3\) according to the prescription of Theorem 3.2. Such a vortex configuration defines a vortex zero-mode \(\varPsi \) of the Dirac operator on \(S^3\) coupled to \(A'= A+ \frac{3}{4} \sigma _3\) according to Theorem 3.5. Implementing the conformal change to \(\mathbb {R}^3\) according to Lemma 2.2 produces the magnetic zero-mode

*H*yields

*C*, which ensures that \(\varPsi _H\) is square-integrable with respect to the Euclidean measure (2.23). Since \(|\varPsi ^\dagger \varvec{\tau } \varPsi |= |\varPsi ^\dagger \varPsi |\) for any spinor \(\varPsi \), it follows that the vector field \( \varPsi _H^\dagger \varvec{\tau } \varPsi _H\) is also square-integrable. The square-integrability of \( \mathbf {B'}_H \) then follows from the square-integrability of \( \mathbf {b}\) and the relation (3.51). \(\square \)

*H*(2.26) and

*U*(3.8) as

*A*so that the given spinor is a zero-mode of the Dirac operator on \(\mathbb {R}^3\) coupled to

*A*. They gave an explicit formula, valid where the spinor does not vanish:

## Lemma 3.7

## Proof

*H*and \(\varOmega \), to deduce

Formula (3.62) is the starting point of several treatments in the literature of magnetic zero-modes, particularly in the papers [2, 3, 4] by Adam, Muratori and Nash (AMN). The AMN construction gives magnetic zero-modes in terms of solutions of the Liouville equation. However, by effectively pulling back local expressions for sections on \(S^2\) to \(S^3\) via the Hopf map it introduces additional singularities which we will discuss in more detail in Sect. 4.2.

### 3.3 Zero-mode combinatorics

*j*. Since such a polynomial satisfies \(iX_3P= j P\) and \(X_+P=0\), it is easy to check that one can combine it with a vortex configuration \((\varPhi ,A)\) of degree \(2n-2\ge 0\) to get a solution

*P*in the spinor adds a multiple of the background field \(\mathbf {b}\) to the solution.

*n*and \(s=-j\), one can write down vortex configurations \((\bar{\varPhi },-A)\) in terms of anti-holomorphic polynomials, and obtain corresponding Dirac zero-modes

## 4 Popov vortices on \(S^2\) and Cartan connections

### 4.1 Popov vortices from vortices on \(S^3\)

We now turn to the promised explanation of the link between our vortex equations on \(S^3\) and vortex equations on \(S^2\) whose solutions are called Popov vortices. Before we write down the equations, we introduce our notation for the round geometry of the 2-sphere.

*z*defined by projection from the south pole, the round metric of a 2-sphere of arbitrary radius \(\lambda \) is

*e*behaves likes \(\bar{\zeta }^2/|\zeta |^2 \hbox {d}\zeta \) near \(\zeta =0\); \(\Gamma \), too, has a singularity at \(z=\infty \). In our chart, the Riemann curvature form is

*U*(1) bundle of degree \(2n-2\) over the 2-sphere. It is a pair \((\phi ,a)\) of a connection

*a*on this bundle and a section \(\phi \) of the associated complex line bundle. With \(a= a_z \hbox {d}z + a_{\bar{z} }\hbox {d}\bar{z} \) and \(f=f_{z\bar{z}} \hbox {d}z\wedge \hbox {d}\bar{z}=da\), the vortex equations in [9] are

*n*which, in our coordinate

*z*, take the form (3.25). The Popov vortices are determined by

We would like to relate the Popov equations and their solutions to vortices on the 3-sphere studied in Sect. 3.1. As reviewed in Sect. 2.1 the Hopf projection \(S^3\simeq SU(2)\rightarrow S^2\) in complex coordinates \((z_1,z_2)\) for \(h\in SU(2)\) (2.4) and the complex stereographic coordinate *z* on \(S^2\) is \(\pi : h \mapsto z_2/z_1\), and a local section of this bundle is given by (2.16).

## Lemma 4.1

The pull-back of the vortex equations (3.3) on \(S^3\) via the section *s* (2.16) yields the Popov equations (4.8) up to a singular gauge transformation.

## Proof

*U*

*s*

*R*that

*a*, we conclude that

### 4.2 Geometrical interpretation and singularities

*g*define a new geometry by rescaling with the Higgs field

*n*. Such singularities can also be thought of as conical singularities with a ‘negative deficit’ angle, i.e. with an excess angle. They resemble a ruffled collar, and are sometimes called ‘Elizabethan geometries’ in the literature.

*a*being a connection on a line bundle of degree \(2n-2\), we know that

*e*under the map

*R*. If

*q*is a zero of \(p_1\), the behaviour near

*q*is

*A*. Near \(z=\infty \), we use again \(\zeta =1/ z\) to write the leading term as

*z*, but has a singularity at \(z=\infty \), where it behaves like

*C*. In this gauge, the full phase rotation of \(4\pi n\) is concentrated at \(z=\infty \).

Our discussion shows that any description of the magnetic zero-modes in terms of the Popov vortex fields invariably has singularities since the Popov vortex is a section of and a connection on a non-trivial bundle, neither of which permits a globally smooth expression. This also applies to the expressions derived in [3, 4], which, in our terminology, express the magnetic zero-modes in terms of the modulus and phase of a scalar Popov vortex field (whose modulus obeys a Liouville equation). While one can shift the location of the singularities with gauge transformations, one cannot remove them on \(S^2\).

### 4.3 Gauge potentials for Cartan connections

Cartan connections combine the frame and spin connection into a non-abelian connection. We now show how the results of the previous section can be expressed in the language of Cartan geometry. We first exhibit a local gauge potential for a Cartan connection constructed from the frame and connection defined by a Popov vortex, and then show how it is related to the gauge potential \(\mathcal {A}\) used for describing vortices on the 3-sphere in Theorem 3.2.

## Lemma 4.2

*su*(2) gauge potential

*R*(3.25) is equivalent to the Popov equations being satisfied by the pair \((\phi ,a)\) defined via (4.9).

In the language of Cartan connections, this lemma says that \(\hat{A}\) is a gauge potential for a Cartan connection describing the round 2-sphere and that \(R^{*}\hat{A}\) is a gauge potential for a Cartan connection describing the deformed geometry defined by the vortex \((\phi ,a)\).

## Proof

*n*on \(S^2\). Given such a divisor we construct a bundle over \(S^2{\setminus } \{q_j\}\) by removing the union of the fibres over the \(q_j\) from

*SU*(2), obtaining the total space

*P*of degree

*n*in \(z_1,z_2\), let

*D*be the divisor of zeros of the associated inhomogeneous polynomial

*p*(so \(P(z_1,z_2)= z_1^n p(\frac{z_2}{z_1})\)). Then, we can define the map

*h*with \((z_{1},z_{2})\) as in (2.4), we have

## Lemma 4.3

*s*defined as in (2.16), the gauge potential for the Cartan connection of the 2-sphere is trivialised by

*s*:

*U*is the bundle map (3.8) covering the rational map \(R=p_2/p_1\), the gauge potential \(R^{*}A\) for the deformed Cartan geometry and the pull-back via

*s*of \(\mathcal {A}=U^{-1}\hbox {d}U\) are related through the singular gauge transformation \(r_{p_1}\):

## Proof

*e*and \(\Gamma \) in terms of

*z*in (4.2) and (4.4). With the map \(U:S^3\rightarrow S^3\) defined in terms of polynomials \(P_1,P_2\) as in (3.8), and the map \(R:S^2\rightarrow S^2\) defined as in (4.13), one checks that

*U*(3.8),

While the 1-form \({\mathcal {A}}= U^{-1} \hbox {d}U\) is manifestly smooth on \(S^3\), its pull-back with *s* is not. The map \(s\circ U\) (4.44) has a singularity of the form \(z^n/|z|^n\) at \(z=\infty \), as one would expect since the pull-backs \(s^*P_1\) and \(s^*P_2\) are local expression for sections of line bundles of degree *n* over \(S^2\) [11]. It follows that the pull-back \(s^* {\mathcal {A}}\) is singular at \(z=\infty \), with the singularity already exhibited at (4.32).

### 4.4 Cartan geometry

Our description of the geometry of the 2-sphere and its pull-back via the rational map *R* in terms of *su*(2) gauge potentials has been entirely local so far. It is time to address the global geometrical structure behind these gauge potentials. We will specify the bundles and the connections for which (4.33) and (4.35) are local gauge potentials in the language of Cartan geometry, but refer the reader to the textbook [16] and particularly to the PhD thesis [17] for general definitions and facts about Cartan geometry.

Cartan connections describe the geometry of manifolds modelled on homogeneous spaces *G* / *H* in terms of a connection on a principal *G*-bundle *Q* over this manifold. In order to recover the geometry of a manifold from a Cartan connection one needs an additional structure, namely a section of an associated *G* / *H* bundle which is transverse to the connection or, equivalently (as explained in [17]), a principal *H* subbundle *P* of *Q* which is transverse to the connection.

*D*of degree

*n*on \(S^2\), and define the quotient

*SU*(2). This is a

*U*(1) bundle over \(S^2{\setminus }\{q_j\}\) with the projection provided by the usual Hopf map \(\pi \) (2.15). It is a Lens space with

*n*circles removed.

*SU*(2) bundle, we define the

*SU*(2)-bundle associated with \(P_D\) via a

*U*(1) action on

*SU*(2):

*SU*(2) bundle over \(S^2{\setminus } \{q_j\}\) with projection

*SU*(2) bundle, we pick a homogeneous polynomial

*P*of degree

*n*and consider the map \(r_P\) in (4.38). Then,

*SU*(2) right action as

*U*satisfying this condition,

*P*, the 1-form

*U*in (3.8) satisfies (4.54). Picking \(P=P_1\) and pulling back \(\omega \) to our stereographic coordinate chart via \(S_U\) lead to the gauge potential

*SU*(2) bundle \(Q_D\). In the trivialisation via (4.55), this section (often called the Higgs field in the physics literature on Cartan connections) is simply the constant map

*su*(2). The geometry is recovered from the covariant derivative

*R*. In other words, the rational map which solves the Popov vortex equations is the ‘transverse Higgs field’ of Cartan geometry in a particular gauge. The geometry is still recovered via the covariant derivative, but since the gauge potential vanishes now, this is simply the exterior derivative \(d \tilde{\varphi }\) which, modulo stereographic projection, indeed reproduces the formula for the frame \(\phi e\) in terms of the derivative of

*R*.

## 5 Summary and outlook

The equations, spaces and maps studied in this paper are summarised in Fig. 3, with magnetic zero-modes on the top left of the picture and the Popov vortex equations on the bottom. The geometry of the 3-sphere, as encoded in the Maurer–Cartan form \(h^{-1}\hbox {d}h\), and its pull-back via the bundle map *U* provides a unifying point of view for both and leads to the explicit and smooth description of both vortex zero-modes and of vortices.

The diagram in Fig. 3 shows that the structures studied in this paper are closely related to many of the most studied topological solitons [18]. Apart from the obvious vortex configurations, the maps *U* and their pull-backs \(U\circ H\) are topologically Skyrme fields on \(S^3\) and \(\mathbb {R}^3\). The rational maps *R* can also be interpreted as ‘lumps’ or baby Skyrmions on the 2-sphere, while the composite maps \(H\circ U\circ \pi = H\circ \pi \circ R\) are topologically Hopfions. As discussed in Sect. 3.2, the magnetic fields on \(\mathbb {R}^3\) obtained by pulling back the area form on the unit 2-sphere via these maps are examples of linked magnetic fields of the kind studied by Rañada and more recently, in more detail and with many pictures, in [14]. Finally, the equation obeyed by vortex zero-modes on \(\mathbb {R}^3\) is related to the Seiberg–Witten equations on \(\mathbb {R}^4\) with a sign flipped [2].

It is clear that much of what we discussed in this paper has a close analogue in a Lorentzian and hyperbolic setting, where the 2-sphere is replaced by hyperbolic 2-space, the 3-sphere by 3-dimensional anti-de Sitter space (or *SU*(2) by *SU*(1, 1)), and Euclidean \(\mathbb {R}^3\) by Minkowski space \(\mathbb {R}^{2,1}\). Spelling this out is the topic of a forthcoming paper [19]. However, one should also consider further generalisations suggested by the origin of the Popov equations in self-duality.

As we briefly mentioned in the Introduction, the Popov equations are symmetry reduction of the self-duality equations for *SU*(1, 1) instantons on \(\mathbb {R}^4\). In fact, there is a whole family of integrable vortex equations recently studied by Manton [20] which have similar links to self-duality equations, with the relevant gauge group depending on the vortex equation [21]. It is intriguing that, for Popov vortices, the non-abelian gauge group *SU*(1, 1) needed for the self-dual connection differs from the *SU*(2) we used in our description in terms of flat Cartan connections. It would be interesting to understand both viewpoints and their relationship systematically for the family of integrable vortex equations studied in [20].

To end, we point out that interactions of spinors with linked magnetic fields of the form (3.56) are currently much discussed in atomic and condensed matter physics, where the spinors arise as an effective description of nearly degenerate states of ultra-cold atomic Bose-Einstein condensates, and the magnetic field as the curvature of a Berry connection, see, for example, the papers [22, 23] for a review. Our explicit expression for more general magnetic zero-modes may prove useful in that context.

## Notes

### Acknowledgements

CR acknowledges an EPSRC-funded PhD studentship. We thank Patrik Öhberg for discussions about possible applications of our ideas.

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