Hyperbolic vortices and Dirac fields in 2+1 dimensions

Starting from the geometrical interpretation of integrable vortices on two-dimensional hyperbolic space as conical singularities, we explain how this picture can be expressed in the language of Cartan connections, and how it can be lifted to the double cover of three-dimensional Anti-de Sitter space viewed as a trivial circle bundle over hyperbolic space. We show that vortex configurations on the double cover of AdS space give rise to solutions of the Dirac equation minimally coupled to the magnetic field of the vortex. After stereographic projection to (2+1)-dimensional Minkowski space we obtain, from each lifted hyperbolic vortex, a Dirac field and an abelian gauge field which solve a Lorentzian, (2+1)-dimensional version of the Seiberg–Witten equations.


Introduction
Vortex configurations consisting of a complex scalar and an abelian gauge field can be given a geometrical interpretation in two dimensions by viewing the modulus of the scalar field as a conformal rescaling of the underlying two-dimensional geometry. As pointed out in [1], this is particularly natural for vortices defined on Kähler geometries and obeying first-order Bogomol'nyi equations. In that case, the rescaled metric has conical singularities at the vortex locations but away from those singularities its curvature can be expressed very simply in terms of the original Kähler form and curvature, and the rescaled Kähler form.
When the first order vortex equations are integrable, the geometrical picture simplifies further. The Popov vortex equations on the sphere [2], for example, can be solved in terms of a rational map [3]. The rescaled metric is then the pullback of the round metric on the sphere via the rational map. It has conical singularities at the ramification points, and these are precisely the locations of the vortices.
Recently, we showed in [4] that one gains further insight into the geometry of Popov vortices by considering the total space S 3 SU (2) of the circle bundle on which they are defined. We showed that the equivariant formulation of the vortex equation on S 3 can be solved in terms of bundle maps of the Hopf fibration, and that each vortex configuration gives rise to a solution The standard geometry of the three dimensional group manifold SU (1, 1) plays a central role in this paper. The lifted vortex configurations can be expressed in terms of bundle maps of π : SU (1, 1) → H 2 , covering holomorphic maps H 2 → H 2 . These bundle maps can in turn be written in terms of two holomorphic functions satisfying an equivariance condition; they fully determine the vortices, the magnetic fields and the Dirac fields.
As in [4], we have found it illuminating to express the geometry defined by a vortex configuration in the language of Cartan geometry. This provides the simplest route to three dimensions, and relates the vortex equations to a flatness condition for a non-abelian connection. The fact that SU (1, 1) is a trivial circle bundle over H 2 simplifies the discussion, but the non-trivial first fundamental group of SU (1, 1) and non-compactness of H 2 add subtleties compared to the Euclidean story, as we will explain. The paper is organised as follows. We begin in Sect. 2 by summarising some results on hyperbolic vortices and their geometric interpretation. Then, following a brief interlude to establish our conventions for the pseudo-unitary group SU (1, 1), we present a Cartan geometry interpretation of hyperbolic vortices. In particular we construct an explicit su(1, 1) gauge potential whose flatness is equivalent to the hyperbolic vortex equations.
Sect. 3 introduces the three dimensional setting, the relationship between AdS 3 and the Lie group SU (1, 1) and an interpretation of both as circle bundles over H 2 . The main result of this section is the equivalence between vortex configurations on SU (1, 1) and flat SU (1, 1) gauge potentials, and expressions for both of these in terms of bundle maps SU (1, 1) → SU (1,1). At this stage the hyperbolic story is arguably richer than its spherical counter part and we are not immediately forced to consider vortex configurations of finite degree. However, configurations with finite equivariant degree give rise to finite charge vortices on H 2 , and we therefore pay them special attention.
In Sect. 4 we introduce stereographic and gnomonic projections from SU (1, 1) to three-dimensional Minkowski space R 1,2 , and use them to relate the gauged Dirac equation on SU (1, 1) and on the interior I ⊂ R 1,2 of a single-sheeted hyperboloid. Then, in Sect. 5, we combine the results of all preceding sections to construct solutions of a Lorentzian version of the Seiberg-Witten equations on AdS 3 and on (2+1)-dimensional Minkowski space. Finally Sect. 6 contains a summary of the paper in the form of Fig. 5, and a discussion.

.1 Hyperbolic vortices and holomorphic maps
First order vortex equations on the Poincaré disk model of hyperbolic space have been studied extensively, with many of the details summarised in [8]. Solutions can be obtained from SO(3) invariant instantons on S 2 ×H 2 [9] and expressed in terms of holomorphic mappings of the disk. We briefly review these solutions here, but should warn the reader that our disk has radius 1 rather than √ 2 as chosen in [8], and that we write the equations in terms of the Riemann curvature form rather than the Kähler form for the disk. We first introduce our notation for the geometry of the Poincaré disk model, which we write as A complexified orthonormal frame field e = e 1 + ie 2 for this metric is This metric is Kähler with Kähler form In terms of the complexified frame the structure equations are given by and its complex conjugate. The structure equations determine the spin connection 1-form Γ to be The Riemann curvature form is and is related to the Gauss curvature, K, and the Kähler form, equation (2.4), through the Gauss equation, A hyperbolic vortex is a pair (φ, a) where a is a connection on a principal U (1) bundle over H 2 , which is necessarily trivial, and φ is a smooth section of the associated complex line bundle. Taking a = a z dz + azdz and F a = da, the vortex equations are (2.9) The first of these requires that φ be holomorphic with respect to the connection a.
It is instructive to compare this equation to a different vortex equation, called the Popov vortex equation, which was introduced in [2] and studied in [3]. A Popov vortex is a pair (φ, a) of a connection a on a degree 2N − 2 principal U (1)-bundle over the two-sphere and a holomorphic section φ of an associated line bundle. In terms of a stereographic coordinate z and curvature two-form R S 2 , the equations are (2.10) In the form presented here, these equations look the same as the hyperbolic equations since the sign difference due to the Gauss curvature is absorbed into the Riemann curvature two-form. From now on we work with hyperbolic vortices but make frequent comparisons with Popov vortices.
Solutions to the hyperbolic vortex equations can be constructed from holomorphic functions f : H 2 → H 2 in the following way. Given the complex frame and spin connection, e, Γ on H 2 , we pull them back via f and define the Higgs field and gauge potential through φe = f * e, a = f * Γ − Γ, (2.11) so that we have an explicit expression for φ in terms of f : It also follows that f * (e ∧ē) = |φ| 2 e ∧ē, (2.13) so that the second vortex equation is a direct consequence: To see that φ is indeed covariantly holomorphic with respect to a as the first vortex equation requires, consider the pullback of the structure equation, (2.33): The final line is the required holomorphicity condition, and equivalent to the first vortex equation.
The pulled back frame f * e degenerates at the zeros of φ. This corresponds to the vortex positions becoming conical singularities of the rescaled metric, with an angular excess related to the charge of the vortex, see [10]. A consequence of the frame being degenerate is that its spin connection,Γ, has singularities. However, f * Γ is the pullback of a smooth spin connection with a smooth map, so is in particular non-singular. The difference between the pulled back spin connection and the spin connection of the degenerate frame is due to singularities at the zeros of φ. This results inR being equal to f * R up to the addition of delta function singularities at the zeros z j of φ: One can view Riemann surfaces of genus g > 1 as the quotient of H 2 by the action of a Fuchsian group Γ < SU (1, 1). Vortices on such Riemann surfaces can therefore be constructed from vortices on H 2 that are invariant under the action of the desired Fuchsian group. In practice, this is not easy. A vortex solution on the Bolza surface (genus 2) is presented in [11]. While these vortices have an infinite number of zeros of the Higgs field on H 2 they have a finite number of zeros within the principal domain of the Fuchsian group.
In the construction of vortices from holomorphic maps between compact Riemann surface via (2.11), the Gauss-Bonnet theorem imposes a constraint on the genus of the surfaces and the vortex number, see [1]. The negative contribution from the singularities in the curvatureR, (2.16), at the zeros of φ plays a key role here, and provides a no-go theorem in some cases.
We now specialise to the case of solutions of the vortex equations on H 2 which satisfy the boundary condition |φ| → 1 as |z| → 1. As explained in [8,12], this boundary condition is required for the vortex to have finite energy. It also means that φ has a finite vortex number associated to it, which is the number of zeros counted with multiplicity. This is analogous to the degree of the line bundle for the case of Popov vortices mentioned above.
As was first observed in [9], solutions of the hyperbolic vortex equations on H 2 which satisfy the boundary condition are obtained from bounded holomorphic functions f : H 2 → H 2 which can be expressed as a finite Blaschke product when working in the Poincaré disk model. For a (N − 1)-vortex solution, the Blaschke product can be written as the ratio of the two holomorphic functions where c k ∈ H 2 , k = 1, . . . , N . As the zeros of f 2 are in the disk of radius 1 the zeros of f 1 , at 1/c k , are not. Thus f has zeros but no poles within the disk. Note that |f (z)| = 1 when |z| = 1, i.e., on the boundary of the disk and that, by the maximum principle, |f (z)| < 1 when |z| < 1, so that f really is a holomorphic mapping of the disk model.
This way of writing the holomorphic function will prove useful later when we introduce and discuss vortex configurations on SU (1, 1), as will the observation about the lack of poles. The pullback of the holomorphic frame field has the explicit form This is a manifestly smooth one-form which vanishes at the zeros of φ, thus illustrating our earlier remarks about the pullback frame.
2.2 Interlude on SU (1, 1) Before we interpret hyperbolic vortices in terms of Cartan geometry we need to make clear our conventions for the pseudo-unitary group SU (1, 1). It is defined as the subgroup of SL(2, C) whose elements h satisfy where τ 3 is the third Pauli matrix. Its Lie algebra su(1, 1) is defined as the set of complex traceless matrices satisfying This forces the diagonal elements to be purely imaginary and the off-diagonal elements to be mutually complex-conjugate. We work with the generators where the τ i are the Pauli matrices. They obey the commutation relation We also frequently use showing that t 0 acts as a complex structure on its complement in su(1, 1), with t + (t − ) as (anti)-holomorphic directions.
The Killing form on su(1, 1) is where η is the 'mostly minus' Minkowski metric: We parametrise an SU (1, 1) matrix h using complex coordinates (z 1 , z 2 ) ∈ C 1,1 as so that we can view SU (1, 1) as a submanifold of C 1,1 : It is the double cover of AdS 3 , the real submanifold of C 1,1 defined by with called the AdS length. The Z 2 quotient identifies (−z 1 , −z 2 ) and (z 1 , z 2 ).
Left invariant one-forms on SU (1, 1) are defined via (2.31) They satisfy We make frequent use of the complex combinations In terms of the complex coordinates we find that The dual left-invariant vector fields X i , i = 0, 1, 2, generate the right action h → ht i and satisfy so that the complex linear combinations, In terms of the complex coordinates they are The only non-zero pairings are The Poincaré disk model of hyperbolic two-space can also be viewed as the coset space where we consider the U (1) generated by t 0 . As H 2 is contractible this means that SU (1, 1) is a trivial circle bundle over H 2 , with X 0 generating translation in the fibre direction.
We can construct a projection from the group manifold SU (1, 1) to H 2 , in an analogous manner to the projection in the Hopf fibration [4]. Using the complex coordinate z ∈ C on H 2 and in terms of the complex coordinates (z 1 , z 2 ) for SU (1, 1) this projection is A global section of this bundle is given by

Hyperbolic vortices as Cartan connections
We now show that the hyperbolic vortex equations can be interpreted as the flatness conditions for a su(1, 1) Cartan connection which encodes the geometry, modified by the vortices.
Proposition 2.1. The frame (2.3) and the spin connection (2.6) for H 2 can be combined into the su(1, 1) gauge potentialÂ The flatness condition forÂ is equivalent to the structure equation (2.5) and Gauss equation In other words, the gauge potentialÂ defines a Cartan connection describing the hyperbolic geometry of the Poincaré disk model of two-dimensional hyperbolic space while f * Â is the gauge potential for a Cartan connection describing the deformed geometry defined by the hyperbolic vortex (φ, a).

Proof. The curvature ofÂ is
The coefficient of t 0 being zero is equivalent to the Gauss equation (2.8), and the coefficients of t ± being zero is equivalent to the structure equation (2.5). For f * Â we can use (2.11) to see that which has curvature The hyperbolic vortex equations (2.9) being satisfied is thus equivalent to the vanishing of this curvature.

Vortex equations and flat SU (1, 1) connections
We now define and solve vortex equations on the group manifold of SU (1, 1) and show that three-dimensional vortex configurations which solve them are equivariant versions of hyperbolic vortices.
To prepare for the equivariant description, we define the space of equivariant functions on SU (1, 1) as with N called the equivariant degree of the function. One finds that where F ∈ C ∞ (SU (1, 1), C) N . From this we deduce the following commutative diagram We will see later that functions with a well defined equivariant degree on SU (1, 1) can be used to construct the lift of a vortex of finite charge from H 2 ; it is these lifts of finite charge vortices that are the analogues of the vortex configurations considered in [4].
Note, that unlike in the Euclidean case considered in [4], we have not imposed any normalisation condition on A and have not fixed the equivariant degree of Φ. However, the vortex equations imply the following equivariance condition: (3.5) The first follows from the Cartan formula L X 0 = dι X 0 + ι X 0 d and the form of F A = dA dictated by the second vortex equation. The equivariance condition for Φ can be obtained by contracting the first vortex equation with (X 0 , X − ). We discuss the case of Φ having equivariant degree 2N − 2, the analogue of the spherical case [4], later.
The vortex equations (3.4) clearly resemble the hyperbolic vortex equations, (2.9), with the complexified left-invariant one-forms, σ andσ replacing the complexified frame, e andē. We will establish the precise relation between the two at the end of this section.
We now come to the central result of this section, which is a three-dimensional analogue of the description of hyperbolic vortices in terms of a flat SU (1, 1) connection given in (2.1). It is also the Lorentzian analogue of Theorem 3.2 in [4], but differs from it in two important respects. In the Euclidean version, the relevant U (1) bundle is the Hopf bundle, and associated line bundles are classified by an integer degree, but the total space SU (2) is simply-connected. Here, the U (1) bundle is trivial, but the total space SU (1, 1) is not simply connected. The generator of the first fundamental form is the curve which enters our condition for vortex configuration to be globally solvable.
Theorem 3.2. A vortex configuration on SU (1, 1) determines a gauge potential for a flat SU (1, 1) connection on SU (1, 1) through the following expression: Conversely, any flat SU (1, 1) connection A on SU (1, 1) with for some n ∈ Z, can be trivialised as where F 1 and F 2 are maps SU (1, 1) → C, with |F 1 | > |F 2 | and in particular F 1 = 0. The vortex configuration (Φ, A) can be computed from the bundle map V through Proof. The proof that the vortex equations (3.4) imply flatness of A given in (3.7) is a simple calculation, see also [4]. Conversely, expanding an su(1, 1)-valued one-form A on SU (1, 1) in terms of the generators t 0 , t + and t − , with coefficients which are linear combinations of the one-forms σ 0 , σ andσ, and imposing (3.8) leads to a gauge potential A of the form (3.7) with Higgs field Φ and abelian gauge field The flatness of A then give the vortex equations (3.4), as already noted.
A connection A on SU (1, 1) can be trivialised in terms of V : In that case one can construct V explicitly from the path-ordered exponential of A along any path, starting at a fixed (but arbitrary) base point, see for example [13].
In our case, the flatness of A ensures the path-independence of the path-ordered exponential for all contractible paths on SU (1, 1), by the non-abelian Stokes Theorem. The condition (3.8) implies that the path-ordered exponential and the exponential of the ordinary integral of A along γ coincide, and finally (3.9) ensures that the path-ordered exponential of A along γ is trivial: Again using flatness of A we conclude that the path-ordered exponential along any closed curve on SU (1, 1) is the identity, thus establishing the path-independence of the path-ordered exponential.
It remains to show that the requirements (3.8) for A = V −1 dV force V to be a bundle map covering a holomorphic map H 2 → H 2 . The first condition becomes for p : SU (1, 1) → R. However, this is precisely the infinitesimal formulation of the requirement that V preserves the fibres of the fibration SU (1, 1) → H 2 , i.e., that V is a bundle map.
The second condition in (3.8) complex conjugates to Applying to X + , the condition (3.15) is thus equivalent to the vanishing of the t − -component in V −1 dV : We need to show that this is equivalent to V covering a holomorphic map.
Using the parameterisation (3.10) of V , we see from (3.14) that the components of V satisfy It follows that the map F = π • V = F 2 /F 1 has equivariant degree zero, and that V covers the map Applying (3.2) to the map F = F 2 /F 1 we deduce that f being holomorphic is equivalent to However, again recalling that F 1 = 0 and using the explicit form (2.35) of σ, this is seen to be equivalent to the condition (3.17) on V . Thus, the requirement (3.8) forces V to be a bundle map covering a holomorphic map, as claimed.
The Theorem and its proof deserve a few comments. First we note that the vortex equations (3.4) are invariant under U (1) gauge transformations of the form (SU (1, 1)). (3.21) The U (1) gauge invariance is implemented at the level of the bundle map V via V →Ṽ = V e βt 0 , β ∈ C ∞ (SU (1, 1)). Secondly, we observe that we can build vortex configurations on SU (1, 1) from a given holomorphic map f : H 2 → H 2 by choosing For this choice of V the connection satisfies A(X 0 ) = 0 and also, by flatness, L X 0 A = 0, so that A is constant along the fibre.
Finally, the condition (3.9) is needed to ensure the existence of a globally defined trivialisation of A. When it is violated, one can still trivialise, but one will in general have to work with local trivialisations, defined in at least two simply connected patches which cover SU (1, 1). We will not consider such trivialisations in this paper, but they may well be of interest.

Vortex configurations of finite equivariant degree
In the following we exhibit a choice of bundle map which is a natural lift of the Blaschke product considered in Sect. 2.1. The resulting vortices on SU (1, 1) are lifts of hyperbolic vortices with a finite number of zeros, and a Lorentzian analogue of the vortices obtained from homogeneous polynomials in the Euclidean version considered in [4].
Before stating our result we define functions F 1 , F 2 : SU (1, 1) → C for given complex numbers c k , k = 1, . . . , N , in the unit disk via and note that F 2 /F 1 is a function of z = z 2 /z 1 and given by the Blaschke product (2.17). In particular, therefore |F 1 | > |F 2 |, and we can use F 1 , F 2 to define a bundle map V covering the Blaschke product (2.17) via (3.10). The vortex configuration (Φ, A) can be given in terms of F 1 , F 2 as Proof. It follows from the explicit form of F 1 , F 2 that so that A(X 0 ) = N t 0 and therefore, by the decomposition (3.7), A(X 0 ) = N − 1. The general equivariance condition (3.5) then implies so that Φ has equivariant degree 2N − 2 by (3.1). To get the explicit expression for Φ we compute To compute A we use to get the claimed result.
Recall that the Blaschke product (2.17) gives rise, via the pull-back construction, to a hyperbolic vortex of charge N − 1 on H 2 . The Corollary above shows that the natural covering V of the Blaschke product defines a vortex configuration on SU (1, 1) of finite equivariant degree 2N − 2.
For finite Blaschke products, the lift (3.24) defines a natural bundle map which we used to construct a three-dimensional vortex configuration with non-zero degree. In general, lifting to a configuration with non-zero equivariant degree is not trivial. For example, the vortices on the hyperbolic cylinder studied in [14,11] require the infinite Blaschke product where the zeros of f are at a j = i tanh jλ 2 with λ defined in [14,11] as λ = πK (k) K(k) for K the elliptic integral of the first kind, K (k) = K( √ 1 − k 2 ) and any 0 < k < 1. These can still be lifted to SU (1, 1) via the lift (3.23), but there does not appear to be any non-trivial natural option.

Lifting Cartan connections for hyperbolic vortices
We have already seen how to lift hyperbolic vortices to vortex configurations on SU (1, 1). Since the latter can be expressed in terms of a flat SU (1, 1) connection, it is natural to expect a link with the Cartan connection encoding hyperbolic vortices according to Proposition 2.1. In this short section, we exhibit this link. (3.33) We use this map as a gauge transform in the following Lemma, which is a Lorentzian analogue of Lemma 4.3 in [4].
Note that, if we work in the gauge where F 1 = 1 and F 2 = f z 2 z 1 , then r f 1 = I , so f * Â and s * (V −1 dV ) agree. More generally, the fact that F 1 and therefore f 1 has no zeros means that the gauge transformation r f 1 is smooth. This is in contrast to the Euclidean case considered in [4] where a singular gauge transformation was needed.
Proof. The proof is a straightforward calculation which proceeds along the lines given in [4].

Magnetic Dirac operators on AdS 3 and Minkowski space 4.1 Notational conventions
We denote three-dimensional Minkowski space by R 1,2 , and use a 'mostly minus' Lorentzian metric η with matrix (2.27) in an orthonormal basis. We write elements of Minkowski space as Our volume element is dx 0 ∧ dx 1 ∧ dx 2 so that dx 0 , dx 1 , dx 2 is an oriented basis of the cotangent space. We use indices i, j . . . in the range 0, 1, 2, raised and lowered using η ij . The scalar and vector product are given by for x, y ∈ R 1,2 , ε 012 = 1 and the summation convention being understood between pairs of raised and lowered indices. The Lorentzian length squared of x is denoted by We also use the notation ∂ i = ∂/∂x i for partial derivatives.
On SU (1, 1) we continue to work with the notation introduced in Sect. 2.2, and use the following oriented orthonormal frame consisting of and orientation Differential forms provide the natural language for our discussion, but occasionally we use the isomorphisms between forms and vector fields which are possible on a three-dimensional manifold with a non-degenerate inner product and volume form. The inner product allows one to identify vector fields with one-forms; denoting the volume form Vol, it establishes a bijection between a vector field X and a two-form F via ι X Vol = F. (4.7) On SU (1, 1), for example, the vector field X 0 generating the fibre translation is mapped to the two-form 1 8 σ 1 ∧ σ 2 via (4.6), and this will play a role in our discussions.
In the following we will be considering the Dirac equation on both SU (1, 1) and R 1,2 , so we need to fix our conventions for the Clifford algebra Cl(1, 2). The algebra is generated by the gamma matrices, γ i which satisfy We pick γ i = 2t i as our representation.

Stereographic projection and frames
We will be using a stereographic projection to relate Dirac operators on AdS 3 SU (1, 1) to Dirac operators on Minkowski space. We now set up our conventions and explain how orthonormal frames on these spaces are mapped into each other via stereographic projection.
To discuss stereographic projection from AdS 3 to R 1,2 , it is helpful to think of AdS 3 as a real manifold. As a subspace of R 2,2 , and with AdS length , it is given by This is just the definition of AdS 3 as a submanifold of C 1,1 from (2.30) written in terms of real coordinates. The real coordinates are related to the complex coordinates for SU (1, 1) (2.28) through (z 1 , z 2 ) = (y 3 + iy 0 , y 2 − iy 1 ). (4.10) Just as stereographic projection from the sphere requires one to single out a north and a south pole, we need to pick special points on AdS 3 to define our stereographic projection. We choose P ± = (0, 0, 0, ± ) ∈ AdS 3 . Moreover, the point of intersection necessarily lies in the subset I ⊂ R 1,2 defined as Geometrically I is the inside of a single sheeted hyperboloid.
We will also need a Lorentzian version of the so-called gnomonic projection discussed and used in [4]. This is the map G : I ⊂ R 1,2 → SU (1, 1) given by which satisfies G 2 = H. The geometric interpretation of this result is shown in Fig. 2 and explained in the caption. Note also the map H is an AdS version of the projection relating a sheet of the two-sheeted hyperboloid to the disk in the Poincaré model. The map G is the AdS analogue of the map to the Beltrami disk model.
(4.21) Figure 2: The lines OG( x) and P − H( x) define the maps G and H. They are analogous to, respectively, the Beltrami and Poincaré map in hyperbolic geometry. The area bounded by the hyperbolic segment P + H( x) and the straight lines OH( x) and OP + is twice that bounded by the hyperbolic segment P + G( x) and the lines OG( x) and OP + . This is the geometry behind the relation H( x) = G 2 ( x).
We find that This means that the ϑ i , i = 0, 1, 2, give a Lorentz-rotated basis for the cotangent space of I .
with the inverse relation expressed as Proof. The proof follows by a calculation which is similar to the corresponding Euclidean version in [4], but differs in important signs. The first statement (4.23) follows from the fact that H = G 2 . For (4.24) we use (4.23) to write it as Then we compute which gives us that we get (4.25) and hence (4.24).

Magnetic Dirac operators
On SU (1, 1), a global gauge potential for the spin connection is given by The Dirac operator in the left-invariant frame is then Minimal coupling to an abelian gauge potential A yields The Dirac operator on R 1,2 minimally coupled to A · d x is In the following, spinors Ψ which satisfy the massless Dirac equation / D A Ψ = 0 on either SU (1, 1) or R 1,2 coupled to an abelian gauge potential are called magnetic Dirac modes, or simply magnetic modes. The following Lemma exhibits the relation between magnetic Dirac modes on SU (1, 1) and R 1,2 . is a magnetic mode of the Dirac operator (4.32) on I ⊂ R 1,2 coupled to the gauge potential H * A.
Proof. The pull-back of the spin connection is Using (4.23) one can show that Then combining (4.27) with gives that Using these results we compute the pull-back of the Dirac operator on SU (1, 1), coupled to both the spin connection and the abelian gauge potential A, to the flat frame in R 1,2 : Proof. First observe that showing that F n has equivariant degree n−1. Next note that X + F n = 0 since F n is holomorphic, that A(X + ) = 0 and that Using this, and the explicit form of / D SU (1,1),A given in (4.31), the equation so Ψ is indeed a magnetic mode.
The following Definition and Theorem are similar to the Euclidean version considered in [4]. However, the non-linear equation in the definition of a vortex magnetic mode has an important overall sign difference.
with the Hodge star operator on SU (1, 1) with respect to the metric (4.5) and orientation (4.6).
We now give a result that enables the construction of magnetic modes from any vortex configuration.
is a vortex magnetic mode on SU (1, 1). For the non-linear equation with a spinor of the form given in (5.7) we have that

Proof. The spinor is a magnetic mode of
On the other hand using (3.4) gives from which the non-linear equation follows.
Both the magnetic two-forms F A and F A are proportional to σ 1 ∧ σ 2 , with a factor of proportionality which is a function on SU (1, 1). The magnetic vector field associated to either of them via (4.7) is therefore similarly proportional to the vector field X 0 , and so the magnetic fields are just the fibres of the fibration π : SU (1, 1) → H 2 .
To visualise these fibres in the embedding of AdS 3 in R 2,2 (with one dimension suppressed), we note that they are in particular geodesics on AdS 3 and therefore can be obtained by intersections of the embedding (4.9) with planes in R 2,2 . This was used to produce the picture of geodesics on AdS 2 , embedded in R 2,1 , in Fig. 3.
We can write down geodesics on AdS 3 explicitly by expressing the right action of e αt 0 , α ∈ [0, 4π), which generates them, in real coordinates. Using the parametrisation (4.16) the orbit of a point (y 0 , y 1 , y 2 , y 3 ) ∈ AdS 3 is (5.12) In other words moving along the fibre is equivalent to a rotation by α 2 in both the y 3 , y 0 and y 1 , y 2 plane.

Dirac modes on Minkowski space
In [5], Loss and Yau used a particular formula to construct gauge potentials for a given spinor so that the spinor is a zero-mode of the Dirac operator coupled to the gauge potential. This formula has a simple Lorentzian analogue, namely where Σ i = 2iΨ † t i Ψ. However, its use is problematic because Lorentzian spinors may be null even when they are not vanishing.
Our construction of magnetic modes proceeds differently. We use Lemma 4.2 to obtain magnetic Dirac modes on I ⊂ R 1,2 directly from the vortex magnetic modes (5.3) on SU (1, 1). Figure 4: The magnetic field lines for the pull-back of vortex magnetic modes to Minkowski space. In particular, they are the magnetic fields lines of the background field b. They are also the images of the fibres illustrated in Fig. 3 under the stereographic projection.
The magnetic field in Minkwoski space is obtained from the magnetic field F A on SU (1, 1) via pull-back with the inverse stereographic projection H. The magnetic field lines are therefore the images, under stereographic projection, of the fields lines on SU (1, 1). While the field lines on SU (1, 1) are all closed, they also leave the domain of the stereographic projection. As a result, the image curves in I are not closed. Instead, they are of the form shown in Fig. 4.
For explicit formulae on Minkowski space, it is convenient to work in vector notation where a one-form is expanded as A = A · d x on I , and where magnetic two-forms are expressed in terms of vector fields according to (4.7). In particular, the inhomogeneous term in the equation (5.6) governing vortex magnetic modes pulls back to the two-form 14) and the corresponding magnetic field is The field lines of b are the fibres of the fibration π : SU (1, 1) → H 2 , and plotted in Fig. 4.
Since vortex magnetic modes on SU (1, 1) satisfy a non-linear equation in addition to the linear Dirac equation, we expect the same to be true for the vortex magnetic modes on Minkowski space. We define them as follows. where B = ∇ × A and b is the background field given in (5.15) The coupled equations in this definition formally resemble the dimensionally reduced Seiberg-Witten equations, perturbed by the background field b. The role of the Seiberg-Witten equations in differential topology makes essential use of a Riemannian metric, and a Lorentzian version like the one defined here does not appear to have been studied.
Combining many of the results derived in this paper, we arrive at the following explicit construction of vortex magnetic modes on I ⊂ R 1,2 : Corollary 5.5. Any given bundle map V : SU (1, 1) → SU (1, 1) covering a holomorphic map f : H 2 → H 2 determines a smooth vortex magnetic mode on I ⊂ R 1,2 . Explicitly, extracting the vortex configuration (Φ, A) on SU (1, 1) from A = V −1 dV via (3.7), the vortex magnetic mode is given by Proof. The result follows by composing the construction of magnetic Dirac modes from vortex configurations with the construction of vortex configurations from bundle maps. We use Theorem (3.2) to construct a vortex configuration (Φ, A) on SU (1, 1) from the bundle map V , then Theorem (5.3) to construct a vortex magnetic mode (Ψ, A ) on SU (1, 1) from (Φ, A). Finally Lemma (4.2) is used to pull it back to I . The confirmation that the magnetic mode thus obtained satisfies the coupled equations (5.16) with gauge field and magnetic field is a straightforward calculation, which is analogous to the one carried out for the Euclidean version in [4].
The Corollary allows one to construct solutions of gauge Dirac equation and to solve initial value problems in Minkowski space. The restriction to I ⊂ R 1,2 is not necessarily a problem in practice since can be chosen arbitrarily. By choosing it large enough, one can capture initial data on any bounded subset of a Cauchy surface.

Summary and outlook
In this paper we presented a lift of hyperbolic vortices to AdS 3 and a construction of massless solutions of magnetic Dirac equations on AdS 3 and on a subset of R 1,2 from the vortices. This provides a new, three-dimensional interpretation of vortices and complements the two-dimensional geometrical interpretation given by Baptista in [1] and the four-dimensional interpretation as rotationally symmetric instantons [9,15].
The summary diagram in Fig. 5 gives a concise presentation of the spaces and equations that we considered here and the maps that relate them. The three dimensional point of view unifies the massless Dirac modes and the hyperbolic vortices and clarifies the geometry underlying this relationship. The story summarised in Fig. 5 is a Lorentzian and hyperbolic analogue of the Euclidean and spherical story told in [4], but there important differences in the details. The triviality of SU (1, 1) as a circle bundle of H 2 , as opposed to the non-triviality of the Hopf bundle, simplifies the topology and allows for global description of all sections. On the other hand, the non-compactness of the base H 2 , as opposed to the compactness of S 2 , leads to a wider variety of vortex configurations than in the Euclidean case, where vortices are in one-to-one correspondence with rational maps and the vortex number of any given configuration is finite.
In the hyperbolic case, we have vortices with a finite vortex number, solved in terms of finite Blaschke products, and configurations with infinitely many zeros of the Higgs field. When the latter are invariant under a Fuchsian group Γ < SU (1, 1) they lead to finite charge solutions of the vortex equations on the Riemann surface H 2 /Γ. The action of Γ on H 2 descends from a left-action of Γ on SU (1, 1) and therefore it makes sense to study vortex configurations and spinors on SU (1, 1) which are left-invariant under Γ. However, the left-action of Γ on SU (1, 1) does not, in general, respect the domain of the stereographic projection (4.12), so does not induce an action of Γ on Minkowski space 1 . As result, there does not appear to be a natural characterisation of the Dirac fields on Minkowski space obtained from hyperbolic vortices on the Riemann surface H 2 /Γ.
We end with a brief outlook on interesting questions for further study. It seems very likely that all of the integrable vortex equations considered in [10] are amenable to a three dimensional interpretation along the lines of this paper and of [4]. However, the cases studied here and in [4] show that there are interesting differences in the details and in the interpretation, and these should be worked out.
As observed in [15], all the integrable vortex equations considered in [10] can be seen as dimensional reductions of the self-duality equations, with gauge groups depending on the type of vortex. For example, Popov vortices arise from su(1, 1)-valued connections in four dimensions and hyperbolic vortices from su(2)-valued connections. The Cartan connections we used are also non-abelian, but exchange the Lie algebras, so use su(2) for Popov vortices and su(1, 1) for hyperbolic vortices. It would be interesting to understand the link between the self-dual and the Cartan view point systematically for all the integrable vortices in [10].