Manton’s five vortex equations from self-duality
We demonstrate that the five vortex equations recently introduced by Manton arise as symmetry reductions of the anti-self-dual Yang–Mills equations in four dimensions. In particular the Jackiw–Pi vortex and the Ambjørn–Olesen vortex correspond to the gauge group , and respectively the Euclidean or the symmetry groups acting with two-dimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the Yang–Mills theory in 4+1 dimensions to construct a metric on vortex moduli spaces. This metric is not positive-definite in cases of non-compact gauge groups.
The Abelian Higgs model at critical coupling admits static solutions modeling vortices which neither attract nor repel each other . The mathematical content of the model consists of a Hermitian complex line bundle over a Riemannian surface , together with a connection and a complex Higgs field satisfying the Bogomolny equations
Here is the Kähler form on , is the covariant –operator (the anti-holomorphic part of the covariant derivative defined by ), is the Abelian Maxwell field and is the anti-holomorphic part of . Setting , and solving the first Bogomolny equation for the one–form reduces the second equation to the Taubes equation 
which is valid outside small discs enclosing the logarithmic singularities of – the locations of vortices on . Here is the Laplace operator on .
In  Manton has considered a two–parameter generalisation of the Taubes equation
The constants and can be rescaled to or , and Manton has argued that only five combinations lead to non–singular vortex solutions
corresponds to the Taubes equation . The magnetic field decays to zero away from vortex center.
corresponds to the magnetic field with constant strength equal to . In  this was called the Bradlow vortex.
The aim of this paper is to show (Theorem 2.1 in §2.3) that Manton’s equation (1.2) for all values of arises as the symmetry reduction of the anti-self-dual Yang–Mills equations (ASDYM) on a four-manifold , where is a surface of constant curvature, and the symmetry group is the group of local isometries of . The value of in (1.2) is determined by the curvature of , and depends on the choice of the gauge group . We shall demonstrate that if , if , and if . The gauge group is if , if and the Euclidean group if . In the integrable cases the Gaussian curvatures of and add up to zero.
In §2.5 we shall show how the five vortex equations are related by a superposition principle which leads to a construction of vortices with higher vortex numbers.
The four-dimensional perspective allows us to derive a canonical expression for the resulting energy of vortices. By considering the kinetic energy of the dynamical Yang–Mills theory on in §3.1 we shall derive integral expressions for moduli space metrics corresponding to various choices of constants in (1.2). If the gauge group is non-compact, then the kinetic energy and the resulting moduli space metric are not positive definite and the moduli space may contain surfaces where they identically vanish. In the integrable cases of equation (1.2), the moduli space metric takes a simple form that generalises the one for integrable Taubes vortices described in .
We thank Nick Manton and Edward Walton for helpful discussions. F.C. is grateful to Cambridge Commonwealth, European and International Trust and CAPES Foundation Grant Proc. BEX 13656/13-9 for financial support. M. D. was supported by the STFC grant ST/L000385/1.
2. Equivariant anti-self-dual connections and symmetry reduction
In this Section we shall formulate the main Theorem. Let us first introduce some notation.
2.1. The group .
The key role will be played by a Lie group which consists of matrices such that
Therefore and – the Euclidean group. The generators of the corresponding Lie algebra ,
satisfy commutation relations
The explicit parametrisation of as well as the left-invariant one-forms are constructed in the Appendix.
In what follows, will denote the Lie group defined in the same way, but changing into .
2.2. The four-manifold
Let be the Riemannian four-manifold given by the Cartesian product with a product metric
where is the Riemann surface introduced in §1, and is a surface of constant Gaussian curvature . Let be a local holomorphic coordinate on , and be a local holomorphic coordinate on so that
where is the conformal factor on . The Kähler forms on and on are given by
We shall choose an orientation on by fixing the volume form .
Let and be Lie groups corresponding, via (2.3) to two real constants and . In the Theorem below will play a role of a gauge group, and will be the symmetry group.
Let be a vector bundle with a connection which, in a local trivialisation, is represented by a Lie-algebra valued one-form . The Lie group is a subgroup of the conformal group on , and acts on isometrically by
We shall impose the symmetry equivariance condition on : it is preserved up to a gauge transformation by the action (2.7) of . The infinitesimal equivariance condition yields
where is the Lie derivative, is any vector field on generating the action (2.7) and is the covariant derivative of a -valued function on .
In the coordinates introduced above we have
Let be –equivariant. Then
There exists a gauge such that
where is defined in (2.6), is a -valued one-form, and is a complex Higgs field on .
The ASDYM equations on are
where . Equivalently, setting ,
We shall split the proof of this theorem into two Propositions
Let be the maximal group of isometries of , where or . The most general –equivariant –connection is gauge equivalent to (2.9).
Proof of Proposition 2.2. Every vector field generating the action on corresponds to a -valued function on called the Higgs field. The symmetry group does not act freely on , which leads to a set of differential and algebraic constraints. These constraints are kinematical, as they arise purely from the symmetry requirement, and do not involve the ASDYM equations. Our analysis of the constraints, and the construction of follows [9, 13], and we refer the reader to these works for details. Another method is presented in .
Consider as the homogeneous space, where stabilises a point with coordinates in . This action is generated by defined in (2.4), and corresponds to a vector field on . The constraint equations are
where the vector fields generate the action, and we set . The map is a homomorphism from to , and equations (2.12) imply that the –components of the potential belong to the image of this homomorphism in the gauge Lie algebra, i.e. . We also find that the components are functions on which do not depend on the coordinates on . Therefore we have arrived at the Abelian Maxwell potential on given by
To solve (2.12) for the Higgs fields, we chose a gauge such that is diagonal.
This is always possible if as Hermitian matrices are diagonalisable by unitary changes
of basis, but needs to be verified by a direct calculation which uses the constraint equations
. In general we find , where . The choice of
amounts to a choice of an orientation
where is a complex valued function on which we shall later identify with the Abelian Higgs field in the vortex equation.
We shall now construct the remaining part of the gauge potential – its components on the surface . Following  define a gauge potential on by
where are right invariant one-forms on given by (Appendix. The group ). In what follows, we use coordinates for described in the Appendix.
We can perform a gauge transformation on such that
where are right-invariant vector fields on such that , and . This yields a gauge potential defined on the quotient , where the subgroup is generated by . It is given by
Notice that and above become hyperbolic functions for . This one-form Lie derives along and thus we can use the diffeomorphism between and to pull back to . This gives , and finally .
A direct calculation shows that this gauge potential is indeed -equivariant. Indeed, it satisfies (2.8) where the vector fields are push forwards of the left-invariant vector fields (Appendix. The group ) on by the projection , and
In order to obtain a neater final expression, perform another gauge transformation by and change into -coordinates using the local coordinate formula (A1) to find the –equivariant –gauge potential on given by (2.9).
Proof of Proposition 2.3. Let be holomorphic coordinates on . The basis of self-dual two forms is spanned by
The ASDYM equations , and take the form
The components of the gauge field are given by
where , , is the covariant derivative with respect to the -connection and
The ASDYM equations (2.13) yield vortex-type equations
This system of non–linear PDEs can be reduced to a single second order equation for one scalar function. In fact, solve the first equation (2.14) for so that and, using the reality of , . Using these expressions for the components of , calculate the Abelian Maxwell field and the second equation (2.15) yields (2.11).
2.4. Integrable cases
Following the integrability dogma [15, 7, 3], a symmetry reduction of ASDYM is integrable if the ASDYM equations are defined on a background with anti-self-dual Weyl curvature. Computing the Weyl tensor of (2.5) shows that conformal anti-self-duality is equivalent to the vanishing of the scalar curvature of . Thus in the integrable cases the Riemann surface on which the vortex equations are defined must have constant Gaussian curvature equal to minus the Gaussian curvature of , i.e. locally,
The local solutions of integrable vortex equations are given explicitly, in a suitable gauge, by 
where is a holomorphic map from to a surface of curvature . The vortices are located at zeros of , which are the zeros of and the poles
The integrable cases on simply-connected Riemann surfaces under the anti-self-duality framework are the following:
The Taubes vortex () is integrable on , in which case it is a symmetry reduction from ASDYM on . In this case, is a Blaschke function
where . This is the original integrable reduction of Witten .
The Popov vortex () is integrable on , in which case it is a symmetry reduction from . In this case, is a rational function , where and are polynomials of the same degree with no common root.
The Bradlow vortex (, ) is integrable on , in which case it is a symmetry reduction from .
The Ambjørn–Olesen vortex (, ) is integrable on , in which case it is a symmetry reduction from .
The Jackiw–Pi vortex (, ) is integrable on , in which case it is a symmetry reduction from .
In each case the symmetry group is and the gauge group is .
2.5. Superposition of vortices
Given a solution to the vortex equation (1.2) define the vortex number to be
This is an integer equal to the first Chern number of the vortex line bundle , and we shall assume that this integer is non–negative.
Let us now explain why there exist only five vortex equations among the nine possible combinations of values of and . Equation (2.14) implies that the vortex number coincides with the number of zeros of counted with multiplicities . Since is holomorphic, is necessarily non-negative. Since is proportional to the magnetic flux, the magnetic field can not be negative everywhere on , but (2.15) implies that this is only possible for the choice of constants
where the last possibility means that the magnetic field is null everywhere and (1.2) is the Laplace equation. We shall call this the Laplace vortex.
The resulting six equations are not disconnected from one another. We shall show that it is possible to construct higher order vortex solutions of one type by superposing two other types of vortices. Let be a vortex solution on with vortex number satisfying
so that has isolated zeros, counting multiplicities. We say that this vortex is of type .
Consider a metric on
which has conical singularities at zeros of . Let be a vortex solution with vortex number on satisfying
and that has zeros. Therefore the superposition of a vortex of type with a rescaled metric on a vortex of type gives rise to a vortex of type with vortex number being the sum of the first two vortex numbers, . In the case this is the Baptista superposition rule .
Taking into account the six possible vortex equations, we make a list of all possible superpositions
where the non-commutative summation means that the first vortex (of type with a rescaled metric) is superposed on the second one (of type ) to result in the vortex on the right hand side of the equality (of type ) with higher vortex number.
3. Energy and Moduli Space Metric
The energy functional of pure Yang-Mills theory on can be reduced to the energy function of an Abelian-Higgs model on using the ansatz (2.9). This can be done by direct calculation,
is the magnetic field on . This expression for the energy is proportional to the one given in .
If we assume that is compact
where is the genus of and we normalised the area of the torus () to . Thus the energy can be written, using the Bogomolny argument along with and the integration by parts (with an additional boundary condition if is not compact) as
where is the vortex number defined by (2.18). If the vortex equations (2.14) and (2.15) are satisfied, then the energy is proportional to the vortex number, characterising a non-interacting theory, and this value is the global minimum of if . However, equation (2.14) cannot be naturally derived from this argument if . In fact the theory corresponding to this energy functional does not involve any Higgs field in this case even though the symmetry reduction of ASDYM necessarily gives rise to a holomorphic Higgs field satisfying (2.14). This is a counter-example to the principle of symmetric criticality, proved under certain conditions in .
If is not compact we can still make sense of energy density (or energy per unit of area of ).
Originally, is the energy functional of pure Yang-Mills theory in four dimensions, which under the ASDYM condition is
where is the instanton number. Comparing both expressions for the energy, we derive a relation between the vortex and instanton numbers,
3.1. Dynamical theory
Yang–Mills instantons on can be regarded as static solitons on with a product metric . Implementing the symmetry reduction of Theorem 2.1, but from five dimensions, leads to vortices on interpreted as stationary solitons in a dynamical theory on . We shall use this approach to find the kinetic term in the total energy functional on , and use it to read-off the metric on the moduli space of static vortices. Let be a –valued Yang–Mills field on . The action functional of pure YM theory with -dependence is
where is defined by the second equality and involves the integral on alone. Under the symmetry reduction of Theorem 2.1, becomes a Lagrangian on ,
The Euler-Lagrange equations, resulting from calculating the variation with respect to and , are, respectively,
This system of second order dynamical equations is satisfied by static solutions to the first order vortex equations (2.14–2.15) in the temporal gauge . In fact, Gauss’ law (3.23) is automatically satisfied. To see that (3.21) is satisfied, use (the complex conjugate of) (2.14) to write and eliminate with (2.15). Finally, equation (3.22) is satisfied upon eliminating in the same way and using .
The kinetic energy can be read off from . In the temporal gauge when it takes the form
where the dots denote -derivatives. This generalises the known kinetic energy for the Taubes vortex () .
In the usual Abelian Higgs model in the critical coupling (yielding Taubes vortices), there is a moduli space of static vortex solutions. All these solutions have the same potential energy, so there are no static forces. The moduli space acquires a metric from the kinetic energy of the theory, and the geodesics of this metric model slow motion of -vortices. There are several ways to obtain the metric for both flat and curved backgrounds [13, 6]. One way to proceed for the integrable vortices is to assume that the vortex positions depend on time, and substitute the explicit solution (2.17) into the kinetic energy (3.24). This, when quotiented out by the gauge equivalence (which is equivalent to imposing Gauss’ law ), gives a quadratic form on . In case of Taubes vortices the resulting metric is positive definite, but we see that (3.24) is not positive definite if , which is the case for Jackiw–Pi, Popov and Ambjørn–Olesen vortices.
Samols derived a localised expression for the metric of the moduli space of Taubes vortices  (see also  and  for the metric of moduli space of hyperbolic vortices), where the moduli are the vortex positions (or zeros of the Higgs field). The moduli space metric of Taubes vortices with simple zeros is, from (3.24),
where and means that the conformal factor is being evaluated at the point .
A similar calculation as the one performed by Samols (cf. also ) adapted to vortices defined by (1.2) gives the following result for the metric on the moduli subspace associated to the simple zeros of the Higgs field
where the constant appears because the calculation involves the use of equation (1.2).
In the integrable case, the background metric is locally given by in (2.16) and the Higgs field is (2.17) in a particular gauge. As above, we assume that admits only simple zeros and that each of them is as zero of and not a pole, which is always the case up to performing the transformation . In this case, can be calculated directly and is given by
where and , , that is to say, the ’s are defined by
where the linear term is absent because we assumed that is a simple zero of .
Now, the moduli space metric (3.25) of simple vortices can be calculated and is given by
This is in agreement with the formula derived for integrable Taubes vortices in . In particular, (3.26) tells us that the metric is zero when depends only holomorphically on the vortex positions, which is the case for some Popov and Jackiw–Pi vortices. In fact, this is what happens for the Popov vortex on corresponding to , and to the Jackiw–Pi vortex on corresponding to .
Appendix. The group
In this Appendix, we denote by , the generators of the Lie-algebra given by (2.4) with replaced by .
A parametrisation of is given by