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Integrability of Vortex Equations on Riemann Surfaces

The Abelian Higgs model on a compact Riemann surface \Sigma of genus g is considered. We show that for g>1 the Bogomolny equations for multi-vortices at critical coupling can be obtained as compatibility conditions of two linear equations (Lax pair) which are written down explicitly. These vortices correspond precisely to SO(3)-symmetric Yang-Mills instantons on the (conformal) gravitational instanton \Sigma\times S^2 with a scalar-flat Kahler metric. Thus, the standard methods of constructing solutions and studying their properties by using Lax pairs (twistor approach, dressing method etc.) can be applied to the vortex equations on \Sigma. In the twistor description, solutions of the integrable vortex equations correspond to rank-2 holomorphic vector bundles over the complex 3-dimensional twistor space of \Sigma\times S^2. We show that in the general (nonintegrable) case there is a bijection between the moduli spaces of solutions to vortex equations on \Sigma and of pseudo-holomorphic bundles over the almost complex twistor space.


Introduction
The Abelian Yang-Mills-Higgs model on R × R 2 at critical value of the coupling constant (the Bogomolny regime) admits static finite energy vortex solutions [1,2] whose existence on R 2 was proved by Taubes [3,4]. They describe magnetic flux tubes (Abrikosov strings [1,2]) penetrating a two-dimensional superconductor. Stability of vortices is ensured by topology [5]. The Taubes results for critically coupled vortices were generalized to compact Riemann surfaces Σ [6,7]. In these investigations the main attention was devoted to study of Kähler geometry of the moduli space of vortices on Σ and to dynamics of vortices in Manton's adiabatic approximation (see e.g. [8]- [12]).
Recall that besides proving the existence theorem, Taubes has also shown that the standard vortex equations on R 2 are equivalent to SO(3)-symmetric SU(2) self-dual Yang-Mills (SDYM) equations on R 2 × S 2 [4]. In other words, there is a one-to-one correspondence between SO(3)equivariant 1 instantons on R 2 × S 2 for the gauge group SU (2) and vortices in the Abelian Higgs model 2 on R 2 . This correspondence works also for vortices on any compact Riemann surface Σ of genus g: there is an equivalence between vortices (A, φ) on Σ and SO(3)-equivariant Yang-Mills instantons A on Σ × S 2 , with the vortex number N (first Chern number of the U(1) connection A on the Hermitian line bundle E over Σ) equals the instanton number (the minus second Chern number of the symmetric connection A on the rank-2 vector bundle E over Σ × S 2 ) [15]. We describe this correspondence in explicit form following [16]. In fact, we consider the pure SU (2) Yang-Mills action on Σ × S 2 and show how it reduces to the action of the Abelian Higgs model on Σ. Then from the standard Bogomolny argument it follows that for a fixed vortex number N ≥ 0 the minimum of the action functional is governed by solutions of the first-order vortex equations on Σ.
The above correspondence can be advanced further. Namely, recall that to each oriented Riemannian 4-dimensional manifold M one can associate a real 6-dimensional manifold Z, the twistor space of M , which has a canonical almost complex structure J [17]. For zero scalar curvature R M = R Σ + R S 2 of M = Σ × S 2 , i.e. for 3 where R is the radius of S 2 , the almost complex structure J is integrable and Z becomes a complex 3-dimensional manifold [18]. Then we can pull back the instanton bundle E over M to a holomorphic bundleÊ over Z [17,21]. We show that in this (integrable) case the vortex equations on Σ can be obtained as the compatibility conditions of linear differential equations defining holomorphic sections of the bundleÊ → Z. Thus, we extend the correspondence between vortices on Σ and Yang-Mills instantons on Σ × S 2 further and show that for g > 1 the vortex equations 4 on Σ are equivalent to the equations defining an integrable holomorphic structure on the smooth complex vector bundleÊ over the complex 3-dimensional twistor space Z.

1
The picture is different for the cases Σ = S 2 (g = 0) 5 , Σ = T 2 (g = 1), Σ with g > 1 and R Σ = −R S 2 as well as for the noncompact case of R 2 . In all these cases an almost complex structure on the twistor space Z of M = Σ × S 2 is not integrable as well as an almost complex structure on the vector bundleÊ → Z which is the pull-back of an instanton bundle E → M . We argue thatÊ with the pulled-back connectionÂ = π * A, for π : Z → M , has the structure of Bryant's pseudo-holomorphic bundle [24] with the curvatureF of type (1,1). Furthermore, we show that the vortex equations on Σ (including the case of R 2 ) are equivalent to the SU(2)-equivariant pseudo-Hermitian-Yang-Mills equations on the almost complex twistor space Z. However, the vortex equations in all these cases cannot be written as compatibility conditions of linear (Lax) equations. In the particular case of g > 1 and equality (1.1), all almost complex structures become integrable, Lax pair appears and the prefix "pseudo-" disappears.

Scalar-flat gravitational instantons and twistors
Here we consider Kähler metrics on a product M = Σ 1 × Σ 2 of two Riemann surfaces of genera g 1 and g 2 and the twistor description of such four-dimensional Riemannian manifolds.
Riemann surfaces. Consider a compact Riemann surface Σ of genus g ≥ 0 with the metric and the volume form given in local (conformal) coordinates w,w by and Since Σ is a Kähler manifold, it follows that Γ w ww = 2 ∂ w log ρ and Γw ww = 2 ∂w log ρ with ρ 2 := g ww , where Γ w ww and Γw ww are nonvanishing components of the Christoffel symbols and R ww is a component of the Ricci tensor.
For the scalar curvature of Σ we have (2.5) Note that for genus g = 1 the Riemann surface Σ has area equal to which follows from (2.5) and the Gauss-Bonnet theorem.
Four-manifold M . Let us consider a smooth oriented real four-dimensional manifold M given by a product of two Riemann surfaces of genera g 1 and g 2 , with the product metric ds 2 M = ds 2 1 + ds 2 2 = 2 g y 1ȳ1 dy 1 dȳ 1 + 2 g y 2ȳ2 dy 2 dȳ 2 (2.8) written in local complex coordinates on Σ 1 and Σ 2 , respectively.
We consider a principal bundle P = P (M, SO(4)) of orthonormal frames on a Riemannian 4-manifold M . For M = Σ 1 × Σ 2 the holonomy group is reduced to U(1)×U(1) ⊂ U(2) ⊂ SO(4) and for the components g y 1ȳ1 = 1/g y 1ȳ1 and g y 2ȳ2 = 1/g y 2ȳ2 of inverse metric we have g y 1ȳ1 = e y 1 1 eȳ 1 1 and g y 2ȳ2 = e y 2 2 eȳ 2 2 , (2.9) where e y 1 1 , . . . , eȳ 2 2 are unitary (local) frame fields. We also introduce a basis of type (1,0) and (0,1) vector fields where e 1 y 1 etc. are inverse to e y 1 1 etc., i.e. e 1 y 1 e y 1 1 = 1 etc. Introducing ρ 2 1 := g y 1ȳ1 (y 1 ,ȳ 1 ) and ρ 2 2 := g y 2ȳ2 (y 2 ,ȳ 2 ) , (2.12) we obtain Γ y 1 y 1 y 1 = 2 ∂ y 1 log ρ 1 , Γ y 2 y 2 y 2 = 2 ∂ y 2 log ρ 2 , Γȳ 1 y 1ȳ1 = 2 ∂ȳ 1 log ρ 1 , Γȳ 2 y 2ȳ2 = 2 ∂ȳ 2 log ρ 2 , (2.13) R y 1ȳ1 = −∂ y 1 ∂ȳ 1 log ρ 2 1 = κ 1 g y 1ȳ1 and R y 2ȳ2 = −∂ y 2 ∂ȳ 2 log ρ 2 2 = κ 2 g y 2ȳ2 (2.14) with all other components vanishing. For the scalar curvature of M we have R M = 2g y 1ȳ1 R y 1ȳ1 + 2g y 2ȳ2 R y 2ȳ2 = 2(κ 1 + κ 2 ) . Equivalent definition of the twistor bundle (2.17) can be obtained by considering the vector bundle Λ 2 M of two-forms on M , associated to the principal bundle P . Using projectors on the subspaces of self-dual Λ 2 + and anti-self-dual Λ 2 − two-forms, one can split Λ 2 M into the direct sum Λ 2 M = Λ 2 + M ⊕Λ 2 − M of subbundles of self-dual and anti-self-dual two-forms on M . Then the twistor space can be introduced as the unit sphere bundle Z = S 1 (Λ 2 − M ) in the vector bundle Λ 2 − M . That is why the Levi-Civita connection on P determines that on Λ 2 M and induces a connection on Z which is the anti-self-dual part Γ − = (Γ i − ), i = 1, 2, 3, of the Levi-Civita connection [17]. This connection generates the splitting of the tangent bundle T Z into the direct sum T Z = H ⊕ V , where V = Ker π * is the subbundle of vectors tangent to fibres CP 1 , and H ∼ = T Z/V consists of vectors horizontal with respect to the connection Γ − on Z. According to the canonical definition, the horizontal liftX of any vector field X on M is defined as where X Γ i − denotes the interior product of a vector field and one-form, and L i 's are vector fields on fibres CP 1 ֒→ Z which give a realization of the generators of the group SU(2). By construction, X is a section of the bundle H → Z, X is a section of T M and π * X = X.
We lift our frame vector fields (2.10) to Z, switch to a complex basis by taking holomorphic parts of vector fields L i and introduce type (0,1) vector fields on Z as V1 :=ẽ1 − λẽ 2 , V2 :=ẽ2 + λẽ 1 and V3 := ∂λ , (2.19) where in local complex coordinates y 1 , y 2 , λ on Z (λ ∈ CP 1 ֒→ Z) we havẽ Almost complex structure on Z. The vector fields (2.19) define an almost complex structure J on Z such that J (Vk) = −iVk (2.21) for k = 1, 2, 3. In fact, the almost complex structure J defined as above on the twistor space Z of M = Σ 1 ×Σ 2 is canonical and does not depend on the choice of local coordinates [17]. For commutators of type (0,1) vector fields (2.19) we have Recall that for integrability of an almost complex structure J on Z it is necessary and sufficient that the commutator of any two vector fields of type (0,1) w.r.t. J is of type (0,1). For our case we see from (2.22) that J is integrable -and Z is a complex manifold -if and only if i.e when the scalar curvature (2.15) of the Kähler manifold Σ 1 × Σ 2 vanishes. Besides the case g 1 = 1 = g 2 (tori) this can be satisfied when one takes g := g 1 ≥ 2 and g 2 = 0 (two-sphere).
3 Vortices on Σ as Yang-Mills instantons on Σ × CP 1 Riemann sphere. Let Σ 2 = CP 1 ∼ = S 2 be the standard two-sphere of constant radius R. In local coordinates on CP 1 the metric reads Note that (3.2) corresponds to the choice of orientation on S 2 inverse to the canonical one, i.e.
That is why for the volume form we have i.e. ω 2 has the inverse sign in comparison with (2.2).
For the two-sphere and we haveẽ which agrees with the general formula (2.6).
Monopole bundles. Consider the Hermitian line bundle L → CP 1 (one-monopole bundle) with a unique SU(2)-equivariant unitary connection Then on the bundle L n := (L) ⊗n with n ∈ Z we have 6 a (n) = n a and f (n) := da (n) = − nR 2 (R 2 + yȳ) 2 dy ∧ dȳ = n 2i R 2 ω 2 . (3.8) The topological charge of this gauge field configuration is given by the first Chern number (equivalently the degree) of the complex line bundle L, This configuration describes |n| Dirac monopoles (n > 0) or antimonopoles (n < 0) sitting on the top of each other. The explicit form of such a connection is known (see e.g. [13,15,16]); it has the form where A is an Abelian connection on the (Hermitian) complex line bundle E over the genus g compact Riemann surface Σ, a is the monopole connection (3.7) on the line bundle L → CP 1 , φ is a section of the bundle E,φ is its complex conjugate and are forms on CP 1 of type (1,0) and (0,1) given by (2.11). In local complex coordinates z := y 1 = x 1 + ix 2 andz :=ȳ 1 = x 1 − ix 2 , we have A = A(z,z) and φ = φ(z,z).
Field strength tensor. In local complex coordinates on Σ × CP 1 the calculation of the curvature F for A of the form (3.10) yields with the non-vanishing field strength components where In (3.14) we have defined Reduction of the Yang-Mills functional. The dimensional reduction of the Euclidean Yang-Mills equations from Σ × CP 1 to Σ can be seen at the level of the Yang-Mills Lagrangian. Substituting (3.14)-(3.16) into the standard Yang-Mills functional and performing the integral over CP 1 , we arrive at the action where ω Σ ≡ ω 1 = i g zz dz ∧ dz (3.19) and x µ are real local coordinates 7 on M = Σ × CP 1 , µ, ν, ... = 1, ..., 4.
For the functional (3.18), using the standard Bogomolny arguments [27], one can obtain the inequality 20) where is the vortex number. In the derivation of (3.20) it is assumed that N ≥ 0 and similar inequality can be obtained for N ≤ 0. Thus, the (minus) second Chern number of the SU(2)-equivariant connection A on the rank-2 bundle E over Σ × CP 1 equals to the first Chern number of the connection A on the line bundle E over Σ.

Vortex equations on Σ. If N ≥ 0, then (3.20) is an equality if and only if the Yang-Mills field
and substitution of (3.14)-(3.16) shows that Eqs. (3.22) are equivalent to the vortex equations on Σ, Note that the case N ≤ 0 corresponds to the anti-self-dual Yang-Mills equations * F = −F which reduce to the anti-vortex equations and we obtain (cf. [6]) the inequality For g = 1 we have N ≤ Vol(T 2 )/2πR 2 . Recall that in our derivation of the vortex equations (3.23) the parameter κ = 1 2 R Σ is not fixed. In particular, one can take g = 0 and g = 1 and obtains vortices on the sphere S 2 or torus T 2 .

Twistor description of vortex equations
In Section 3 we discussed the relation between vortices on Riemann surfaces Σ and Yang-Mills instantons on the manifolds Σ × CP 1 . Here, we push this correspondence further and show that to any solution (A, φ) of vortex equations on Σ there corresponds a connectionÂ defining a pseudoholomorphic structure on a rank-2 complex vector bundleÊ over an almost complex twistor space Z for Σ × CP 1 . We also discuss the integrability of this pseudo-holomorphic structure and the corresponding vortex equations.
Pull-back to the twistor space. Consider an SU(2)-equivariant rank-2 complex vector bundle E → Σ × CP 1 with a connection A described in Section 3. Using the projection (2.17), we pull E back to a bundleÊ := π * E over Z: In accordance with the definition of the pull-back, the connectionÂ := π * A onÊ is flat along the fibres CP 1 x of the bundle π : Z → M and we can set the componentsÂ λ andÂλ of the restriction ofÂ to CP 1 x ֒→ Z equal to zero. Thus, restrictions of smooth vector bundleÊ to fibres CP 1 x of the projection π are naturally holomorphic and holomorpically trivial for each x ∈ M = Σ × CP 1 .
Note that a lift of the generators of the group SU(2) acting on CP 1 = SU(2)/U(1) to Z can be defined analogously to the flat Euclidean case [28]. Namely, we lift the SU(2)-generators to Z so that an almost complex structure on Z is invariant under the action of the group SU(2) generated by these lifted vector fields. Then one can impose SU(2)-equivariance conditions on the vector bundleÊ → Z and connectionÂ onÊ.
Connection along T 0,1 Z. Recall that the twistor space Z of M has a canonical almost complex structure J [17]. In the considered case it is defined by the type (0,1) vector fields (2.19). Hence, we can introduce a (0,1) part∇ 0,1 of the covariant derivative∇ onÊ by formulaê where∇ X denotes the covariant derivative along the vector field X. The components A 1 etc. can be extracted from (3.10), where A 1 = e z 1 A z , A1 = ez 1 Az, A 2 = e y 2 A y , A2 = eȳ 2 Aȳ and similarly for A 1 , A1 and a 2 , a2.
Pseudo-holomorphic bundles. Let us consider an almost complex manifold (Q, J ) and a complex vector bundle V over Q endowed with a connectionÂ. According to Bryant [24], a connection A on V is said to define a pseudo-holomorphic structure on V if it has curvatureF of type (1,1). In principle, one can also endow V with a Hermitian metric and chooseÂ to be compatible with the Hermitian structure on V [24]. If, in addition, η is an almost Hermitian metric on (Q, J ) and tr η (F ) = iγ IdÊ with γ ∈ R , (4.5) the connectionÂ is said to be pseudo-Hermitian-Yang-Mills. Here, we will consider the equations 8 (4.5) with γ = 0.
Consider the SU(2)-equivariant complex vector bundleÊ = π * E from (4.1) with a connection A = π * A. In [17], it was shown that pulling-back a real structure τ : Z → Z toÊ, one can endoŵ E with a Hermitian structure. Then pseudo-holomorphicity of the bundleÊ is equivalent to the equationsF where∇ Vī for i = 1, 2, 3 are given in (4.2) and (4.3).
Pseudo-holomorphicity and instantons. After simple calculations we see that the only nontrivial components of the tensorF 0,2 readŝ Thus, we see that equations (4.6) on Z, defining pseudo-holomorphic structure onÊ, reduce to the equationF which is equivalent to the SDYM equations on Σ × CP 1 , Fzȳ = 0 , g zz F zz + g yȳ F yȳ = 0 and F zy = 0 . (4.10) Conversely, every pseudo-holomorphic SU(2)-equivariant vector bundleÊ over Z such that it is holomorphically trivial on each CP 1 x ֒→ Z, x ∈ Σ × CP 1 , is the pull-back to Z of an SU(2)equivariant bundle E with a self-dual connection on Σ × CP 1 . 9 Recall that equations (4.10) are equivalent in turn to vortex equations on Σ as discussed in Section 3.
Pseudo-Hermitian-Yang-Mills equations on Z. One can check by direct calculation that the connectionÂ = π * A satisfies the equationŝ which also reduce to the SDYM equations (4.10). Together, (4.9) and (4.11) constitute pseudo-Hermitian-Yang-Mills equations on Z. However, in our concrete case of twistor space Z equation (4.11) does not impose additional restrictions onÂ in comparison with (4.9). This is in conformity with [24]. Integrable case. In the general case of the twistor space Z of M = Σ × CP 1 , the subbundle T 0,1 Z of T C Z is not integrable since the distribution of vector fields of type (0,1) is not closed under the Lie bracket. However, for Σ with genus g ≥ 2, one can always rescale the metric on Σ to fulfill the condition In this case the almost complex structure J on the twistor space Z of M becomes integrabledistribution of (0,1) vector fields (2.19) is closed under the Lie bracket -and the pseudo-holomorphic bundleÊ → Z becomes holomorphic. In other words, the almost complex structure onÊ defined by (0,1)-type connection (4.2) and (4.3) becomes integrable and one can introduce holomorphic sections ofÊ.
Linear system. Recall that a (local) section χ of the complex vector bundleÊ is said to be holomorphic if∇ Accordingly, the bundleÊ → Z is said to be holomorphic if equations (4.14) are compatible, i.e. the (0,2) components of the curvatureF of the bundleÊ vanish. This condition yields equations (4.6) (and (4.10)) but now with vector fields Vī which are closed under the Lie bracket.
Let us now introduce a 2 × 2 matrix ψ of fundamental solutions of eqs. (4.14), i.e. such that columns of ψ form smooth frame fields forÊ. Then we obtain two linear equations (Lax pair) since the third linear equation∇ V3 ψ = 0 is trivially solved for ψ which does not depend onλ. Here λ ∈ U + = CP 1 \{∞} ⊂ CP 1 .
Explicit Lax pair. In our case, the explicit form of the Lax pair (4.15) reads are equivalent to the vortex equations (3.23) on Σ. Note that for span{Vī} closed under the Lie bracket, (4.17) is equivalent to (4.6) but this is not true for the nonintegrable distribution T 0,1 Z.
Riemann-Hilbert problems. Let us consider an open subset U of the manifold Σ × S 2 , restrict Z | U ∼ = U × CP 1 and consider a two-set open covering {U + , U − } of the fibre CP 1 in π : U × CP 1 → U, U + = CP 1 \{∞} and U − = CP 1 \{0}. Then the restriction ofÊ to Z | U will be described by a transition 2× 2 matrix f +− on U × U + ∩ U − , holomorphic in λ, whose restriction to U + ∩ U − ֒→ Z | U is splitted, the vortex equations on Σ written in the form (4.17) will have no solutions. However, one can rescale the metric on Σ, g zz → t 2 g zz , and obtain inequalities For any N the condition (4.22) can be satisfied for sufficiently large t and then the moduli space of vortices will be nonempty. However, this rescaling of metric on Σ leading to (4.21) brings us back to the case of pseudo-holomorphic bundlesÊ over Z, i.e. to the case of nonintegrable almost complex structures. This is the price to be paid for having nonempty vortex moduli space when N > 2(g − 1). The value N = 2(g − 1) of the vortex number separates the holomorphic and nonholomorphic cases.

Conclusions
In this paper, we have shown the equivalence between vortex equations on a compact Riemann surface Σ, the Yang-Mills instanton equations on Σ × CP 1 and pseudo-Hermitian-Yang-Mills equations on the twistor space Z of Σ × CP 1 , summarized in the diagram (4.12). We have shown that in a special case, when the twistor geometry becomes integrable (holomorphic), the vortex equations on Σ appear as the commutator (4.17) of two auxiliary linear differential operators (with a 'spectral' parameter) having clear geometric sense. This brings us to a situation when one can, in principle, apply methods of integrable systems to finding solutions of vortex equations.
We considered vortices in the Abelian Higgs model. There are various non-Abelian generalizations of this model (see e.g. [30]- [35] and references therein). Results of this paper can be extended to the non-Abelian case. The simplest generalization can be obtained if one replaces the scalar function φ in (3.10) by p × q matrix and substitutes 1 p ⊗ a and −1 q ⊗ a for 1 ⊗ a and −1 ⊗ a, respectively. We will consider this case in more detail elsewhere.
The constructions of this paper can also be generalized to N -extended supersymmetric SDYM theory [36,37] defined on Σ × CP 1 together with a reduction to supersymmetric vortex equations on Σ. In this case vortices on Σ will correspond to supersymmetric Yang-Mills instantons in the background of gravitational instantons Σ × CP 1 of conformal (super)gravity. It is of interest since conformal supergravity interacting with Yang-Mills supermultiplets arises in the twistor string theory proposed recently [38,39]. Vortices on Σ represent the simplest and very natural interaction of Yang-Mills instantons and gravitons in conformal gravity which cannot be reduced to Einstein gravity. These Yang-Mills/gravity configurations could be a good test background for calculation of open/closed twistor string amplitudes especially because the moduli spaces of Riemann surfaces and vortices are studied very well. It would be also interesting to look at our vortices from the cosmic string perspective.