Instanton Solutions from Abelian Sinh-Gordon and Tzitzeica Vortices
DDAMTP-2014-84
Instanton Solutions from Abelian Sinh-Gordonand Tzitzeica Vortices
Felipe Contatto , , Daniele Dorigoni CAPES Foundation, Ministry of Education of Brazil,Bras´ılia - DF 70040-020, Brazil. Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K.
Email: [email protected], [email protected]
February 2015
Abstract
We study the Abelian Higgs vortex solutions to the sinh-Gordon equation and the ellipticTzitzeica equation. Starting from these particular vortices, we construct solutions to theTaubes equation with higher vortex number, on surfaces with conical singularities.We then, analyse more general properties of vortices on such singular surfaces and pro-pose a method to obtain vortices on conifolds from vortices on surfaces of revolution. Weapply our method to construct explicit vortex solutions on the Poincar´e disk with a conicalsingularity in the centre, to which we refer as the “hyperbolic cone”.We uplift the Abelian sinh-Gordon and Tzitzeica vortex solutions to four dimensionsand construct cylindrically symmetric, self-dual Yang-Mills instantons on a non-self-dual(nor anti-self-dual) 4-dimensional K¨ahler manifold with non-vanishing scalar curvature. Theinstantons we construct in this way cannot be obtained via a twistorial approach. a r X i v : . [ h e p - t h ] S e p Introduction
The Abelian Higgs model is a gauge field theory on a 2 + 1-dimensional manifold, Σ × R , where R parametrizes the time and Σ is a surface with Riemannian metric. This model describestype I and type II superconductors [1] depending on the value of the coupling constant, whosecritical value separates both types of superconductivity. For example, the model in a non-planar geometry is physically relevant in describing thin superconductors of curved shape. Inthe critically coupled regime, the theory admits topological solitons called vortices, finite energysolutions to the Bogomolny equations [2].The Bogomolny equations are not integrable in general, and we do not have an analytic formfor the vortex profile function. Only in few lucky cases we can construct explicit solutions. Inparticular [3, 4, 5], when Σ is a Riemann surface of constant Gauss curvature − , the Bogomolnyequations reduce to the Liouville equation and analytic solutions can be found [6]. Similarly,when the conformal factor on Σ is of a very particular type, with a conical singularity at theorigin, the Bogomolny equations reduce either to the sinh-Gordon or to the Tzitzeica equation[7]. Solutions to these equations are not known in explicit forms, but since the radial reductionsof the sinh-Gordon and the Tzitzeica equations are both special cases of Painlev´e III ODEs, wecan obtain an explicit asymptotic expansion for the solutions close to the vortex centre and faraway from it, without having to rely on numerical simulations [8, 9].Even if the vortex equations are non linear, it is still possible to superpose multiple vortexsolutions [10]. This non-linear superposition rule allows us to obtain multi-vortex solutions ontop of the sinh-Gordon and Tzitzeica vortices by solving an auxiliary Taubes equation on adifferent conically singular manifold. Singular Abelian vortex equations arise naturally as aneffective tool to study vortex solutions that are invariant under the action of a symmetry groupor when other type of constraints are present [11].In [12], Popov proved that there is a one-to-one correspondence between vortices on Σ andcylindrically symmetric instantons on Σ × S . In particular, once we find an (anti-)vortex so-lution, we can find a solution to the (anti-)self-dual Yang-Mills, (A-)SDYM, equations on this4-dimensional manifold. This correspondence is a particular type of symmetry reduction ofSDYM or ASDYM [13, 14]. When the metric on Σ × S is K¨ahler and it has vanishing scalarcurvature, we can use the twistor transform to construct instanton solutions. In this case, the up-lifting of the vortex solutions correspond to rank-2 holomorphic vector bundles over the complextwistor space of Σ × S .We can apply this equivariant reduction to uplift the vortices arising from the sinh-Gordon andTzitzeica equations and obtain instanton solutions in four dimensions. An interesting aspect ofthese instantons is that their backgrounds are K¨ahler manifolds of non-vanishing scalar curvature(they are not “scalar-flat”) and thus the twistor transform cannot be used.The paper is organised as follows. In Section 2 we introduce the Abelian Higgs model, theBogomolny equations and present the vortex solutions from the sinh-Gordon and the Tzitzeicaequations. Thanks to the work of Baptista [10], in Section 3 we superpose additional vortices,in a non-linear way, on top of the sinh-Gordon and Tzitzeica solutions and construct multi-vortex solutions on particular conically singular spaces. We generalise the results of the first twosections to vortices on conifolds in Section 4 and, as an application of our analysis, we use theexplicit analytic expression for vortices on the Poincar´e disk to construct vortex solution on aconically singular hyperbolic space. Finally, in Section 6, we briefly review the correspondencebetween vortices and instantons and use it to obtain cylindrically symmetric solutions of theSDYM equation by uplifting the vortices constructed in the first part.2 Abelian vortices
Let us consider a Riemann surface Σ with metric written in isothermal coordinates ds = Ω( z, ¯ z ) dzd ¯ z, where z = x + iy is a local holomorphic coordinate. The function Ω : Σ → R + is called theconformal factor of the metric.Let L be a Hermitian complex line bundle over Σ. For a certain trivialization, the Abelian U (1) connection on L is given by A = A x dx + A y dy = A z dz + A ¯ z d ¯ z and its curvature is themagnetic field B = ∂ x A y − ∂ y A x = 2 i ( ∂ ¯ z A z − ∂ z A ¯ z ), where we defined ∂ z = ( ∂ x − i∂ y ) / , ∂ ¯ z =( ∂ x + i∂ y ) / A z = ( A x − iA y ) / , A ¯ z = ( A x + iA y ) / τ = 1, is V = i (cid:90) Σ dz ∧ d ¯ z (cid:20) Ω − B + D µ φD µ φ + Ω4 (1 − ¯ φφ ) (cid:21) , where the Higgs field φ is a smooth global section of L and D µ = ∂ µ − iA µ is the covariantderivative. The Bogomolny equations result from completing the square in the potential V .They are B = Ω2 (cid:0) − φ ¯ φ (cid:1) (1) D ¯ z φ = 0 . (2)Vortices are defined as finite energy solutions of (1)–(2).We set φ = e h + iχ , where h is a real function defined on Σ and the phase χ is a real functiondefined on each open patch depending on the gauge choice. We can calculate A z and A ¯ z = A z from (2) and substitute them in (1), yielding the Taubes equation∆ h + Ω (cid:0) − e h (cid:1) = 0 , (3)where ∆ = 4 ∂ z ∂ ¯ z is the flat Laplacian.The Higgs field φ vanishes at N isolated points { Z i } , the vortex locations, and precisely atthese points h possesses logarithmic singularities. The centres Z i are not necessarily distinct, thenumber of vortices, or the number of zeros of φ counted with multiplicity, is equal to the firstChern number of the bundle N = i π (cid:90) Σ dz ∧ d ¯ z B . (4)By integrating (1) over Σ, Bradlow [15] showed that on a surface of finite area, N is bounded by A Σ = i (cid:90) Σ dz ∧ d ¯ z Ω > πN , (5)where A Σ is the area of Σ. We can interpret (5) as saying that the effective area of a vortex is4 π . Whenever | φ | = 0, h has a logarithmic singularity and this implies that (3) is only valid awayfrom the zeroes { z i } of | φ | and Taubes equation should be corrected with delta-function sourcesas ∆ h + Ω (cid:0) − e h (cid:1) = 4 π N (cid:88) i =1 δ ( z − z i ) . (6)3or smooth and geodesically complete metrics, when all the vortices are located at the samepoint, i.e. z i = z for all i , h can be expanded around z = z as h ( z, z ) ∼ N log | z − z | + a ( z i , z ) + 12 b ( z , z )( z − z ) + 12 b ( z , z )( z − z )+ c ( z , z )( z − z ) + d ( z , z )( z − z )( z − z ) + c ( z , z )( z − z ) + · · · . (7)Apart from the leading logarithmic term, this expansion is a Taylor series in z − z and its conju-gate. The Taubes equation (6) requires that d ( z , z ) = − Ω( z , z ) /
4, but the other coefficientsshown here are not determined purely locally, but only from the complete 1-vortex solution.If we look for a circular symmetric solution of the form h = h ( r ), with r = | z | , then (7) allowsus to expand h ( r ) around r = 0 as h ( r ) ∼ N ln r + a − Ω(0)4 r + O ( r ) . (8)We underline that equation (8) is true only when the metric is smooth and geodesicallycomplete, in particular we require Ω(0) to be well-defined. In what follows we will give examplesof metrics with conical singularities at the origin r = 0; the vortex solutions to Taubes equationwith these singular conformal factors will have asymptotic series in the origin in fractional powersof r , also known as Puiseux series.We will be working mainly with surfaces of revolution, which admit z (cid:55)→ ze iϕ as a oneparameter group of isometries. We point out that since there is a unique solution to (6) once wefix all the vortex positions z i and since (6) is invariant under isometries of the manifold, vorticesat the origin of a surface of revolution are necessarily rotationally invariant. This translates theintuitive fact that, in a surface of revolution, there is no preferred radial direction.Noticing that h vanishes at r → ∞ , equation (6) reduces to a Bessel equation whose solutionhas the asymptotic behavior h ( r ) ∼ Λ √ r e −√ Ω as r , (9)where Ω as = lim r →∞ Ω and Λ is a constant, called the vortex strength.Given the two asymptotic forms (8) –(9), Taubes equation uniquely determines the constants a and Λ but, since generically an explicit solution is not known, they have to be computednumerically in most situations. In the flat case, Ω = 1, it is possible [16] to relate the twoasymptotic expansions and effectively reduce the problem of solving Taubes equation to a systemof transcendental algebraic equations relating a and Λ. A particularly special case is when Σ isa hyperbolic surface of constant Gauss curvature − , for which we can obtain an exact solutionto the Taubes equation by reducing the problem to a Liouville equation [3, 4, 5]. Other twointegrable cases, focus of the present work, are when the Taubes equation reduces to the sinh-Gordon or to the Tzitzeica equations [7]. When the conformal factor is chosen in a very peculiarway, we can reduce the radially symmetric Taubes equation to particular cases of Painlev´e IIIODEs, and even if an explicit solution is not known for all r , we can recover analytically the twoasymptotic forms (8)–(9) for h , and the connection formulas for a and Λ. In this Section, following [7], we describe how to reduce Taubes equation to the sinh-Gordonequation and present the solution in the two asymptotic regimes (8–9).Let us consider the vortex equations on the surface Σ = C whose metric, in isothermalcoordinates, has the conformal factor Ω = e − h/ and reads g Σ = e − h ( z, ¯ z ) / dzd ¯ z. ( h/
2) = sinh( h/ . (10)Looking for a solution with rotational symmetry, we assume that h depends only on theradial component r = | z | . This implies that the vortex position, the point where the Higgs field φ vanishes, must be the origin.An N sinh-Gordon vortex would be a solution to (10) with a logarithmic singularity, closeto r ∼
0, of the form h ∼ N log r , with N > h → r → ∞ . As in(6) the logarithmic singularity in h corresponds to the zero of | φ | and it adds a delta functionsingularity on the right hand side of (10).It is possible [7] to map equation (10) to a particular type of Painlev´e III ODE, with param-eters (0 , , , − h has to vanish for r → ∞ , together with the Painlev´e property [9], fix uniquelythe asymptotic forms of the solution. There is a unique solution to (10) yielding a vortex solution[7]. This sinh-Gordon vortex has N = 1 and for r ∼ h sG ( r ) ∼ r ) + 4 ln β sG − rβ sG + O ( r ) , (11)where β sG = 2 − / / / ≈ . β sG . The asymptoticsolution for r → ∞ is also uniquely determined h sG ( r ) ∼ − Λ sG K ( r ) , (12)where K n denotes the modified Bessel functions of the second kind, which decay exponentiallywith r precisely as (9), and the sinh-Gordon vortex strength is denoted by Λ sG = 8 λ ∼ . λ = √ π .We notice in the expansion of h sG close to the origin, a linear term in r , or equivalently | z | ,which should not be present according the expansion (8). The reason for this is the presence of apole in the conformal factor of the metric (13) at r ∼
0, the metric is not geodesically completeat the origin and the expansion (8) does not apply. The metric close to the origin takes the form g Σ ∼ rβ sG ( dr + r dθ ) . (13)The change ρ = √ r of the radial coordinate shows that, close to the origin, Σ possesses a flatmetric g Σ ∼ β sG ( dρ + 14 ρ dθ ) , which presents a conical singularity with deficit angle π .The cone is graphically visible by performing an isometric immersion of the surface Σ into R re iθ ∈ Σ (cid:55)→ ( X ( r, θ ) , Y ( r, θ ) , Z ( r, θ )) = (cid:16) √ r cos θ, √ r sin θ, √ r (cid:17) ∈ R whose image satisfies the equation of a cone with aperture π/ Z = (cid:112) X + Y ) . (14)Note that the linear term − r/β sG in (11), when expressed using the coordinate ρ = √ r ,takes precisely the form − ρ Ω(0) / ρ , for which the metric is flat and Ω(0) well defined and non-vanishing, that we canrecover (8) from (11). This situation will arise whenever our background metric has a conicalsingularity and we insist in inserting a vortex exactly at the tip of the cone: the right coordinatesto use are the ones for which the metric is flat, despite the angular variable not having periodicity2 π , in this way we will recover precisely the expansion (8), see Section 4 for more details.Far from the origin, the metric is perfectly smooth and takes the form g Σ ∼ e λK ( | z | ) dzd ¯ z , which means that for large r the metric is actually flat since for r → ∞ , K ( r ) ∼ (cid:112) π r e − r . A similar analysis of the Taubes equation can be carried along for vortices coming from theTzitzeica equation and analytic asymptotic solutions can be obtained in a similar fashion [7].Let us consider the vortex equations on the surface Σ = C whose metric has the conformalfactor Ω = e − h/ . The metric in isothermal coordinates reads g Σ = e − h ( z, ¯ z ) / dzd ¯ z. Once again the Riemannian background metric is fixed by the vortex profile function on thisparticular metric.In this case, (3) becomes the elliptic Tzitzeica equation∆ u + 13 (cid:0) e − u − e u (cid:1) = 0 , (15)where h = 3 u .A story similar to the one presented in the previous Section can be repeated for the Tzitzeicavortex [7]. We can map (15) to a Painlev´e III ODE, this time with parameters (1 , , , − h has a 2 N log r singularity, with N integer, and thenvanishes asymptotically for r → ∞ , together with the Painlev´e property [8], fix uniquely thesolution. As in the sinh-Gordon case, there is a unique Tizteica vortex, which also has vortexnumber N = 1, and, for r ∼
0, takes the asymptotic form h T T ( r ) = 3 u ( r ) ∼ r ) + β T T − e − β TT / r / + O ( r / ) , (16)where all the higher order terms are fixed in terms of β T T , which can be read off from equation(19) of [8]: β T T = 3 log (cid:20) − ν +1 / Γ( + ν )Γ( ν )Γ( − ν )Γ( − ν ) (cid:21) , where ν = 3 (cid:0) − pπ (cid:1) and p has to be set to 8 π/
9, so β T T ≈ . r (cid:29) h T T ( r ) ∼ √ π (cid:18) cos p + 12 (cid:19) K ( r ) = − Λ T T K ( r ) , (17)by substituting p = 8 π/ T T ≈ . h T T close tothe origin is not of the form (8). Instead, it has a power series in r / . The reason is once againa conical singularity for the metric at r ∼
0. The metric close to the origin takes the form g Σ ∼ e − β TT / r / ( dr + r dθ ) . (18)With the change of variables ρ = r / , we see that Σ possesses a flat metric close to the origin g Σ ∼ e − β TT / ( dρ + 19 ρ dθ ) , with a conical singularity with deficit angle 4 π/
3. The cone is embeddable into R as re iθ ∈ Σ (cid:55)→ ( X ( r, θ ) , Y ( r, θ ) , Z ( r, θ )) = (cid:16) r / cos θ, r / sin θ, √ r / (cid:17) ∈ R whose image satisfies the equation of a cone with aperture 2 cot − √ Z = (cid:112) X + Y ) . (19)Also in the Tzitzeica case, if we use the right coordinate ρ = r / , for which the metric isflat and Ω(0) well defined, we can rewrite the term − e − βTT / r / in (16) as − ρ Ω(0) / g Σ ∼ e − Λ TT K ( r ) ( dr + r dθ ) . In this Section we briefly review a non-linear rule [10] for superposing vortices in order to cre-ate higher vortex number solutions on (Σ , g ) by solving instead a lower vortex number Taubesequation on a modified background ( ˜Σ , ˜ g ).Let us suppose that h satisfies the Taubes equation (6) on (Σ , g ), with vortex number N andvortex centres { Z i } , and that ˜ h satisfies a second Taubes equation∆ ˜ h + ˜Ω (cid:16) − e ˜ h (cid:17) = 4 π M (cid:88) j =1 δ (cid:16) z − ˜ Z j (cid:17) , where ˜Ω( z, ¯ z ) = e h ( z, ¯ z ) Ω( z, ¯ z ) is the conformal factor of a degenerate metric, vanishing at { Z i } .We call ˜Σ the surface with metric ˜ g = ˜Ω dzd ¯ z .Now, it is straightforward to verify the identity∆ (cid:16) ˜ h + h (cid:17) + Ω (cid:16) − e ˜ h + h (cid:17) = 4 π N (cid:88) i =1 δ ( z − Z i ) + 4 π M (cid:88) j =1 δ (cid:16) z − ˜ Z j (cid:17) , which shows that h + ˜ h satisfies the Taubes equation on (Σ , g ), with vortex number N + M andvortex locations at { Z i } ∪ { ˜ Z j } .So, as a non-linear rule for superposing vortices, instead of looking for a N + M vortex on (Σ , g )we can look for a M vortex solution ˜ h on ( ˜Σ , ˜ g ), with conformal factor ˜Ω = e h Ω. The combination h + ˜ h is now the vortex solution on (Σ , g ) we were looking for. Note that generically, even if(Σ , g ) is smooth and geodesically complete, ( ˜Σ , ˜ g ) will not be so: the logarithmic singularities of h will induce conical singularities in ˜ g . 7 (cid:87) (cid:72) r (cid:76) Figure 1: The conformal factor Ω = e − h sG / for the sinh-Gordon vortex, upper plot, and therescaled one ˜Ω = e h sG Ω = e h sG / , lower plot. Let us apply now the superposition rule to obtain multi-vortex solutions on top of the sinh-Gordon vortex h sG , which is defined on the surface Σ = C with metric g = e − h sG ( z, ¯ z ) / dzd ¯ z .Firstly, we Weyl rescale the metric g by | φ | = e h sG to find the metric ˜ g = e h sG g on thesurface ˜Σ. In the limits r → r → ∞ , respectively, it is given by˜ g ∼ β sG r (cid:0) dr + r dθ (cid:1) = 4 β sG (cid:18) d ˜ ρ + 94 ˜ ρ dθ (cid:19) ( r →
0) (20)˜ g ∼ e − λK ( r ) (cid:0) dr + r dθ (cid:1) ( r → ∞ ) (21)where ˜ ρ = r / .The two conformal factors, Ω and ˜Ω, are shown in Figure 1. Note that Ω diverges in the originas r − while ˜Ω goes to zero as r , both factors tend to 1 for large r . In a neighbourhood of theorigin, ˜Σ looks like a cone in Minkowskian R , as explicitly shown by the isometry re iθ ∈ ˜Σ (cid:55)→ ( ˜ X ( r, θ ) , ˜ Y ( r, θ ) , ˜ Z ( r, θ )) = (cid:32) r / cos θ, r / sin θ, √ r / (cid:33) ∈ R . It is indeed an isometry as d ˜ X + d ˜ Y − d ˜ Z = r (cid:0) dr + r dθ (cid:1) .In order to find a multi-vortex solutions with N + 1 vortices located at the origin of Σ, wecould try to solve the Taubes equation (3) with conformal factor Ω = e − h sG / , with h given by(11) and (12), however, as we have seen in Section 2.1, Ω is actually diverging for r →
0. Toby-pass the complications of an ill-defined conformal factor we can use the superposition rulejust explained. Instead of looking for an N + 1 vortex solution on (Σ , g ) we study an N vortexproblem on ˜Σ whose metric has the conformal factor ˜Ω = e h sG Ω = e h sG / which is well-defined8t the origin, ˜Ω(0) = 0. The problem of finding N + 1 vortices at the origin in (Σ , g ) reduces to∆ h sG = 2 sinh (cid:18) h sG (cid:19) + 4 πδ ( z ) , (22)∆ ˜ h + e h sG / (cid:16) − e ˜ h (cid:17) = 4 πN δ ( z ) , (23)where h sG satisfies (11) and (12), while ˜ h has the asymptotic expansions˜ h ( r ) ∼ N ln r + ˜ a − β sG r + O ( r ) ( r →
0) (24)˜ h ( r ) ∼ ˜Λ K ( r ) ( r → ∞ ) , (25)where all the higher orders are uniquely determined in terms of ˜ a (or equivalently ˜Λ) and β sG .Note that both ˜ a and ˜Λ are constants in r but depend actually on the vortex number N . Througha numerical analysis of the system (22–23), see the Appendix for more details, we obtained theseconstants ˜ a and ˜Λ for various vortex numbers N . Note, by the way, that since the area of Σ isinfinite, the vortex number N is not limited by the Bradlow inequality (5). For the N = 1 case,our numerical analysis gave ˜ a = − . , ˜Λ = − . ρ = r / , the metric ˜ g becomesflat and with a non-vanishing conformal factor ˜Ω(0) = 4 β sG /
9. The term − r β sG / − ˜ ρ ˜Ω(0) / h T T , defined on the surfaceΣ = C with metric g = e − h TT ( z, ¯ z ) / dzd ¯ z . Rescaling the metric g by | φ | = e h TT we find thenew conformal factor ˜Ω = e h TT Ω = e h TT / of a new surface ˜Σ. In the limits r → r → ∞ ,respectively, this metric is given by˜ g ∼ e β TT / r / (cid:0) dr + r dθ (cid:1) = 9 e β TT / (cid:18) d ˜ ρ + 169 ˜ ρ dθ (cid:19) ( r →
0) (26)˜ g ∼ e − Λ TT K ( r ) (cid:0) dr + r dθ (cid:1) ( r → ∞ ) (27)where ˜ ρ = r / .The two conformal factors, Ω and ˜Ω, are shown in Figure 2, note that Ω diverges in the originas r − / while ˜Ω goes to zero as r / , both factors tend to 1 for large r . In a neighbourhood ofthe origin, ˜Σ looks like a cone in Minkowskian R , as explicitly shown by the isometry re iθ ∈ ˜Σ (cid:55)→ ( ˜ X ( r, θ ) , ˜ Y ( r, θ ) , ˜ Z ( r, θ )) = (cid:32) r / cos θ, r / sin θ, √ r / (cid:33) ∈ R . It is an isometry since d ˜ X + d ˜ Y − d ˜ Z = r / (cid:0) dr + r dθ (cid:1) .As we did before, instead of studying the problem of finding N + 1 vortices, located at theorigin in (Σ , g ), we first solve for the Tzitzeica vortex and then we look for an N vortex solutionon ( ˜Σ , ˜ g ): ∆ h T T + e − h TT / (cid:0) − e h TT (cid:1) = 4 πδ ( z ) , (28)∆ ˜ h + e h TT / (cid:16) − e ˜ h (cid:17) = 4 πN δ ( z ) , (29)9 (cid:87) (cid:72) r (cid:76) Figure 2: The conformal factor Ω = e − h TT / for the Tzitzeica vortex, upper plot, and therescaled one ˜Ω = e h TT Ω = e h TT / , lower plot.where h T T satisfies (16) and (17), while ˜ h has the asymptotic expansions˜ h ( r ) ∼ N ln r + ˜ a − e β TT / r / + O ( r / ) ( r →
0) (30)˜ h ( r ) ∼ ˜Λ K ( r ) ( r → ∞ ) , (31)where, once again, all the higher orders are uniquely determined in terms of ˜ a (or equivalently˜Λ) and β T T . For the N = 1 case, our numerical analysis gave ˜ a = − . , ˜Λ = − . ρ = r / , forwhich the metric (26) has a non-vanishing conformal factor in the origin ˜Ω(0) = 9 e β TT / /
16, werecover precisely the term − ˜ ρ Ω(0) / In this Section we derive the asymptotic expansion for the vortex profile function h , close to thevortex centre, when the background surface (Σ , g ) is a cone. For simplicity we will assume thatthe metric g takes the form g = r α (cid:0) dr + r dθ (cid:1) = | z | α dz d ¯ z , (32)with 1 + α > r ∈ R + , not just close to r ∼ ρ = r α the metric becomes flat g = 1(1 + α ) (cid:0) dρ + ρ (1 + α ) dθ (cid:1) = dZ d ¯ Z , (33)where Z = z α , but the new angle variable Θ = (1 + α ) θ is now periodic with period 2 π (1 + α ),denoting precisely a conical singularity with deficit angle 2 πα , embeddable as a cone in R for − < α < R for α >
0. For example in the sinh-Gordon case (13) α = − / α = 1 /
2, similarly for the Tzitzeica vortex (18) α = − / α = 1 / C / Z , embedded in C , and all its images under the orbifoldinggroup Z .For generic α , the change of variables Z = z α maps the complex plane into an infinitelymany sheeted Riemann surface. Let us assume for simplicity (and to make contact with thevortex solutions described above) that 1 + α = − /n with n ∈ N , i.e. for the sinh-Gordon vortex n = 2 while for the Tzitzeica vortex n = 3. In this case the change of variables that flattens themetric is simply given by z = Z n where Z is a complex coordinate on the orbifold C / Z n , i.e. Z ∈ C and Z ∼ e πi/n Z is an equivalence relation. Note that the origin Z = 0 has a non-trivialisotropy group under the orbifolding group Z n , this is precisely the reason why vortex solutionslocated at this special singular point have different properties from standard vortices on smoothmanifolds [11] and need to be treated separately.We can easily unfold the orbifold by considering n copies of the original manifold C / Z n ,modulo the identification Z ∼ e πi/n Z ; in Figure 3 we show this unfolding for the orbifold C / Z . For the multi sinh-Gordon and multi Tzitzeica case the change of variables that flattensthe metric at the origin is given by z = Z / and z = Z / respectively. When the change ofvariables takes the form z = Z n/m , with general n, m ∈ N ∗ the unfolding of the cone can still beperformed on a multi-sheeted Riemann surface with finitely many sheets, although a pictorialdescription of the unfolding of the cone in this case would get rather messy.As we can see from Figure 3, to find a vortex solutions when the centre is located in theinterior of the cone, away from the singularity, we can simply embed the cone in flat C andthen look for a solution with centres located at the finitely many images under the orbifoldinggroup action Z n of the original vortex location. In this way, the solution on C has manifest Z n symmetry and yield a solution on C / Z n . It is clear now that, when the vortex centre coincidewith the tip of the cone, the situation becomes more subtle because all the images under theorbifolding group Z n degenerate to a single point with non-trivial isotropy group.Let us write the second Bogomolny equation in Z = z /n coordinates, which can be obtainedfrom the pull-back of (2): ∂ ¯ Z φ − iA ¯ Z φ = 0 , (34)where A ¯ Z = n ¯ z − /n A ¯ z is the anti-holomorphic component of the connection 1-form in Z coor-11inate. It is clear from the definitions that the gauge and Higgs fields satisfy strict periodicity A ¯ Z ( Ze πin ) = A ¯ Z ( Z ) e πin and φ ( Ze πin ) = φ ( Z ) . (35)At first glance one could say that the condition (35) is too restrictive for a gauge field theoryand that periodicity should be respected up to a gauge transformation. However, (35) is aconsequence of the fact that we are seeking vortices on the cone and not on its n -covering. Thisis what it means to start from the equation (2) and not directly from (34). Strict periodicity isnecessary to obtain integer vortex numbers, as we show below. On the orbifold C / Z n we canimpose that (35) hold only up to a constant gauge transformation, in this way one can constructsolutions with fractional vortex numbers stuck at the conical singularity [17].While fractional vortices are necessarily fixed at the origin, integral vortices can move aroundand they possess a moduli space of solutions. We stress that, even if the background manifoldhas a conical singularity, the vortex moduli space is still a K¨ahler manifold with a well definedmetric [11]. The metric on the moduli space of these singular vortices is not known explicitly.The gaussian curvature has generically a delta function singularity at the tip of the cone and thisprevents us from using, in a straightforward way, the expansion for the moduli space metric ofvortices moving on surfaces of small curvature, obtained in [18]. On the other hand, as we havejust shown, once we unfold the orbifold into n copies living in C , we obtain a smooth backgroundmanifold, even at the origin, then it is conceivable that the moduli space metric could be studied,maybe numerically, starting from n vortices moving on the smooth, unfolded cone.A simple modification of a well-known result of Jaffe and Taubes [19] (c.f. Proposition 5.1 inChapter III) allows us to show that if A ¯ Z and φ form a smooth solution to (34), then the Higgsfield can be written, close to a vortex position Z k , as φ ( Z ) = ( Z − Z k ) N k ϕ k ( Z ) , (36)where N k ∈ N ∗ and ϕ k is C ∞ and non-vanishing in a neighbourhood of Z k . Furthermore, when Z k = 0, the smooth function ϕ k is invariant under Z (cid:55)→ Ze πin hence, for a vortex at the origin,periodicity of φ implies that N k ∈ n N and the actual vortex number (i.e. the winding of the phase χ of the Higgs field) turns out to be N k n once we go back to the original coordinates z = Z n . Thisis not surprising, since, as we can easily see from Figure 3, every neighbourhood of the origin in C is an n -covering of the region Z ∼ C / Z n , and the same vortex is counted n times. When we relax (35) and assume periodicity only up to a gauge transformation, the aboveargument ceases to hold and it is possible to find fractionalized vortex solutions [17].We can use (36) to expand h = ln | φ | around Z k , and we find that our original expansion(7) still holds also on conically singular spaces once we use the right set of coordinates Z : h ( Z, ¯ Z ) = 2 N k ln | Z − Z k | + a ( Z k , ¯ Z k ) + ¯ b ( Z k , ¯ Z k ) ( Z − Z k ) + b ( Z k , ¯ Z k ) (cid:0) ¯ Z − ¯ Z k (cid:1) + · · · . (37)Of course, we have assumed in equation real analyticity of ϕ k ( Z ), as it should be possible toprove by following similar steps of [19].Since h should be invariant under Z (cid:55)→ Ze i πn and Z k (cid:55)→ Z k e i πn , we have a ( Z k e i πjn ) = a ( Z k )and b ( Z k e i πjn ) = b ( Z k ) e i πjn for j = 0 , ..., n − Z k dependence. Ifwe consider the positions of the n vortices on the n -covering of the cone as in figure 3, namely, { Z k e i πjn } j =0 ,...,n − , we have n − (cid:88) j =0 b ( Z k e i πjn ) = n − (cid:88) j =0 b ( Z k ) e i πjn = 0 , z k → Z k = 0 and the vortex sits at the origin of the orbifold,we recover (8) where the correct radial variable to use is ρ = | Z | = r /n and the actual vortexnumber N = N k /n ∈ N ∗ , generalising the particular expansions (11) and (16) found for thesinh-Gordon and Tzitzeica vortices. Explicitly, by denoting ρ = | Z | = r /n , h ( ρ ) ∼ n N log ρ + a − Ω(0)4 ρ + O ( ρ ) , where the conformal factor Ω(0) = α ) = n can be read off from (33). Translating thisasymptotic form back to the original variable r = ρ n , with r = | z | , we get that close to r ∼ h ( r ) ∼ N log r + a − n r /n + O ( r /n ) , (38)matching exactly the leading asymptotic forms discussed previously for the sinh-Gordon vortex(11), n = 2, and the Tzitzeica vortex (16), n = 3. Note that generically, | φ | is not a C ∞ functionof r , as expected from the work of Baptista on singular vortices [11].A similar discussion holds for the multi-vortex case. As described above, to obtain theexpansion (8), the variable to use is ρ = r m/n , the conformal factor (33) at the origin takes theform Ω(0) = n /m and the vortex profile function, written in the original coordinate r , can beexpanded as h ( r ) ∼ N log r + a − n m r m/n + O ( r m/n ) , (39)matching precisely the leading terms in the multi sinh-Gordon case, m/n = 3 / m/n = 4 / r ) = e αh ( r ) ∼ r α (cid:0) a + O ( r β ) (cid:1) , where a is a non-vanishing constant and the exponent β > β = 1 in the (multi) sinh-Gordon case, while β = 2 / ρ = r α flattens only the leading order r α of the metric. For this reason the corrections to the vortex profile function (39) will not begenerically of order O ( r m/n ) but rather O ( r β +2 m/n ), i.e. O ( r ) in the multi sinh-Gordon case(24) and O ( r / ) in the multi Tzitzeica case (30). In this section we first rederive the construction of explicit, radially symmetric vortex solutionson the entire Poincar´e disk. Then, we take the Z n orbifold of the Poincar´e disk, so that theorigin becomes a conical singularity and the base manifold becomes a “hyperbolic cone”. Onthis particular conically singular space we can obtain an explicit expression for the Higgs fieldfollowing the discussion of Section 4. 13 .1 Vortices on the Poincar´e disk It is convenient to define u such that the conformal factor of the metric is Ω = e u . If u satisfiesthe Liouville’s equation ∆ u = e u , (40)then the Gauss curvature of this surface is K = − ∆ log Ω = − e − u ∆ u = − . In this case, theTaubes equation (3) reduces to another Liouville’s equation∆( u + h ) = e u + h . (41)Therefore, vortices on surfaces of constant curvature K = − can be obtained by solving two Li-ouville’s equations and, using the integrability properties of Liouville’s equation, we can constructexplicit vortex solutions on the entire surface.We begin with the study of solutions to (40). The Liouville’s equation ∆ ψ = e ψ , where ψ is a real function on R , has 3 types of solutions[21] ψ (1) = ln (cid:34) v x + v y ) v (cid:35) (42) ψ (2) = ln (cid:34) v x + v y )sinh v (cid:35) (43) ψ (3) = ln (cid:34) v x + v y )sin v (cid:35) , (44)where the function v satisfies Laplace’s equation ∆ v = 0.Let us restrict to the radially symmetric case v = v ( r ) so that ∆ v = r ( v (cid:48) r ) (cid:48) = 0 implies v ( r ) = C ln( r ) + D, (45)where C, D ∈ R are constants. Then the three types of solutions (42–44) become, respectively, ψ (1) = ln (cid:20) C r ( C ln( r ) + D ) (cid:21) (46) ψ (2) = ln (cid:20) C r sinh ( C ln( r ) + D ) (cid:21) (47) ψ (3) = ln (cid:20) C r sin ( C ln( r ) + D ) (cid:21) . (48)The initial conditions will uniquely fix the type of solution: ψ (1) , ψ (2) or ψ (3) . In fact, theinitial data set includes 3 disjoint subsets, each one of them corresponding to one solution among(46–48). To see this, let us impose some general initial conditions to ψ . Let r > v , v ∈ R ,and suppose ψ ( r ) = v ψ (cid:48) ( r ) = v . (49)14hen substituted in (46–48), conditions (49) impose the following constraints, respectively:2 e − v (cid:18) v r (cid:19) = 12 e − v (cid:18) v r (cid:19) = cosh ( C ln( r ) + D ) > e − v (cid:18) v r (cid:19) = cos ( C ln( r ) + D ) < . Note that the two inequalities must be strict otherwise ψ ( r ) is not well-defined. This meansthat, given the initial data (49), depending on the quantity 2 e − v (cid:16) v + r (cid:17) − If u ( r ) takes one of the forms (46)–(48), we can choose coordinates to set C = 1 and D = 0,and the surface Σ with conformal factor Ω = e u is, respectively, the once-punctured disk, thehyperbolic unit disk and a hyperbolic annulus. The only surface over which we can perform a Z n orbifold and consider a vortex located at the origin is the hyperbolic unit disk, since it is theonly surface containing this point. Vortices on hyperbolic surfaces constructed from holomorphicmaps have been studied in [5] .We then choose the solution (47), u ( r ) = ln (cid:104) − r ) (cid:105) , yielding the metric of the hyperbolicdisk ds = − r ) ( dr + r dθ ). Vortices on this surface should satisfy the boundary conditions h ( r ) ∼ N ln r + const. + O ( r ) for r →
0, where N is the vortex number, and h ( r ) → r →
1. The only solution obtained through our method and consistent with the above boundarycondition is given by h ( r ) = ln (cid:104) A r sinh ( A ln r + B ) (cid:105) − u ( r ), with A = N + 1 and B = 0. Hence h = 2 ln (cid:20) ( N + 1) r N (1 − r )1 − r N +1) (cid:21) , corresponds to a multi-vortex solution located at the origin of the Poincar´e disk, with any vortexnumber N > h = 2 ln (cid:20) ( N + 1) r N/n (1 − r /n )1 − r N +1) /n (cid:21) is a solution to the Taubes equation on the Poincar´e disk with a conical singularity at the originand metric ds = 8 r /n − n (1 − r /n ) (cid:0) dr + r dθ (cid:1) = 8(1 − ρ ) (cid:18) dρ + ρ n dθ (cid:19) , where ρ = r /n . One can check by direct calculation, that this function h ( r ) is indeed an exactsolution to Taubes equation on the hyperbolic cone with the above form for the metric.For the Higgs field to be uniquely valued on the orbifold, we need N ∈ n N and the solutiongives Nn vortices at the origin. If N is not an integer multiple of n or if n is non-integer, the15 - - - - - - Figure 4: Unfolding of the Z hyperbolic cone in the Poincar´e disk (left) and in the upper-halfplane (right). The images of the vortex centre under the Z orbifolding group are represented asred dots.solution gives a non-integer vortex number solution at the expense that the Higgs field is definedon the universal cover of the orbifold, i.e., it is periodic up to a gauge transformation.We can easily map this solution from the Poincar´e disk D = { w ∈ C | | w | ≤ } to theupper-half plane H = { z ∈ C | Im z ≥ } by means of the M¨obius transformation w = z − iz + i , which maps the origin of the Poincar´e disk w = 0 to the point z = i in the upper-half plane.The orbifold group action Z n in the Poincar´e disk geometry takes the form w ∼ e πi/n w , whilein the upper-half plane it becomes z ∼ ( az + b ) / ( cz + d ) where Z n is realized as a subgroup ofthe natural SL (2 , R ) action on H , generated by the elements A k = (cid:18) a bc d (cid:19) = − (cid:18) cos( πk/n ) sin( πk/n ) − sin( πk/n ) cos( πk/n ) (cid:19) . As depicted in Figure 4, we can unfold the hyperbolic cone in the Poincar´e disk or equivalentlyin the upper-half plane. To find a vortex solution with centre in the interior of the cone we cansimply take the explicit solution in the Poincar´e disk or upper-half plane with centres located atthe images, under the action of the orbifolding group, of the original vortex location. Since theorigin w = 0 of the Poincar´e disk, or similarly the point z = i in the upper-half plane, is a fixedpoint of the orbifolding group, when we insert a vortex exactly at the conical singularity all theimages collapse to the vortex centre and we have to rely on the analysis carried out before. Historically, the first multi-instanton solutions were found by Witten [3], who sought SO (3)-equivariant solutions to the self-dual Yang-Mills equations (SDYM) on R . SO (3)-equivariancemeans invariance up to gauge transformations of the connection 1-form under 3-dimensionalrotations acting with 2-dimensional orbits, we will precise this notion below. As we will describein this Section, there is a close relation between cylindrically symmetric instantons and Abelian Our two Liouville’s equations (40–41) correspond to equations (3.2) and (3.5) of this reference. R by S ∼ CP and write its metric g in complexand spherical coordinates g = 4 R ( R + y ¯ y ) dyd ¯ y = R ( dθ + sin θdϕ ) , where y = R tan (cid:18) θ (cid:19) e − iϕ , ¯ y = R tan (cid:18) θ (cid:19) e iϕ , ≤ θ < π, ≤ ϕ < π. Let
E → M be a rank-2 complex vector bundle over M = Σ × S , where Σ is a Riemannsurface, and A be an SO (3)-equivariant su (2)-valued connection on E . It means that, under theaction of SO (3), A is invariant up to a gauge transformation: for any R ∈ SO (3), there exists amatrix valued function g R ∈ SU (2) such that R ji A j ( R x ) = g R ( x ) A i ( x ) g R ( x ) − − ∂ i g R ( x ) g R ( x ) − . (50)The left hand side are the components of the pullback 1-form R ∗ A . The group SO (3) actstrivially on Σ and through its left action on S (for R ∈ SO (3), x ∈ S (cid:55)→ R x ).Cylindrical symmetry imposes the following explicit form to the connection [12, 24, 25] A = (cid:18) A ⊗ ⊗ b φ ⊗ ¯ β − ¯ φ ⊗ β − A ⊗ − ⊗ b (cid:19) , (51)where A = − i ( A z dz + A ¯ z d ¯ z ) is an (Abelian) U (1) connection on a Hermitian complex line (rank-1) bundle L over Σ, φ is a section of this bundle, b is the monopole connection over a complexline bundle L over S given by b = 12( R + y ¯ y ) (¯ ydy − yd ¯ y ) , and finally β = √ R R + y ¯ y dy and ¯ β = √ R R + y ¯ y d ¯ y are differential forms on S .The connection 1-form in (51) is a matrix whose entries are 1-forms. The tensor productsin this expression can be regarded as the usual multiplication between scalar functions anddifferential forms.Explicitly, the components of A = A z dz + A ¯ z d ¯ z + A y dy + A ¯ y d ¯ y are A z = − i A z σ , A ¯ z = − i A ¯ z σ , A y = ¯ y R + y ¯ y ) σ − R R + y ¯ y ¯ φ √ σ − , A ¯ y = − y R + y ¯ y ) σ + R R + y ¯ y φ √ σ + , (52)where σ = (cid:18) − (cid:19) , σ + = (cid:18) (cid:19) , σ − = (cid:18) (cid:19) . F of A is F = d A + A ∧ A = F − (cid:16) R − φ ¯ φ (cid:17) β ∧ ¯ β ( dφ − iAφ ) ∧ ¯ β − ( d ¯ φ + iA ¯ φ ) ∧ β − F + (cid:16) R − φ ¯ φ (cid:17) β ∧ ¯ β with non-vanishing components F z ¯ z = 12 F z ¯ z σ , F y ¯ y = − R ( R + y ¯ y ) (cid:18) R − φ ¯ φ (cid:19) σ , F ¯ z ¯ y = 1 √ R R + y ¯ y ( ∂ ¯ z φ − iA ¯ z φ ) σ + , F z ¯ y = 1 √ R R + y ¯ y ( ∂ z φ − iA z φ ) σ + , F zy = − √ R R + y ¯ y ( ∂ z ¯ φ + iA z ¯ φ ) σ − , F ¯ zy = − √ R R + y ¯ y ( ∂ ¯ z ¯ φ + iA ¯ z ¯ φ ) σ − , (53)where F = dA = F z ¯ z dz ∧ d ¯ z = − i ( ∂ z A ¯ z − ∂ ¯ z A z ) dz ∧ d ¯ z .The Hodge operator ∗ is defined by ∗F µν = √ | det g | (cid:15) σηµν g σα g ηβ F αβ , where g is the metricof the background. By applying the Hodge operator we verify that F is self-dual, i.e, ∗F = F ,if and only if φ and A satisfies the vortex equations on Σ2 F z ¯ z = Ω2 (cid:18) R − φ ¯ φ (cid:19) D ¯ z φ = 0 , which are equivalent to (1–2) when R = √
2, since B = 2 F z ¯ z .The conclusion is that the SDYM equations on M = Σ × S are reduced to vortex equationson Σ once we impose an SO (3) symmetry on the field. Similarly the anti-self-dual Yang-Millsequations ∗F = −F can be reduced to anti-vortex equations. Conversely, given a(n) (anti-)vortexon Σ, it can be lifted to a cylindrically symmetric Yang-Mills (anti-)instanton on Σ × S . In this Section we derive a solution of the SDYM equations from the vortices described in Sections2.1, 2.2 and 3.1. Not to overcrowd this Section with too many equations, we will give explicitformulas only for the sinh-Gordon vortex, similar results can be derived in a straightforwardmanner for the Tzitzeica case as well as for the multi-vortex case.From our sinh-Gordon vortex solution h sG we can reconstruct the Higgs field φ = e h sG / iχ ,where χ is a real function defined on each open patch depending on the gauge choice, whileusing the Bogomolny equation (2) we can obtain the gauge field from the Higgs field: A ¯ z = − i∂ ¯ z log φ = − i∂ ¯ z h sG / ∂ ¯ z χ . Close to the origin we can simply use the asymptotic expansion(11) and obtain φ ( z, ¯ z ) ∼ β sG | z | e iχ (1 + O ( | z | )) , (54) A ¯ z ( z, ¯ z ) ∼ − i zz ¯ z (cid:18) | z | β sG + O ( | z | ) (cid:19) + ∂ ¯ z χ. (55)We can use (52) to uplift this vortex solution to an instanton solution on Σ × S , provided18hat R = √
2, and the components of the connection for the instanton solution close to z ∼ A z = (cid:18)
14 ¯ zz ¯ z − β sG ¯ z | z | − i ∂ z χ (cid:19) σ , A ¯ z = (cid:18) −
14 ¯ zz ¯ z + 18 β sG ¯ z | z | − i ∂ ¯ z χ (cid:19) σ , A y = ¯ y y ¯ y ) σ − √
22 + y ¯ y | z | β sG e − iχ σ − , A ¯ y = − y y ¯ y ) σ + √
22 + y ¯ y | z | β sG e iχ σ + , where we omitted higher terms in | z | .The components of the curvature two-form can be easily calculated and the only non-vanishingcomponents are F z ¯ z = 18 β sG | z | σ , F y ¯ y = − y ¯ y ) σ , F ¯ zy = − √
22 + y ¯ y (cid:18) β sG | z | ¯ z − z (cid:19) e − iχ σ − , F z ¯ y = √
22 + y ¯ y (cid:18) β sG | z | z − ¯ z (cid:19) e iχ σ + , , where we neglected, once again, higher terms in | z | .The corresponding expansion for the instanton connection A and field strength F as z → ∞ can be calculated in the same way from the expansion as r → ∞ for φ and A given by (12).We want to check now that the uplifting of our sinh-Gordon vortex does indeed correspondto a 1 − instanton solution on Σ × S . The instanton number N is defined as the integral N = − (cid:90) Σ × S C , where C = 18 π (Tr( F ∧ F ) − Tr F ∧ Tr F ) = d (cid:18) π Tr( d A ∧ A + 23
A ∧ A ∧ A ) (cid:19) is the second Chern form. The wedge product ∧ between matrices indicates the usual multipli-cation of matrices but applying the wedge product between the entries.It is easy to check that the Chern forms splits in the product of the first Chern class for thevortex field strength and the first Chern class for the monopole connection, so that the instantonnumber can be rewritten as N = − (cid:90) Σ × S C = i π (cid:90) Σ F · i π (cid:90) S db . (56)Since the sinh-Gordon vortex has vortex number one and the monopole has magnetic chargeone, it follows that the instanton number is also N = 1. This means that the SDYM solution onΣ × S obtained from the uplifting on the sinh-Gordon vortex on Σ corresponds precisely to a1 − instanton located at the origin of Σ and spread along the S .Note that even if F is singular close to z ∼
0, the instanton number is still finite and integer.This follows from the Bogomolny equation (1): the sinh-Gordon (and Tzitzeica) vortex solutionhas a diverging magnetic field close to the origin of Σ because of the diverging conformal factorΩ, nonetheless this singularity is integrable and the vortex has a finite and quantised magneticflux.Similar results can be derived from the Tzitzeica vortex, furthermore, higher instanton num-ber solutions can be obtained by uplifting in a similar fashion our multi-vortex solutions ofSection 3.1. 19
Conclusion
In this paper we first reviewed the construction of Abelian vortex solutions from the sinh-Gordonequation and the elliptic Tzitzeica equation. These solutions are not known in explicit formsover the entire background but only in the asymptotic regimes r → r → ∞ .Using the non-linear superposition rule described by Baptista, we constructed multi-vortexsolutions on top of the sinh-Gordon and the Tzitzeica vortex and analysed their properties withvarious numerical simulations. The vortices constructed with this procedure are all defined onsurfaces with conical singularities, and, for this reason, the usual expansion for the vortex profilefunction (7) ceases to hold.For this reason we analysed the problem of finding vortex solutions on conically singularspaces and we showed that with a careful change of coordinates (from z to Z ), for which themetric becomes flat and the cone can be unfolded in the complex plane, the problem reducessimply to the study of vortex solutions invariant under the action of an orbifold symmetry. Inparticular we see from equation (37) that, close to the vortex location, the Higgs field | φ | is realanalytic in the new coordinates Z .When the vortex is located away from the conical singularity, the solution close to the vortexcentre is smooth in the original coordinates z as well. On the contrary, when the vortex islocated precisely at the conical singularity, the asymptotic form of the profile function (38),when expressed in the original coordinates, takes the form of a Puiseux series in | z | /n , whichmeans that generically the Higgs field | φ | is only C as a function of r = | z | , as already expectedfrom Theorem 2.1 of [11]. However, we note from equation (11) that, for the sinh-Gordon vortex, | φ | is C ∞ as a function of the original coordinate r .As an additional example of our analysis, we discuss the Taubes equation on the Poincar´edisk. On this surface, explicit vortex solutions can be obtained from the Liouville equation andfrom them we can analytically construct vortex solutions on the hyperbolic cone. Once we mapthe Poincar´e disk to the upper half plane H , the orbifold group can be realized as a discretesubgroup of the natural SL (2 , R ) acting on H . When uplifted to 4-dimensions, vortex solutionson the upper-half plane H become instantons with cylindrical symmetry on R ∼ H × S , itwould be interesting to understand what kind of instantons can be obtained by uplifting thesevortex solutions on H/ Z n to 4-dimensions.It would also be interesting to apply our approach to vortices on conifolds to the case ofcompact surfaces, for example, a sphere with one or more conical singularities [26], to analysethe effect of the orbifold action and the compact nature of the background on the global propertiesof the vortex.In the final part of our work, we described how to uplift our multi-vortex solutions, definedon the conically singular surface Σ, to instanton solutions, with cylindrical symmetry, on thebackground M = Σ × S with a product metric. The four dimensional manifold M is K¨ahlerwith K¨ahler form ω = i Ω dz ∧ d ¯ z + i y ¯ y ) dy ∧ d ¯ y. (57)Moreover, this metric has non-vanishing scalar curvature and therefore has no self-dual or anti-self-dual Weyl tensor (c.f. Proposition 10.2.2 of [14]). In this case, the 6-dimensional twistor spaceof M has a non-integrable almost complex structure [27, 28] and the Penrose-Ward transformdoes not apply. These solutions go beyond the analysis of integrable (anti-)self-dual backgrounds,in which case the vortex equations would arise as compatibility conditions of linear differentialequations on the twistor space. 20 cknowledgements We are grateful to Maciej Dunajski, Nick Manton, Norman Rink and Alexander Cockburn foruseful discussions. F.C. is grateful for the support of Cambridge Commonwealth, European &International Trust and CAPES Foundation Grant Proc. BEX 13656/13-9. D.D. is gratefulfor the support of European Research Council Advanced Grant No. 247252, Properties andApplications of the Gauge/Gravity Correspondence.
Appendix. Numerical Analysis
The asymptotic forms for the solution of the Taubes equation (6) can be fixed analytically onlyin radial reductions of the sinh-Gordon, Ω = e − h sG / , or the Tzitzeica case, Ω = e − h TT / , byexploiting the Painlev´e property of the two ODEs. Unfortunately for a generic metric no suchmethods exist and if we want to compute multi-vortex solutions to (23) or (29), we have to relyon a numerical calculation.Furthermore, when we apply our superposition rule for vortices, we need to use the modifiedconformal factors ˜Ω = e h sG / or ˜Ω = e h TT / , but we do not have the explicit solutions to thesinh-Gordon and Tzitzeica vortices for all values of r , so we will have to obtain numericallythe sinh-Gordon and Tzitzeica vortex solutions interpolating between the two known asymptoticforms. This problem and all the subsequent studies for multi-vortex solutions have been solvednumerically in the following way, first instead of working for r ∈ R we cut away the r → ∞ and the singular point r ∼ r ∈ [ (cid:15), R ] and checking that the solution does notchange as we send (cid:15) → R → ∞ .Secondly instead of working with the profile function h it is better to strip away the log − likesingularity by working with: h ( r ) = u ( r ) + 2 N log( r/R ) , in this way the δ function on the right-hand side of Taubes equation disappears and u satisfies: ∇ u + Ω (cid:18) − r N R N e u (cid:19) = 0 . From the asymptotics of h we can read those of u : u ( (cid:15) ) ∼ a +2 log R + O ( (cid:15) α ), where α > r ∼ R the log term that we added vanishes (but notits derivative) so h ( R ) = u ( R ) ∼ Λ e − R (remember that all our metrics are asymptotically flat,i.e. Ω → r → ∞ ). To obtain a solution for u we implemented both a shooting and a coolingmethod and the two solutions coincide within numerical errors.Let us first construct the sinh-Gordon and Tzitzeica vortices. We know from Sections 2.1-2.2that the vortex number can only be N = 1 and we need to solve for d u sG dr + 1 r du sG dr + e − u sG / r (cid:18) − r R e u sG (cid:19) = 0 , (A1) d u T T dr + 1 r du T T dr + e − u TT / r / (cid:18) − r R e u TT (cid:19) = 0 , (A2)with boundary conditions u sG ( (cid:15) ) = 4 log β sG + 2 log R − (cid:15)β sG + O ( (cid:15) ) , (A3) u sG ( R ) = Λ sG K ( R ) + O ( e − R ) , (A4)21 (cid:72) r (cid:76) Figure 5: Plot of the magnet field B ( r ) times the radial coordinate r for the sinh-Gordon vortex,finite at r = 0, and the Tzitzeica vortex, diverging at r = 0.and similarly u T T ( (cid:15) ) = β T T + 2 log R − e − β TT / (cid:15) / O ( (cid:15) / ) , (A5) u T T ( R ) = Λ T T K ( R ) + O ( e − R ) . (A6)We fixed (cid:15) = 10 − and R = 30, so that the higher terms in the boundary conditions arenumerically negligible.We can see in Figure 5 that the magnetic field B ( r ), for the two numerical solutions, is obviouslylocalized in a region close to the origin and decays exponentially to 0 for large r . We note thatin the Tzitzeica case, due to the diverging conformal factor present in the Bogomolny equation(1), we have rB ( r ) ∼ r − / close to r ∼
0, however this singularity at the origin is integrableand the magnetic flux is actually finite and quantized.With the sinh-Gordon and Tzitzeica vortex solutions in our hands, we are now in the positionto study the multi-vortex problem. Let us focus for simplicity on the multi sinh-Gordon problem(23), which translated to the ˜ u ( r ) = ˜ h ( r ) − N log( r/R ) variable takes the form: d ˜ udr + 1 r d ˜ udr + e h sG / (cid:18) − r N R N e ˜ u (cid:19) = 0 , (A7)˜ u ( (cid:15) ) = ˜ a + O ( (cid:15) ) , ˜ u (cid:48) ( (cid:15) ) = O ( (cid:15) ) . To obtain the behaviour (24) of u close to r ∼ u ( r ) = (cid:80) n ≥ a n r n , and imposingthat (A7) is satisfied order by order, fixes all the coefficients a n in terms of ˜ a = a . By choosing (cid:15) = 10 − we can set u (cid:48) ( (cid:15) ) ∼
0, so that we can perform a shooting method where the shootingparameter ˜ a is chosen in such a way that u ( R ) = h ( R ) ∼ ˜Λ e − R →
0, where we set R = 30. Wechecked the precision of our numerical simulations by evaluating the magnetic flux as a functionof the vortex number N and the shooting parameter ˜ a .22
10 15 20 N (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:76) Figure 6: Values for ˜ a , left plot, and the vortex strength ˜Λ, right plot, as a function of the vortexnumber N for the multi sinh-Gordon vortex.In Figure (6) we summarize the results of our multi sinh-Gordon vortex numerical analysis byplotting the values of ˜ a and ˜Λ as a function of the vortex number N ∈ { , ..., } : once thevortex number is fixed the solution is uniquely determined by ˜ a or equivalently by the vortexstrength ˜Λ.We repeated an identical numerical analysis for the multi Tzitzeica vortex and obtainedsimilar plots for ˜ a, ˜Λ as functions of the vortex number N . References [1] A. A. Abrikosov. On the magnetic properties of superconductors of the second group.
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