THE BERRY CONNECTION OF THE GINZBURG-LANDAU VORTICES. Introduction

Transcription

1 THE BERRY CONNECTION OF THE GINZBURG-LANDAU VORTICES ÁKOS NAGY Abstrat. We analyze 2-dimensional Ginzburg-Landau vorties at ritial oupling, and establish asymptoti formulas for the tangent vetors of the vortex moduli spae using theorems of Taubes and Bradlow. We then ompute the orresponding Berry urvature and holonomy in the large area limit. Introdution The Ginzburg-Landau theory is a phenomenologial model for superondutivity, introdued in [GL50]; for a more modern review see [AK02]. The theory gives variational equations the Ginzburg-Landau equations for an Abelian gauge eld and a omplex salar eld. The gauge eld is the EM vetor potential, while the norm of the salar eld is the order parameter of the superonduting phase. The order parameter an be interpreted as the wave funtion of the so-alled BCS ground state, a single quantum state oupied by a large number of Cooper pairs. This paper fouses on ertain stati solutions of the 2-dimensional Ginzburg-Landau equations alled τ-vorties. Physiists regard the number τ as a oupling onstant, sometimes alled the vortex-size. Mathematially, τ is a saling parameter for the metri. The geometry of τ-vorties have been studied sine [JT80, B90], and there is a large literature on the subjet; f. [MN99, MS03, CM05, B06, B11, DDM13, BR14, MM15]. Families of operators in quantum physis arry anonial onnetions. This idea was introdued by Berry in [B84], and generalized by Aharonov and Anandan in [AA87]. These so-alled Berry onnetions were used, for example, to understand the Quantum Hall Eet [K85]. In gauge theories inluding the Ginzburg-Landau theory the Berry onnetion an be understood geometrially as follows: the spae P of solutions of gauge invariant equations is an innite dimensional prinipal bundle over the part of moduli spae M where the ation of the gauge group G is free. Thus if all solutions are irreduible, then P is an innite dimensional prinipal G-bundle P G M Date: April 8, Key words and phrases. Berry onnetion, Ginzburg-Landau vorties. 1

2 over M. The anonial L 2 -metri of P denes a horizontal distribution the orthogonal omplement of the gauge diretions whih denes the Berry onnetion. A onnetion on a prinipal G-bundle P X denes parallel transport: for eah smooth map Γ : [0, 1] X, parallel transport around Γ is a G-equivariant isomorphism from the ber at Γ0) to the ber at Γ1). If Γ is a losed loop, then Γ0) = Γ1), and the orresponding parallel transport is alled holonomy. If G is Abelian, holonomy is given by the ation of an element in G. Holonomies of the Berry onnetion are gauge transformations, whih have a physial interpretation: they desribe the adiabati evolution of the state of the system, that is its behavior under slow hanges in the physial parameters suh as external elds, or oupling onstants. This paper investigates the Berry onnetion of the τ-vortex moduli spae assoiated to a degree d hermitian line bundle over a losed, oriented Riemannian surfae Σ. The Berry holonomy assigns a gauge transformation g τ to eah losed urve Γ in M τ. These gauge transformations are U1)- valued smooth funtions on Σ. When d is positive and τ is greater than the geometry-dependent onstant τ 0 = 4πd AreaΣ), then the moduli spae, M τ, is identied with the d-fold symmetri power of the surfae Sym d Σ). A losed urve Γ in Sym d Σ) denes a 1-yle in Σ, alled shγ), the shadow of Γ, whih is onstruted by hoosing a lift of Γ to Σ d Sym d Σ), and taking the union of the non-onstant urves appearing in the lift see equation 5.2) for preise denition). The main theorem of this paper gives a omplete topologial and analytial desription of these gauge transformations in terms of the shadow: Main Theorem. [The Berry holonomy of the τ-vortex prinipal bundle] Let g τ G be the Berry holonomy of a smooth urve Γ in the τ-vortex moduli spae M τ, and shγ) be the losed 1-yle in Σ dened in equation 5.2). Then the following properties hold as τ : 1) [Convergene] g τ 1 in the C 1 -topology on ompat sets of Σ shγ). 2) [Crossing] Let j : [0, 1] Σ be a smooth path that intersets shγ) transversally and positively one, and write g τ j = exp2πiϕ τ ). Then ϕ τ 1) ϕ τ 0) ) [Conentration] As a 1-urrent, dg τ onverges to the 1-urrent dened by shγ). The map Γ g τ indues a pairing dened in 5.5). 2πi g 1 τ hol : H 1 Σ; Z) H 1 Σ; Z), 4) [Duality] For all τ > τ 0, the homomorphism hol is Poinaré duality. When Γ is a positively oriented, bounding single vortex loop, or a positively oriented vortex interhange see Setion 5 for preise denitions) our main theorem implies that the orresponding holonomy an be written as g τ = exp2πif τ ), for a real funtion f τ on Σ. Moreover, f τ an be hosen so that it onverges to 1 on the inside of the urve, and to 0 on the outside. This makes physiists' intuition about the holonomy preise; f. [I01]. As mentioned above, τ is a saling parameter for the metri: one an look at the Ginzburg- Landau theory with τ = 1 xed, but the Kähler form ω saled as ω t = t 2 ω. Our results, inluding 2

3 the Main Theorem above, an be reinterpreted as statements about the large area limit i.e. t ), whih an be more diretly related to physis see Setion 6 for details). The paper is organized as follows. In Setion 1, we give a brief introdution to the geometry of the τ-vortex equations on a losed surfae, derive the tangent spae equations of the τ-vortex moduli spae, and then reast them in a ompat form. In Setion 2, we use theorems of Taubes and Bradlow to prove a tehnial result, Theorem 2.3, that establishes asymptoti formulas for the tangent vetors of the of the τ-vortex moduli spae. In Setion 3, we introdue the Berry onnetion assoiated to this problem. We then prove asymptoti formulas for the Berry urvature in Setion 4. In Setion 5, we prove our Main Theorem; the proofs are appliations of Theorem 2.3. Setion 6 disusses the large area limit. Aknowledgment. I wish to thank my advisor Tom Parker for guiding me to this topi and for his onstant help during the preparation of this paper. I am grateful for the support of Je Shenker via the NSF grant DMS I also beneted from the disussions with David Dunan, Manos Maridakis, and Tim Nguyen. Finally I thank the Referee for the inredibly helpful revision. 1. Ginzburg-Landau theory on losed surfaes 1.1. The τ-vortex equations. Let Σ be a losed surfae with Kähler form ω, ompatible omplex struture J, and Riemannian metri ω, J )). Let L Σ be a smooth omplex line bundle of positive degree d with hermitian metri h. For eah unitary onnetion, and smooth setion φ, onsider the Ginzburg-Landau free energy: E λ,τ, φ) = Σ F 2 + φ 2 + λw 2) ω, 1.1) where λ, τ R + are oupling onstants, F is the urvature of, and w = 1 2 τ φ 2). 1.2) The Euler-Lagrange equations of the energy 1.1) are the Ginzburg-Landau equations: d F + i Imhφ, φ)) = 0 φ λwφ = a) 1.3b) When λ = 1, the energy 1.1) an be integrated by parts and rewritten as dierent sum of nonnegative terms, and get the lower bound 2πτ d. The minimizers satisfy the τ-vortex equations: iλf w = 0 φ = 0, 1.4a) 1.4b) 3

4 where ΛF is the inner produt of the Kähler form ω and the urvature of, and = 0,1 is the Cauhy-Riemann operator orresponding to. Solutions, φ) to the rst order equations 1.4a) and 1.4b) automatially satisfy the seond order equations 1.3a) and 1.3b) The τ-vortex moduli spae. As is standard in gauge theory, we work with the Sobolev W k,p -ompletions of the spae of onnetions and elds. Let C L be the W 1,2 -losure of the ane spae of smooth unitary onnetions on L and Ω 0 L be the W 1,2 -losure of the vetor spae of smooth setions of L. Similarly, let Ω k, and Ω k L be the W 1,2 -losure of k-forms, and L-valued k-forms respetively. The orresponding gauge group G is the W 2,2 -losure of AutL) in the W 2,2 -topology. The gauge group is anonially isomorphi to the innite dimensional Abelian Lie group W 2,2 Σ, U1)), whose Lie algebra is W 2,2 Σ; ir). Elements g AutL) at on C L Ω 0 L as g, φ) = g g 1, gφ ), and this denes a smooth ation of G on C L Ω 0 L. Finally, the energy 1.1) extends to a smooth funtion on C L Ω 0 L. The spae P τ of all ritial points of the extended energy is an innite dimensional submanifold of C L Ω 0 L. Due to the gauge invariane of energy 1.1), G ats on P τ, and every ritial point is gauge equivalent to a smooth one by ellipti regularity. The τ-vortex moduli spae is the quotient spae M τ = P τ /G. Elements of P τ are alled τ-vortex elds, while elements of M τ gauge equivalene lasses of τ-vortex elds) are alled τ-vorties. For brevity, we sometimes write τ-vortex elds as υ =, φ) P τ and the orresponding τ-vorties as [υ] = [, φ] M τ. There is a geometry-dependent onstant τ 0 = 4πd AreaΣ), alled the Bradlow limit, with the property that if τ < τ 0, then the moduli spae is empty and if τ > τ 0, then there is a anonial bijetion between M τ and the spae of eetive, degree d divisors; f. [B90, Theorem 4.6]. This spae is also anonially dieomorphi to the d-fold symmetri produt of the surfae Sym d Σ), whih is the quotient of the d-fold produt Σ d = Σ... Σ by the ation of the permutation group S d. Although this ation is not free, the quotient is a smooth Kähler manifold of real dimension 2d. For eah value of τ > τ 0, there is a anonial L 2 -Kähler struture see Setion 1.3). In the borderline τ = τ 0 ase, the φ-eld vanishes everywhere and the moduli spae is in one-to-one orrespondene with the moduli spae holomorphi line bundles of degree d [B90, Theorem 4.7]. Aordingly, we fous on the τ > τ 0 ase in this paper. When τ > τ 0, Bradlow's map from M τ to Sym d Σ) is easy to understand: By integrating equation 1.4a), one sees that the L 2 -norm of φ is positive. On the other hand, φ is a holomorphi setion of L by equation 1.4b). Sine φ is a non-vanishing holomorphi setion it denes an eetive, degree d divisor, giving us the desired map. The inverse of this map is muh harder to understand an involves non-linear ellipti theory. Muh of this piture arries over to open surfaes, even with innite area for example Σ = C), if one imposes proper integrability onditions; f [T84]. For simpliity we will always assume that Σ is ompat. Furthermore the moduli spae is empty for d < 0, and a single point for d = 0. Thus we will always assume that d > 0 in this paper. 4

5 1.3. The horizontal subspaes. The tangent spae at any point of the ane spae C L Ω 0 L is the underlying vetor spae iω 1 Ω 0 L. The tangent spae of P τ is desribed in the next lemma. Lemma 1.1. The tangent spae of P τ at the τ-vortex eld υ =, φ) is the vetor spae of pairs a, ψ) iω 1 Ω 0 L that satisfy iλda + Rehψ, φ)) = 0 ψ + a 0,1 φ = a) 1.5b) Proof. The linearization of equations 1.4a) and 1.4b) in the diretion of a, ψ) is: 1 t iλf +t a 1 2 τ φ + t ψ 2)) = iλda + Rehψ, φ)) lim t 0 1 lim t 0 t +t a φ + t ψ) ) = ψ + a 0,1 φ, where we used that υ is a τ-vortex eld. This ompletes the proof, sine the tangent spae is the kernel of the linearization of equations 1.4a) and 1.4b). The ane spae C L Ω 0 L has a anonial L2 -metri given by a, ψ) a, ψ ) = a a + Re h ψ, ψ )) ω ) 1.6) Σ where is the onjugate-linear) Hodge operator of the Riemannian metri of Σ. One an hek that the restrition of the L 2 -metri 1.6) to the solutions of equations 1.5a) and 1.5b) makes P τ a smooth, weak Riemannian manifold, and that gauge transformations at isometrially on P τ. The pushforward of the tangent spae T 1 G by the gauge ation is alled the vertial subspae of T υ P τ. We dene the horizontal subspae of T υ P τ to be the orthogonal omplement of the vertial subspae by the L 2 -metri 1.6). Sine M τ = P τ /G, the horizontal subspae is anonially isomorphi to the tangent spae T [υ] M τ of the moduli spae. The next lemma shows that the horizontal subspae is also the kernel of a rst order linear ellipti operator. Lemma 1.2. The horizontal subspae of T υ P τ, at the τ-vortex eld υ =, φ) P τ, is the vetor spae of pairs a, ψ) iω 1 Ω 0 L that satisfy iλd + d )a + hψ, φ) = 0 ψ + a 0,1 φ = a) 1.7b) Proof. The real part of equation 1.7a) is equation 1.5a) and equation 1.7b) is equation 1.5b); thus solutions of equations 1.7a) and 1.7b) are in T υ P τ. To nish the proof, we must hek that a pair a, ψ) in T υ P τ is orthogonal to the vertial subspae at υ exatly if equations 1.7a) and 1.7b) hold. The pushforward of if LieG) at υ is given by X f υ) = idf, ifφ), hene horizontal vetors are pairs a, ψ), that satisfy the following equation for every f C Σ; R): 0 = a, ψ) idf, if φ) = a idf) + Rehψ, ifφ))ω). Σ 5

6 Integrating the right-hand side by parts yields 0 = d a + i Imhψ, φ)))ifω. Σ Beause this holds for all f, we onlude that a, ψ) is orthogonal to the vertial subspae at υ exatly if d a + i Imhψ, φ)) = 0 holds. Adding this purely imaginary) equation to the purely real) equation 1.5a) gives equation 1.7a). Equations 1.7a) and 1.7b) depend on the hoie of υ, but if a, ψ) is a solution of equations 1.5b) and 1.7a) for υ and g G, then a, gψ) is a solution of equations 1.5b) and 1.7a) for gυ) = + gdg 1, gφ ). Sine gauge transformations at isometrially, the L 2 -metri on the horizontal subspaes of T P τ desends to a Riemannian metri on M τ. Let K be the anti-anonial bundle of Σ, and Ω 0,1 = Ω 0 K, the W 1,2 -ompletion of the spae of smooth setions of K. We reast equations 1.7a) and 1.7b) in a more geometri way in the next lemma. Lemma 1.3. Equations 1.7a) and 1.7b) are equivalent to the following pair of equations on α, ψ) Ω 0,1 Ω 0 L : 2 α hφ, ψ) = 0 1.8a) 2 ψ + αφ = b) Moreover, the unitary bundle isomorphism ) a, ψ) 2 1 a + i a), ψ 1.9) interhanges solutions of equations 1.7a) and 1.7b) with solutions of equations 1.8a) and 1.8b). Proof. A omplex 1-form α is in Ω 0,1 exatly if α = i α. For a iω 1, dene the unitary map u by ua) = 1 2 a + i a). 1.10) Using 2 a = a = a, we see that ua) = iua), and thus ua) Ω 0,1. Set α = ua). With this notation a 0,1 = α 2, whih proves the equivalene of equations 1.7b) and 1.8b). The Kähler identities yield iλd + d )a = 2 α, whih is equivalent to equation 1.8a). The vetor spae of solutions to equations 1.8a) and 1.8b) has a anonial almost omplex struture oming from the omplex strutures of K and L, and this denes an almost omplex struture for M τ. Mundet i Riera [R00] showed that this struture is integrable, and together with the L 2 -metri it makes M τ a Kähler manifold. To put equations 1.8a) and 1.8b) in a more ompat form, note that they are equivalent to the single equation L υ a, ψ) = 0, 6

7 where L υ = D + A φ is dened as D : Ω 0,1 Ω 0 L Ω 0 Ω 0,1 L ; A φ : Ω 0,1 Ω 0 L Ω 0 Ω 0,1 L ; α, ψ) ) 2 α, 2 ψ α, ψ) hφ, ψ), αφ), The operator D is a rst order ellipti dierential operator, and the operator A φ is a bundle map. Straightforward omputation shows that for all Z Ω 0,1 Ω 0 L A φ A φz) = φ 2 Z. 1.11) Thus A φ is non-degenerate on the omplement of the divisor of φ. Note that D, A φ, and hene L,φ) make sense for any pair, φ) C L Ω 0 L. Lemma 1.4. Let υ =, φ) C L Ω 0 L be a pair suh that φ = 0. Then the operator D A φ + A φ D is identially zero. Proof. The adjoint operators are D f, ξ) = ) 2 f, 2 ξ A φf, ξ) = hφ, ξ), fφ) 1.12a) 1.12b) for any f, ξ) Ω 0 Ω 0,1 L. The lemma follows from equations 1.12a) and 1.12b), the holomorphiity of φ, and the denitions of D and A φ. Corollary 1.5. Let υ =, φ) C L Ω 0 L be a pair suh that φ = 0. Then kerl υ) is trivial. Proof. From Lemma 1.4 and equation 1.11), we obtain L υ L υ = D D + A φ A φ. Hene if Z is in the kernel of L υ, then 0 = L υz) 2 L 2 = D Z) 2 L 2 + A φz) 2 L 2, whih implies that both terms on the right vanish. By equation 1.11), Z vanishes where φ does not, whih is the omplement of a nite set. But then Z vanishes everywhere by ontinuity. Hene the kernel of L υ is trivial. 2. The asymptoti form of horizontal vetors In this setion we will use of the following results of Taubes and Bradlow about the large τ behavior of τ-vortex elds. Reall from equation 1.2) that w = 1 2 τ φ 2 ). Theorem 2.1. [Bradlow and Taubes] There is a positive number = Σ, ω, J, L, h) suh that eah τ-vortex eld υ =, φ) P τ satises φ 2 τ ) 2.1a) w + φ τ exp, 2.1b) 7 τdistd

8 where dist D is the distane from the divisor D = φ 1 0), and w is dened in equation 1.2). Proof. In [B90, Proposition 5.2] Bradlow showed inequality 2.1a), using the fat that τ-vortex elds satisfy the ellipti equation + φ 2) w = φ ) The right-hand side is positive away from a nite set, so the maximum priniple and equation 1.2) implies inequality 2.1a). Inequality 2.1b) was proved in [T99, Lemma 3.3]. We all a divisor simple if the multipliity of every divisor point is 1. Lemma 2.2. Fix a simple divisor D Sym d Σ) and a orresponding τ-vortex eld υ =, φ) with τ > τ 0. The smooth funtion depends only on D and τ, but not on the hoie of υ. Moreover, h D,τ = 1 2πτ φ 2 + 2w 2) 2.3) lim τ h D,τ = p D δ p, 2.4) in the sense of measures, where δ p is the Dira measure onentrated at the point p Σ. Proof. Every term in equation 2.3) is gauge invariant, whih proves the independene of the hoie of υ for D. Using equation 2.2) we get h D,τ = 1 2πτ w + τw), hene for any smooth funtion f: h D,τ fω = 1 2πτ w + τw)fω Σ = 1 2πτ Σ Σ w f)ω + 1 2π Σ wfω. By [HJS96, Theorem 1.1], w onverges to 2πδ D in the sense of measures as τ. Thus the rst term onverges to 0, and the seond term onverges to p D fp), whih ompletes the proof. The spae of simple divisors, Sym d sσ), is an open dense set in Sym d Σ), and its omplement is alled the big diagonal. When D is simple, a tangent vetor in T D Sym d Σ) an be given by speifying a tangent vetor to Σ at eah divisor point. Thus the rank d omplex vetor bundle K Sym d sσ) dened by K D = K p p D is isomorphi to T 0,1 Sym d sσ). We next use ideas of [T99, Lemma 3.3] to onstrut an almost unitary isomorphism from K to T 0,1 Sym d sσ). Fix a simple divisor D, and let υ be a orresponding τ-vortex eld. Dene ρ D = min{distp, q) p, q D & p q} {injσ, ω)}), 2.5) 8

9 where injσ, ω) is the injetivity radius of the metri. Let χ be a smooth funtion on [0, ) that satises 0 χ 1, χ [0,1] = 1, and χ [2, ) = 0, and set χ p = χ 2distp ρ D ). 2.6) For eah Θ = {θ p } p D K D let θ p be the extension of θ p to the open ball of radius injσ, ω) entered at p using the exponential map. Dene a smooth setion σ Θ of K supported in neighborhood of D by setting σ Θ = p D χ p θp 2.7) and extending by 0 to all of Σ. Note that σ Θ satises σ Θ p) = θ p & σ Θ = Odist D ) p D. 2.8) Finally, for eah suh υ and Θ dene Y υ,θ as ) Y υ,θ = 1 2wσΘ 2πτ, iλσ Θ φ) Ω 0,1 Ω 0 L, 2.9) where again w = 1 2 τ φ 2). Note that Y υ,θ is gauge equivariant, that is, for every g G: g Y υ,θ = Y gυ),θ. 2.10) The following analyti result is the key ingredient needed to ompute the asymptoti urvature in Theorem 4.1 and holonomies in the Main Theorem. Theorem 2.3. [The asymptoti form of horizontal vetors] For every υ P τ and Θ K D as above, there is a unique Z υ,θ Ω 0 Ω 0,1 L suh that X υ,θ = Y υ,θ L υz υ,θ ) 2.11) is a horizontal tangent vetor at υ. Moreover, the following asymptoti estimates hold: 1) [L 2 -estimate] Y υ,θ 2 L 2 Σ) p D θ p 2 as τ. 2) [Pointwise bound] X υ,θ Y υ,θ = O τ 1/2 exp τdistd from D, and is the positive number from Theorem 2.1. )), where dist D is the distane Equation 2.11) denes a bundle map from K to T 0,1 Sym d sσ) by D, Θ) D, Π X υ,θ )), 2.12) where υ is any τ-vortex eld orresponding to the divisor D, and Π is the projetion from P τ to M τ = Sym d Σ). Equation 2.10) implies that 2.12) does not depend on the hoie of υ. Furthermore, this map is almost unitary by Statements 1) and 2). Similar results have only been known for at metris [T99, Lemma 3.3]. 9

10 Proof of Theorem 2.3: Fix Y υ,θ as in equation 2.9). Sine L υ is ellipti, and kerl υ) = {0}, by Corollary 1.5, the operator L υ L υ is has a bounded inverse L υ L υ) 1. Thus the equation has a unique solution for Z υ,θ given by Consequently X υ,θ in equation 2.11) is horizontal. The pointwise norm of Y υ,θ satises L υ Y υ,θ L υz υ,θ )) = ) Z υ,θ = L υ L υ) 1 L υ Y υ,θ )) 2.14) Y υ,θ 2 = h D,τ σ Θ 2, where h D,τ is dened in equation 2.3). Using equation 2.4), one gets Statement 1). In order to prove Statement 2), we put Z υ,θ = f υ,θ, ξ υ,θ ) Ω 0 Ω 0,1 L in equation 2.13) to obtain the equations φ 2) f υ,θ = 1 8πτ w σ Θ, φ 2) ξ υ,θ = i 4πτ Λ 0,1 σ Θ φ ). 2.15a) 2.15b) Let G [φ] be the Green's operator of the non-degenerate ellipti operator H [φ] = φ 2 on Ω 0. Both H [φ] and G [φ] depend only on the gauge equivalene lass of φ. By an abuse of notation G [φ] will also denote the orresponding Green's funtion, whih is a positive, symmetri funtion on Σ Σ with a logarithmi singularity along the diagonal. With this denitions, we an write f υ,θ as f υ,θ = 1 8πτ G [φ] w σ Θ ω. 2.16) Standard ellipti theory gives the following bounds on the Green's funtion: G [φ] x, y) 1 + ln τdistx, y) ) ) ) exp τdistx,y) ) dg [φ] x, y) τdistx,y) exp Σ τdistx,y) 2.17a) 2.17b) for some R + independent of τ or D see [T99, Equation 6.10)]). Using equation 2.16) and inequalities 2.17a) and 2.17b), together with the bound on w in inequality 2.1b) and on σ Θ in equation 2.8) we get after possibly inreasing ) f υ,θ ) τ exp τdistd fυ,θ τ exp τdistd 2.18a) ). 2.18b) 10

11 Before turning our attention to equation 2.15b), note that we have the following two salar identities Using the Kähler identity on Ω 0,1 L ξ υ,θ 2 = 2Re ξ υ,θ ξ υ,θ ) 2 ξ υ,θ a) ξ υ,θ 2 = 2 ξ υ,θ ξ υ,θ 2 d ξ υ,θ 2, 2.19b) = 2 iλf = τ φ 2), 2.20) in equation 2.15b), together with the Cauhy-Shwarz inequality in equation 2.19a), and Kato's inequality f. [FU91, Equation 6.20)]) gives us d ξ υ,θ ξ υ,θ 2.21) τ) ξ υ,θ τ σ Θ φ. 2.22) Equation 2.22), together with the bound on φ in inequality 2.1b) and the bound on σ Θ in equation 2.8) gives us again after possibly inreasing ) ξ υ,θ ) τ exp τdistd. 2.23) Applying to equation 2.15b) gives an ellipti equation on ξ υ,θ. Similarly to the previous omputation we get the following inequality: τ) ) ξ υ,θ τ exp τdistd. Thus after possibly inreasing one last time) ξ υ,θ τ exp τdistd Finally, inequalities 2.18a), 2.18b), 2.23) and 2.24) give us ). 2.24) X υ,θ Y υ,θ = L υz υ,θ ) 2 f υ,θ + φ f + 2 ξ + φ ξ )) = O τ 1/2 exp, whih ompletes the proof of Statement 2). τdistd 3. The Berry onnetion The τ-vortex prinipal bundle is the prinipal G-bundle Π : P τ M τ desribed in Setion 1, with Πυ) = [υ]. In Lemma 1.1 we onstruted a horizontal distribution on the τ-vortex prinipal bundle, whih is the orthogonal omplement of the kernel of Π. This distribution is G-invariant, so is a onnetion in the distributional sense f. [KN63, Chapter II]), whih we all the Berry 11

12 onnetion. The orresponding onnetion 1-form is the unique LieG)-valued 1-form A that satises the three onditions: 1) kera υ ) is the horizontal subspae at υ P τ, 2) g A) gυ) = ad g A υ ), for all g G, 3) AX f ) = if, for all if W 2,2 Σ; ir) = LieG), where X f υ) = idf, ifφ), as dened in Lemma 1.2. The next lemma gives a formula for A υ. Reall that for eah τ-vortex eld υ =, φ), the Green's operator G [φ] is the inverse of the non-degenerate ellipti operator H [φ] = φ 2. Lemma 3.1. The LieG)-valued 1-form on P τ dened as A υ a, ψ) = 1 2 G [φ]d a + i Imhψ, φ))) 3.1) is the onnetion 1-form orresponding to the Berry onnetion. Proof. The right-hand side of equation 3.1) is the omposition of the non-degenerate Green's operator and a LieG)-valued 1-form. In the proof of Lemma 1.1 we saw that the kernel of this 1-form is exatly the horizontal subspae. This proves Condition 1) above. Beause G is Abelian, the adjoint representation of G is trivial, and hene the Condition 2) redues to g A) gυ) = A υ. Sine g a, ψ) = a, gψ), we have g A gυ) a, ψ) = 1 2 G [φ]d a + i Imhgψ, gφ))) = 1 2 G [φ]d a + i Imhψ, φ))), thus g A gυ) = A υ. This proves Condition 2). Finally, we show that A is the anonial isomorphism between the bers of the vertial bundle and the Lie algebra of G, that is AX f ) = if for every f C Σ; R): A υ X f ) = 1 2 G [φ]d idf) + i Imhifφ, φ)))ω ) = ig 1 [φ] 2 f φ 2) f = if, thus Condition 3) holds. We an use Lemma 3.1 to ompute the urvature 2-form of the Berry onnetion. Sine G is Abelian, the urvature, alled the Berry urvature, is a LieG)-valued 2-form whih desends to the base spae M τ. Theorem 3.2. The urvature 2-form of the Berry onnetion at [υ] M τ is Ω [υ] X, Y ) = G [φ] i Imhψ X, ψ Y ))) 3.2) where a X, ψ X ) and a Y, ψ Y ) are the horizontal lifts of X and Y, respetively, at υ. Moreover, equation 3.2) does not depend on the hoie of the τ-vortex eld υ representing [υ]. 12

13 Proof. The laim about the independene of the hoie υ is immediate sine everything on the right-hand side is gauge invariant. The urvature is the unique LieG)-valued 2-form Ω on M τ that satises Π Ω) = da, where Π is the projetion from P τ to M τ. Thus it is enough to ompute da υ a X, ψ X ), a Y, ψ Y )) and ompare it with equation 3.2). Reall, that the formula for the exterior derivative ) da X, X = X )) )) ]) AỸ A Ỹ X A[ X, Ỹ, 3.3) where X and Ỹ are smooth loal extensions of a X, ψ X ) and a Y, ψ Y ) respetively. Choose the extensions so that their Lie braket vanishes at υ. Let Υ t be the loal ow generated by X, so Υ t υ) = υ + t a X, ψ X ) + O t 2) ). Sine AỸ = 0 at Υ 0 υ) = υ, we have )) ) 1 X υ AỸ = lim t 0 t A Υ tυ)ỹ Υt υ)). Note that Ỹ Υ tυ)) = Υ t ) a Y, ψ Y )) + O t 2) [ ], beause X, Ỹ = 0. Finally, let us write G [φ+t ψx +Ot 2 )] = G [φ] + t G X [φ] + O t 2). Keeping only the linear terms, we obtain )) ) 1 X υ AỸ = lim t 0 t A Υ tυ)ỹ Υt υ)) = lim 1 t 0 2t G [φ+t ψ X +Ot 2 )] d a Y + i Imhψ Y, φ + t ψ X )) + O t 2)) = lim 1 t 0 2t itg[φ] Imhψ Y, ψ X ))) + t G X [φ]d a Y + i Imhψ Y, φ))) ) = i 2 G [φ]imhψ Y, ψ X ))), where we used the fat that d a Y + i Imhψ Y, φ)) = 0 for tangent vetors. Interhanging X and Ỹ hanges sign, sine Imhψ Y, ψ X )) is skew. Substituting these into equation 3.3), and noting that the ommutator vanishes, gives equation 3.2). υ 4. The asymptoti Berry urvature In this setion we use Theorems 2.3 and 3.2 to analyze the Berry urvature in the large τ limit. As before, let D be a simple divisor, and υ =, φ) be th orresponding τ-vortex eld. For eah p D, hoose Θ p = {θ p,q } q D K D, so that θ p,q = δ p,q. Let σ p = σ Θp be the orresponding setion dened by equation 2.7), and let X υ,θp = a p, ψ p ), as dened in Theorem 2.3. By equation 2.11), X υ,θp = Y υ,θp L ) υ Zυ,Θp. 4.1) where Z υ,θp = f p, ξ p ) Ω 0 Ω 0,1 L. It is easy to see that in Statement 2) of Theorem 2.3 we an now replae dist D with dist p, the distane from the single point p. 13

14 ) The set {X p } p D, where X p = Π Xυ,Θp T[υ] M τ, is an asymptotially orthonormal basis for the horizontal subspae at υ, in the sense that as τ )) X p X q = δ p,q + O exp δ p,q. τρd Finally, for eah tangent vetor X, let X = X be the metri-dual ovetor. Theorem 4.1. [The asymptoti Berry urvature] There is a positive number = Σ, ω, J, L, h) suh that if τ > τ 0 = 4πd AreaΣ) and [υ] is a simple τ-vortex, then the Berry urvature satises Ω [υ] = χp ) iw δ p,q X p ix q ) + ibτ p,q Xp Xq + icτ p,q ix p ) ix q ) ), 4.2) p,q D πτ + iap,q τ where χ p as dened in Equation 2.6), and A p,q τ A p,q τ + B p,q τ + Cτ p,q = O, Bτ p,q, and Cτ p,q are real funtions, with τ 1 exp τρd )). 4.3) Proof. For p q, Theorems 2.1 and 2.3 imply that hψ p, ψ q ) = ψ p ψ q exp τdistp+dist q) ) exp τρd ). This inequality together with the fat that G [φ] 1) = O τ 1) from inequality 2.17a), gives equation 4.3) in this ase. In general, for every p D, Theorem 3.2 shows that 1 i Ω [υ]x p, ix p ) = G [φ] ψ p 2). 4.4) This is non-negative, beause G [φ] is given by onvolutions with the positive Green's funtion. By equation 4.1), we an write ψ p = 1 2πτ iλσ p φ) 2 ξ p + f p φ. 4.5) Applying the bounds in Theorems 2.1 and 2.3 to equations 4.4) and 4.5), we obtain 1 i Ω [υ]x p, ix p ) = 1 2πτ G [φ] σ p 2 φ 2) ))) τdistp + G [φ] O exp. 4.6) By inequality 2.17a), and the positivity of the Green's funtion the last term is ))) G [φ] O exp = O τ 1 exp τdistp τdistp )). 4.7) Theorem 2.1 and equation 2.2) gives us H [φ] χ p w) = 1 2 χ p φ 2 )) τdistp + O τ exp. 4.8) Thus we an write the main term in equation 4.6) as ) ) σp G 2 φ 2 χp [φ] 2πτ = G φ 2 [φ] 2πτ + O G [φ] τdist 2 p exp w = χ p πτ τ + O 1 τdistp exp 14 ))) τdistp )), 4.9)

15 sine σ p 2 χ p = O dist 2 p) by 2.8). Combining equations 4.6), 4.7) and 4.9) yields 1 i Ω w [υ]x p, ix p ) = χ p πτ τ + O 1 )) τdistp exp. This ompletes the proof of equations 4.2) and 4.3). 5. The asymptoti Berry holonomy A onnetion on a prinipal G-bundle P X denes the notion of parallel transport; f. [KN63, Chapter II]. Parallel transport around a loop is alled holonomy. Holonomy an be viewed as a map from the loop spae of X to the spae of onjugay lasses of G. For Abelian G, the later spae is anonially isomorphi to G. In our ase, the τ-vortex prinipal bundle, P τ M τ, is a prinipal G-bundle equipped with the Berry onnetion. The physial interpretation is that if one adiabatially moves the divisor points along a urve Γ in Sym d Σ), then the orresponding τ-vortex eld evolves by the parallel transport dened the Berry onnetion f. [K50], and [B84]). In partiular, when Γ is a loop, the holonomy of the Berry onnetion, alled the Berry holonomy, is a gauge transformation. In this setion, we give analyti and topologial desriptions of the gauge transformations that arise as Berry holonomies. Sine the Berry holonomy is a map from the loop spae of τ-vortex moduli spae, we reall some well-known properties of loops in M τ = Sym d Σ). We all a loop Γ in Sym d Σ) a single vortex loop if only one of the divisor points moves, and all other divisor points are xed. In other words, single vortex loops are indued by loops in Σ that are based at one of the divisor points. Every loop in Sym d Σ) an be deomposed up to homotopy and thus homology) to a produt of single vortex loops. Moreover, H 1 M τ ; Z) = H 1 Sym d Σ); Z) = H1 Σ; Z), 5.1) where the last isomorphism is given by sending single vortex loops to their homology lasses by the Hurewiz homomorphism. Reall that the omplement of Sym d sσ) is alled the big diagonal. A loop in Sym d Σ) is regular if it is a smooth, embedded immersed, if d = 1) loop that does not interset the big diagonal. The big diagonal is empty when d = 1. When d > 1 the big diagonal is a subvariety of odimension at least 2, thus every smooth loop in Sym d Σ) an be made regular after a small smooth perturbation. Now onsider the anonial overing map Σ d s Sym d sσ), where Σ d s is the spae of ordered d-tuples in Σ without repetition. Given a regular loop Γ that starts at the simple divisor D = Γ0) Sym d sσ), eah lift D Σ d of D determines a unique lift Γ of Γ. The lift Γ an be regarded as a d-tuple γ 1,..., γ d ) of urves not neessarily loops) in Σ. The shadow of Γ, shγ) Σ is shγ) = imageγ i ), 5.2) 15

16 where the union is over all non-onstant γ i. The set shγ) has a natural orientation oming from the orientation of Γ. Sine Γ is regular, shγ) is a union of immersed, oriented loops, hene it is an integer 1-yle in Σ. The homology lass in H 1 Σ; Z) represented by shγ) is independent of the hoie of the lift D. We denote this lass by [Γ]. It is easy to hek the homotopy lass of Γ in Sym d Σ) is sent to the homology lass [Γ] by the isomorphism 5.1). In general, for a single vortex loop, only one of the γ i 's is not onstant, say γ, and [Γ] = [γ] H 1 Σ; Z). Example 5.1. An example of shγ) is seen on Figure 1, where [Γ] = [γ 1 ] + [γ 2 ] + [γ 3 ] = [γ 3 ], sine both γ 1 and γ 2 are null-homologous. Thus Γ is homologous to a single vortex loop. We all a loop a positively oriented) vortex interhange if, as in Figure 2, only two γ i 's, say γ 1 and γ 2, are not onstant, and the omposition Γ = γ 1 γ 2 is the oriented) boundary of a disk. γ 1 γ 1 γ 2 γ 3 γ 2 Figure 1. Single vortex loops: One divisor point moves along one of the γ i 's. All other divisor points are xed. Figure 2. Vortex interhange: One divisor point moves along γ 1 and another divisor point moves along γ 2. All other divisor points are xed. Sine the Berry holonomy has values in the gauge group G, we also reall a ouple well-known properties of gauge transformations. Elements g G represent lasses in H 1 Σ; Z) as follows: For a losed manifold X and a nitely generated Abelian group G, H n Σ; G) is anonially isomorphi to the spae [X, KG, n)] of homotopy lasses of ontinuous maps from X to the Eilenberg-MaLane spae KG, n) f. [H02, Theorem 4.57]). Sine KZ, 1) = U1) and G is homotopy equivalent to [Σ, U1)], we get that H 1 Σ; Z) is anonially isomorphi to π 0 G), whih is also a group, beause G is. In fat, if G 0 is the identity omponent of G, then π 0 G) G/G 0, and the short exat sequene {0} G 0 G H 1 Σ; Z) {0} 5.3) is non-anonially split. The isomorphism between π 0 G) and H 1 Σ; Z) an be understood on the o)yle level; sine Σ is a losed, oriented surfae, H 1 Σ; Z) is anonially isomorphi to HomH 1 Σ; Z), Z). An element g G denes an element [g] HomH 1 Σ; Z), Z) via [g][γ]) = gγ) = 1 2πi g 1 dg Z. 5.4) 16 γ

17 The Berry holonomy an be viewed as a map from the loop spae ΩM τ of M τ to G. It then indues a map hol on the onneted omponents: π 1 M τ ) π 0 ΩM τ ) hol π 0 G) H 1 Σ; Z). Sine ohomology groups are Abelian, the above map fators down to the homology, and thus denes a homomorphism: hol : H 1 Σ; Z) H 1 M τ ; Z) hol H 1 Σ; Z), 5.5) where the rst isomorphism is from 5.1). Using equation 5.4), an expliit formula for hol an be given as follows: if g = holγ), then hol [Γ]) evaluates on any 1-yle γ by hol [Γ])[γ]) = 1 2πi g 1 dg. 5.6) γ Finally, reall that a k-urrent is a ontinuous linear funtional on Ω k. A 1-form a Ω 1 denes a 1-urrent by C a b) = a b. 5.7) Similarly, a smooth 1-hain γ denes a 1-urrent by C γ b) = Σ γ b. 5.8) We say that the 1-urrents in equations 5.7) and 5.8) are the 1-urrents dened by a and γ respetively. Now we are ready to prove our main theorem about the Berry holonomy, stated in the introdution. The proof of the Main Theorem. Sine every smooth path an be made regular by an arbitrarily small smooth perturbation, it is enough to hek regular loops, Γ. We prove Statement 1) rst: Let υ be a τ-vortex eld orresponding to D, and Γ be the horizontal lift of Γ starting at υ P τ. Sine Γ is regular, Γt) is simple for all t, thus we an apply Theorem 2.3 to the veloity vetor Γ t) = a t, ψ t ), whih is horizontal, and hene obtain By denition of the holonomy, a t + ψ t τ 1 g τ υ) = υ + 0 d γ it) exp i=1 Γ t)dt = τdistγi t) 1 a t dt, φ + 0 ). 5.9) ψ t dt.

18 On the other hand, by the denition of the gauge ation for Abelian groups), Thus we have and g τ υ) = + g τ dg 1 τ, g τ φ ). g τ dg 1 τ = g τ 1)φ = a t dt, 5.10) ψ t dt. 5.11) Let V Σ be any ompat set in the omplement of shγ). Sine distshγ), V) > 0, Theorem 2.1 shows that φ τ 2 on V for all large τ. Hene equations 5.9) and 5.11) imply that for x V φ g τ 1 x τ 2 g τ 1 x 1 0 ψ t x) dt τ d 1 i=1 0 γ it) exp Using g τ = 1, dg 1 = dg τ, and equations 5.9) and 5.10), we also obtain τ dg τ x d 1 i=1 0 a t x) dt τ d 1 i=1 0 γ it) exp ) τdistγi t),x) dt. 5.12) ) τdistγi t),x) dt. 5.13) Combining the last two inequalities gives τ gτ 1 x + dg τ x ) τ exp τdistshγ),v), 5.14) for all large τ, whih implies that g τ 1 and dg τ onverge to 0, uniformly on V, as τ. This proves Statement 1). In order to prove Statement 2), we rst assume that Γ is a single vortex loop, for whih shγ) is an embedded loop, that bounds an embedded disk B in Σ, and D = Γ0) = Γ1) has no divisor points in the interior B, and let p be the divisor point in D that is moved by Γ. There is a anonial embedding of B into M τ that sends a point x B to the divisor x + D p). The image of this map, B, is an oriented) disk in M τ, whose oriented) boundary is Γ. We will denote this embedding by π B : B B. Sine Γ is null-homotopi, g τ is in the identity omponent of G, and so an be written as g τ = exp2πif τ ), where f τ is a smooth, real funtion on Σ. By Stokes' Theorem, f τ = 1 2πi Ω. 5.15) 18 B

19 If j is a path as in Statement 2) of the Main Theorem, then ϕ τ = f τ j. Using equation 5.15) we see that ϕ τ 1) ϕ τ 0) = f τ j1)) f τ j0)) = 1 2πi Ωj1)) Ωj0))), where j1) is in B, and j0) is not. To evaluate this integral, we reparametrize using π B. By Theorem 2.3, if ω τ is the pullbak of the Kähler lass of M τ from B to B using π B, then ω τ = πτω + O1). For eah x B, let w x be the funtion w, dened in equation 1.2), orresponding to the divisor D = π B x), and let d x = distx, {j0), j1)}). By [HJS96, Lemma 1.1], w x B onverges to 2πδ x, in measure, sine D B = {x}. Now using Theorem 4.1, the last integral above equals to ))) ϕ τ 1) ϕ τ 0) = 1 wxj1)) wxj0)) 2π πτ + O τ 1 exp 2 τd x πτ + O1))ω x = B B δx, j1)) + O = 1 + O τ 1). exp τdx B )))ω x + O τ 1) This implies Statement 2) in the ase where Γ is a simple vortex loop that bounds a disk. In the general ase, let I2ε) be the tubular 2ε-neighborhood of I = imagej), where ε is small enough so that I2ε) shγ) is a single embedded ar. Let Γ be a single, embedded bounding loop in Σ, as in the previous ase, for whih shγ) and shγ ) oinide on I2ε). Let Γ out and Γ out denote the parts of Γ and Γ, respetively, for whih shγ out ) and shγ out) lie in the omplement of I2ε). Similarly Γ in = Γ in is their ommon part. Now one an join Γ out with the reverse of Γ out to get a pieewise smooth loop Γ new, suh that shγ new ) is disjoint from I2ε), and furthermore, the loop sum Γ Γ new and Γ y dier only by Γ out and its reverse, thus g τ hol Γ new = hol Γ. On Iε) we have that hol Γ new onverges to 1 in the C 1 -topology as τ. Sine the rossing formula holds for Γ, it must hold for Γ as well. This establishes the general ase of Statement 2). In order to prove Statement 3) we pik a nite over U of Σ by oordinate harts suh that for every U, Ψ) U the preimage of the intersetion Ψ 1 shγ) U) R 2 is either i) empty, ii) the x-axis, or iii) the union of the two axes. By using a subordinate partition of unity, it is enough to prove Statement 3) for 1-forms that are supported inside one of these harts. Fix one suh hart U, Ψ) U, and let b Ω 1 a 1-form with suppb) U. Let us also write Ψ a τ = df τ 5.16a) Ψ b = Adx + Bdy, b)

20 where A and B are ompatly supported funtions on R 2. If Ψ 1 shγ) U) is empty, then the support of b and shγ) are disjoint, hene C shγ) b) = b = b = b = 0. shγ) shγ) suppb) On the other hand, as τ, a τ 0 on suppb) by Statement 1), and hene whih proves Statement 3) in the ase i). C aτ b) max suppb) { a τ } b L 1 0, Next, if Ψ 1 shγ) U) is the x-axis, then by equation 5.16b), We also have C shγ) b) = C aτ b) = Σ Ψ 1 shγ) U) a τ b = Again using equation 5.16b), this beomes C aτ b) = df τ Adx + Bdy) = R 2 R 2 R 2 U Ψ b = a τ b = R 2 df τ Adx + f τ Bdy) Ax, 0)dx. 5.17) df τ Ψ b. 5.18) R 2 f τ ) B x A y dx dy. The rst integral on the right-hand side is zero by Stokes' Theorem and the fat that A and B are ompatly supported. By Statement 2), we hoose f τ so that it onverges to 0 when y is positive and to 1 when y is negative. As τ, we then have ) C aτ b) = f A τ y B x dx dy = 0 A B y x, y) x )dydx x, y) Ax, 0)dx. Together with equation 5.17) this proves Statement 3) for ase ii). Finally, if Ψ 1 shγ) U) is the union of the two axes, then again by equation 5.16b), C shγ) b) = Ψ 1 shγ) U) Ψ b = 20 Ax, 0)dx + B0, y)dy. 5.19)

21 As before, we also have C aτ b) = R 2 f τ ) A y B x dx dy. 5.20) But now Statement 2) shows that, as τ, f τ onverges to 0 in the upper left quadrant, to 1 in the upper right and the lower left quadrants, and to 2 in the lower right quadrant. Hene by equation 5.20), C aτ b) 2 R + R + R + R + + = R R A B y x, y) x )dydx x, y) A B y x, y) x )dydx x, y) A B y x, y) x )dydx x, y) Ax, 0)dx + B0, y)dy, where the last step is an elementary omputation. Together with 5.19), this proves Statement 3) for the ase iii). To prove Statement 4), aording to equation 5.6) and the denition of the Poinaré duality, we need to show that for any [γ] H 1 Σ; Z), hol [Γ])[γ]) = 1 2πi gτ 1 dg τ = [Γ] [γ]. 5.21) γ Sine everything in 5.21) is homotopy invariant, we have the freedom to hange Γ by a homotopy. Reall from 5.1) that Γ an be deomposed, up to homotopy, to a produt of single vortex loops; thus it sues to hek Statement 4) for a basis of π 1 Sym d Σ) ) = H1 Σ; Z). We use a sympleti basis: a set of simple losed urves {α i, β i } 1 i gσ) suh that α i intersets β i transversally and positively one, and α i β j = α i α j = β i β j = for i j. There is always suh a set, and {[α i ], [β i ]} 1 i gσ) is a basis of H 1 Σ; Z), with [α i ] [α j ] = 0, [β i ] [β j ] = 0, [α i ] [β j ] = δ i,j, where is the homology intersetion. Denote the orresponding single vortex loops in Sym d Σ) by { α i, β i } 1 i gσ). To prove Statement 4), we need only to verify 5.21) for every pair in the basis. When γ {α i, β i }, and Γ { α j, β j } with i j, we have by Statement 1) that igτ 1 dg τ = O τ exp γ 21 τdistγ,shγ)) )). 5.22)

22 When γ = α i and Γ = α i, for some i, we an hose another representative α i for [γ] = [α i] that is disjoint from α i. Thus, again by Statement 1), we have )) igτ 1 τdistαi,α dg τ = O τ exp i). 5.23) γ Thus all integrals in 5.22) and 5.23) onverge to 0 as τ. On the other hand, these integrals are integer multiples of 2π, so they had to be 0 for all τ > τ 0. Finally, assume that i = j and γ = α i and shγ) = β i, or γ = β i and shγ) = α i. In order to prove the rst of these ases, let us x a small embedded segment j on γ = β i that intersets shγ) = α i one positively. Suh paths exist by the onstrution of the basis. Write g τ I = exp2πiϕ τ ), and so gτ 1 dg τ I = 2πidϕ τ. Thus by equation 5.6), hol [Γ])[γ]) = I dϕ τ + 1 2πi γ I gτ 1 dg τ τ = ϕ τ 1) ϕ τ 0) + O exp τdistshγ),{j0),j1)}) By Statement 2), this onverges to 1 as τ. On the other hand, hol [Γ])[γ]) is an integer, so it had to be 1 for all τ > τ 0. The same argument an be used in the ase of shγ) = β i and γ = α i, whih ompletes the proof of Statement 4) and the Main Theorem. )). Corollary 5.2. For all τ > τ 0, the spae P τ is an innite-dimensional vetor bundle over a onneted, oriented and smooth manifold without boundary, M τ. This manifold has real dimension 2d+1 and is a U1)-prinipal bundle over the universal over of M τ. In partiular P τ is homotopy retrats to M τ. Proof. First we will impose the Coulomb gauge: x a τ-vortex eld υ 0 =, φ) P τ. We say that υ =, φ ) is in Coulomb gauge with respet to υ 0 if the 1-form a = is satises: d a = 0. For eah υ P τ there is in fat a gauge transformation g that is in the identity omponent of G, and is unique up to onstant gauge transformations fators in U1)), suh that gυ) is in Coulomb gauge with respet to υ 0. A proof of this, whih applies to our ase too, an be found, for example, in [EN11, Lemma 2.1]. The set M τ P τ, alled the Coulomb slie, onsisting the τ-vortex elds that are in Coulomb gauge with respet to υ 0 intersets eah ber. Fix then a point x Σ, and require gx) = 1. Suh a g = g υ is then unique, moreover, an be written as g υ = expif υ ), and f υ is also unique if one presribes f υ x) = 0. Set g t,υ = expitf υ ). Then the map dened as rt, υ) = g t,υ υ) is a homotopy retration of P τ to the Coulomb slie. The intersetion of eah ber with the Coulomb slie is a olletion of irles, due to the U1) ambiguity mentioned above. Moreover, these irles are in bijetion with π 0 G) = π 1 M τ ). Thus M τ = M τ /U1) is a π 1 M τ )-over of M τ, whih is the universal over if onneted. 22

23 Our Main Theorem implies that P τ is onneted, by the following argument: let υ and υ be two arbitrary τ-vortex elds. Sine simple divisors are dense in Sym d Σ), we an assume, that the orresponding divisors are simple. Join the two divisors by a regular path Γ 0. Then hol Γ0 υ) is equal to gυ ) for some g G. If g is not in the identity omponent of G, then it represents a nonzero ohomology lass [g] H 2 Σ; Z). Let γ be a smooth loop based at a divisor point of υ that represents the Poinaré dual of [g]; let γ be the indued single vortex loop, and set Γ = γ 1 Γ 0. Now υ and hol Γ υ) = υ are onneted by the path in P τ given by parallel transport. On the other hand, υ and υ are gauge equivalent, and the onneting gauge transformation is in the identity omponent of G, whih means that there is a path from υ to υ. Thus P τ is onneted, but then so is M τ, whih ompletes the proof. 6. The large area limit Consider the energy 1.1) for the ritial oupling onstant λ = 1 and with τ = 1. Bradlow's riterion for the existene of irreduible vorties in this ase beomes τ 0 = 2πd AreaΣ) < 1, 6.1) using the area with respet to the given area 2-form ω. Even when inequality 6.1) does not hold for ω, it still holds for ω t = t 2 ω if t > t 0 = τ 0. Let P t be the spae of all solutions of the 1-vortex equations with Kähler form ω t for t > t 0. A pair, φ) C L Ω 0 L is in P t if iλ t F = φ 2) 6.2a) φ = 0, 6.2b) where Λ t = Λ/t 2. Let M t be the orresponding moduli spae P t /G. Bradlow's Theorem still holds, hene M t = Sym d Σ), where the dieomorphism is again given by the divisor of the φ-eld. By [B90, Proposition 5.1], the following diagram is ommutative when t 2 = τ: P t Φ t P τ, Sym d Σ) where Φ t is the isomorphism of prinipal bundles given by Φ t, φ) =, tφ). The L 2 -metri on P t is dened by equation 1.6), but with Hodge operator and area form given by ω t. For all X T P t, we now have: Φ t ) X Pτ = t X Pt. 6.3) Thus the L 2 -metri of P t is onformally equivalent to the pullbak of the L 2 -metri of P τ via the bundle isomorphism Φ t. 23

24 The Berry onnetion on P t M t is again dened as the orthogonal omplement of the vertial subspaes, hene it is the same as the pullbak of the Berry onnetion on P τ via Φ t. Thus the results of Theorems 2.3 and 4.1 and our Main Theorem hold in the large area limit t ): Main Theorem for the large area limit. The onlusions of the Main Theorem in the introdution hold for the prinipal G-bundle P t M t with τ replaed everywhere by t 2. Referenes [AA87] Y. Aharonov and J. Anandan. Phase hange during a yli quantum evolution. Phys. Rev. Lett., 58: , [AK02] I. S. Aranson and L. Kramer. The world of the omplex Ginzburg-Landau equation. Rev. Mod. Phys., 74:99143, [B06] J. M. Baptista. Vortex equations in abelian gauged sigma-models. Commun. Math. Phys., 261:161194, [B11] J. M. Baptista. On the L 2 -metri of vortex moduli spaes. Nul. Phys., B, 844:308333, [B84] M. V. Berry. Quantal phase fators aompanying adiabati hanges. Proeedings of the Royal Soiety of London A: Mathematial, Physial and Engineering Sienes, ):4557, [B90] S. B. Bradlow. Vorties in holomorphi line bundles over losed Kähler manifolds. Commun. Math. Phys., 1351):117, [BR14] M. Bokstedt and N. M. Romao. On the urvature of vortex moduli spaes. Math. Z., 277:549573, [CM05] H. Y. Chen and M. S. Manton. The Kähler potential of abelian Higgs vorties. J. Math. Phys., 46:052305, [DDM13] D. Dorigoni, M. Dunajski, and N. S. Manton. Vortex motion on surfaes of small urvature. Annals Phys., 339:570587, [EN11] G. Etesi and Á. Nagy. S-duality in abelian gauge theory revisited. J. Geom. Phys., 61:693707, [FU91] Daniel S. Freed and Karen K. Uhlenbek. Instantons and four-manifolds, volume 1 of Mathematial Sienes Researh Institute Publiations. Springer-Verlag, New York, seond edition, [GL50] V. L. Ginzburg and L. D. Landau. On the theory of superondutivity. Zh. Eksp. Teor. Fiz., 20: , [H02] A. Hather. Algebrai topology. Cambridge University Press, Cambridge, [HJS96] M. Hong, J. Jost, and M. Struwe. Asymptoti limits of a Ginzburg-Landau type funtional. In Geometri analysis and the alulus of variations, pages Int. Press, Cambridge, MA, [I01] D. A. Ivanov. Non-abelian statistis of half-quantum vorties in p-wave superondutors. Phys. Rev. Lett., 862):268, [JT80] A. Jae and C. H. Taubes. Vorties and monopoles. Progress in Physis. Birkhäuser, Boston, Mass., [K50] T. Kato. On the Adiabati Theorem of Quantum Mehanis. Jour. Phys. So. Japan, 56):435439, [K85] M. Kohmoto. Topologial invariant and the quantization of the Hall ondutane. Annals of Physis, 1602): , [KN63] S. Kobayashi and K. Nomizu. Foundations of dierential geometry. Vol I. Intersiene Publishers, New York-London, [MM15] R. Maldonado and N. S. Manton. Analyti vortex solutions on ompat hyperboli surfaes. J. Phys., A, 4824):245403, [MN99] N. S. Manton and S. M. Nasir. Volume of vortex moduli spaes. Commun. Math. Phys., 1993):591604,

25 [MS03] N. S. Manton and J. M. Speight. Asymptoti interations of ritially oupled vorties. Commun. Math. Phys., 236:535555, [R00] I. Mundet i Riera. A Hithin-Kobayashi orrespondene for Kähler brations. J. Reine Angew. Math., 528:4180, [T84] C. H. Taubes. On the Yang-Mills-Higgs equations. Bull. Amer. Math. So. N.S.), 102):295297, [T99] C. H. Taubes. GR = SW: ounting urves and onnetions. J. Di. Geom., 523):453609, Ákos Nagy) Department of Mathematis, Mihigan State University, East Lansing, MI address: nagyakos@math.msu.edu URL: akosnagy.om 25