Ginzburg-Landau equations on Riemann surfaces of higher genus
aa r X i v : . [ m a t h . A P ] A p r Ginzburg–Landau equations on Riemann surfacesof higher genus
D. Chouchkov ∗ , N. M. Ercolani † , S. Rayan ‡ , I. M. Sigal § April 23, 2019
Abstract
We study the Ginzburg–Landau equations on Riemann surfaces of arbi-trary genus. In particular, we– construct explicitly the (local moduli space of gauge-equivalent) solu-tions in the neighbourhood of the constant curvature ones;– classify holomorphic structures on line bundles arising as solutions tothe equations in terms of the degree, the Abel-Jacobi map, and sym-metric products of the surface;– determine the form of the energy and identify when it is below theenergy of the constant curvature (normal) solutions.Nous ´etudions les ´equations de Ginzburg-Landau d´efinies sur des surfacesde Riemann de genre arbitraire. En particulier,– nous construisons explicitement l’espace des modules locaux des solu-tions (´equivalentes par transformation de jauge) dans le voisinage dessolutions de courbure constante;– nous classifions les structures holomorphiques dans les fibr´es en droitesqui apparaissent comme solutions de ces ´equations, en fonction de leurdegr´e, de l’application d’Abel-Jacobi, et des produits sym´etriques desurface;– nous obtenons une expression pour l’´energie et identifions dans quellesconditions elle est inf´erieure l’´energie des solutions (normales) de cour-bure constante. ∗ Dept. of Math., U. of Toronto, Toronto, ON, M5S 2E4, Canada † Dept. of Math., U. of Arizona, Tucson, AZ, 85721-0089, USA ‡ Dept. of Math. & Stats., U. of Saskatchewan, Saskatoon, SK, S7N 5E6, Canada § Dept. of Math., U. of Toronto, Toronto, ON, M5S 2E4, Canada inzburg–Landau equations on Riemann surfaces, April 23, 2019 Let X be a Riemann surface with a complex structure given by a hermitianmetric h and let E be a smooth, unitary line bundle over X . The Ginzburg-Landau equations on X involve a section, ψ , and a connection one-form a on E and are written as ( ∆ a ψ = κ ( | ψ | − ψ,d ∗ da = Im( ψ ∇ a ψ ) , (1.1)where ∇ a and − ∆ a = ∇ ∗ a ∇ a are the covariant derivative and Laplacianassociated with a unitary connection form a (locally, ∇ a = d + ia ) and κ isthe Ginzburg-Landau parameter. The adjoints here are taken with respectto the inner products on sections and bundle-valued one-forms, respectively,induced by fixing a smooth hermitian inner product, k (which determinesthe U (1)-structure of E ) on the fibers of E and a hermitian metric, h on X (see e.g. [12, 25]).Equations (1.1) lie at the foundation of the macroscopic theory of su-perconductivity. Soon after their birth, they migrated to other areas ofcondensed matter physics and then to particle physics. They gave the firstexample of gauge field theory, led to the Yang-Mills-Higgs equations and, to-gether with the latter, formed a foundation of the standard model of particlephysics. (For some background, see [24, 29, 38].)Equations (1.1) are the Euler-Lagrange equation of the Ginzburg-Landauenergy functional E ( ψ, a, h ) = k∇ a ψ k + k da k + κ k ( | ψ | − k . (1.2)On R (the cylindrical geometry, in the physics literature), equations(1.1) are translation, rotation and gauge invariant (see (1.13) below). How-ever, for low magnetic fields, its ground state - the solution with the lowestenergy per unit area (see below) - turns out not to be gauge - translationinvariant. This was discovered by A.A. Abrikosov who suggested that forType II superconductors it has the symmetry of a lattice.The Abrikosov’s prediction was confirmed in experiments and was rec-ognized by a Nobel Prize. The corresponding state is called the Abrikosov,or vortex, lattice. As alluded above, it is the ground state for Type II su-perconductors. This phenomenon is related to the one of the crystallizationand proving the vortex lattice solutions are ground states is a major openproblem.The Abrikosov solutions can be reformulated as solutions of (1.1) on aflat torus. Hence, mathematically, it is natural to go one step further andconsider (1.1) on an arbitrary smooth Riemann surface. (On the physicsside, one can imagine superconducting thin membranes with surfaces of inzburg–Landau equations on Riemann surfaces, April 23, 2019 higher genera. In this case, (1.1) would have to be modified, but the presentmathematical theory would still apply.)We fix a genus g Riemann surface with a fixed homology basis (referredto as a marking). Recall that a hermitian metric, h , determines the Rie-mann metric, volume form (as the real and imaginary parts of h ) and thecomplex structure (relating the first two). We allow h to vary keeping thecomplex structure fixed within its conformal class. By a solution of (1.1)we understand the triple ( ψ, a, h ), (or the equivalence classes - or moduli -of such triples related by the gauge transformation, see below).To fix ideas, in what follows, we allow variations of Hermitian metricsonly along a family, h = rh ′ , r > , where h ′ is an arbitrary fixed metric.To formulate our result, we need some notation and definitions:The curvature of a connection a is defined as F a = da and a connection a on E is said to be of constant curvature iff F a is of the form F a = bω ,where b is a constant and ω is the symplectic volume form on X . By theChern-Weil relation (see (2.1) below), b = πn | X | , where | X | denotes the totalarea of X ;deg( E ) denotes the degree (a topological invariant with values in Z ) of E ; we assume E has the degree n and denote this topological bundle, uniqueup to homeomorphism, by E n . A cc,n denotes the space of constant curvature unitary (with respect to k ) connections on E n ; H rn and ~ H r denote the order r Sobolev spaces of sections of E n andco-closed (i.e. d ∗ α = 0) one-forms, α ∈ Ω (for r <
1, the derivative d ∗ α isdefined in the weak sense); O and O stand for error terms in the sense of the norms in H and ~ H , respectively.The main results of this paper can be summarized as follows Theorem 1.1.
Let ≤ n ≤ g and a c ∈ A cc,n be a regular value of theAbel-Jacobi map Φ (see (1.14) below). Assume r is close to πn/ | X | and ( r − b )( κ − κ c ( a c )) > . (1.3) with κ c ( a c ) defined below. Then the space, H , of holomorphic sectionsof the holomorphic bundle corresponding to a c (see Theorems 1.6 and E.1below) has the dimension ; the Ginzburg–Landau equations (1.1) on E n have the branch of solutions ( ψ s , a s , r s ) = (cid:0) sφ + O ( s ) , a c + O ( s ) , πn/ | X | + O ( s ) (cid:1) , (1.4) where φ = g ˆ φ , with g a gauge transformation (constructed explicitly) and ˆ φ ∈ H , normalized as k φ k H / | X | = 1 , and s ∈ R is given by the equation,with β ( a c ) defined below, s = κ ( r − b ) β ( a c )( κ − κ c ( a c )) + O ( κ ( r − b ) ); (1.5) inzburg–Landau equations on Riemann surfaces, April 23, 2019 up to gauge equivalence, these are the only solutions in H × ~ H × R in asmall neighbourhood of the solution branch { (0 , a c , r ) : r > } . Note that (1.3) follows from (1.5). Taking s = 0 in (1.4) gives a normalsolution (0 , a c , h ). In fact, it is well known, see e.g. [37], that(0 , a, h ) is a solution of (1.1) on E n iff a ∈ A cc,n (1.6)The solution (0 , a, h ), where a is a constant curvature connection on E ,will be called the normal state (irreducible solution in some mathematicsliterature).Now, we define the Abrikosov function β ( a c ) and the parameter threshold κ c ( a c ), a c ∈ A cc,n , used in the theorem, by: β ( a c ) := min φ ∈ Null ∂ ′′ ac , h| φ | i =1 h| φ | i , (1.7)where ∂ ′′ a stands for the (0 , − part of the unitary connection ∇ a , and h f i := | X | R X f , and κ c ( a c ) := s (cid:18) − β ( a c ) (cid:19) (1.8)Note κ c ( a c ) < / √ κ = 1 / √ β ( a c ) is invariant under the gauge transformations (see Proposition 1.4below) and therefore is in fact defined on the space of holomorphic structureson E n (see Theorem E.1 of Appendix E).Furthermore, if a c ∈ A cc,n is a regular value of the Abel-Jacobi map Φ,then Null ∂ ′′ a c is one-dimensional (see Proposition 1.7) and β ( a c ) := h| φ | i , (1.9)with φ ∈ Null ∂ ′′ a c and normalized as h| φ | i = 1.For a review and references in the torus case see [33]. For general compactRiemann surfaces, the existence of solutions of (1.1) for κ = 1 / √ inzburg–Landau equations on Riemann surfaces, April 23, 2019 Theorem 1.2.
Let b = πn | X | . We have, under the conditions of Theorem1.1, | X | E ( ψ s , a s ) = κ πn ) − κ ( r − b ) β ( a c )( κ − κ c ( a c )) + O ( κ ( r − b ) ) , (1.10) Note that κ + (2 πn ) is the energy of the normal state (0 , a ) per unit area. Corollary 1.3.
We have for s sufficiently small E ( ψ s , a s ) < E (0 , a c ) ⇐⇒ κ > κ c ( a c ) , (1.11)inf E E ( ψ, a ) < inf E E ′ ( ψ ′ , a ′ ) ⇐⇒ β ( a c ) < β (( a c ) ′ ) . (1.12)We say that a solution u ∗ = ( ψ ∗ , a ∗ ) is stable iff Hess E ( u ∗ ) ≥
0. Itfollows from our results on the linearized problem (see Proposition 4.1 andEq (5.9)), see also [30], that the solution (0 , a, h ), where a is a constantcurvature connection on E with F a = πn | X | ω , is stable iff | X | ≤ πn/κ .Following the Hessian computations in [32], one can show that ( ψ s , a s , r s )is stable iff r s ≥ πn/ | X | .We discuss Theorem 1.1 and its proof. We begin with a key fact that(1.1) is a gauge theory. In particular, it has the gauge symmetry, which canbe formulated as following proposition which can be checked directly. Proposition 1.4. If g : E → E ′ is a U (1) -equivariant isomorphism ofsmooth line bundles over a Riemann surface X and ( ψ, a ) solve (1.1) on E ,then T gauge g ( ψ, a ) = ( gψ, a + ig − dg ) (1.13) is a section and connection pair on E ′ and it solves (1.1) on E ′ . For E ′ = E maps (1.13) are called the gauge transformations. We usethe latter term for general g : E → E ′ . For E ′ = E , the above meansthat a single solution is simply a representative of an infinite dimensionalequivalence class of solutions. Therefore when we describe a solution withsome property, we mean that this property holds up to a gauge equivalence(with E ′ = E ).Topologically equivalent classes of smooth line bundles over a Riemannsurface are determined by their degree with values in Z . However, a bundle, E ≡ E n , of a given degree n may be given a variety of distinct holomorphicstructures.A unique equivalence class of holomorphic structures on E n is determinedby ∂ ′′ a c (see Appendix E, Theorem E.1) and a c ∈ A cc,n .We derive Theorem 1.1 from some general facts from the theory of Rie-mann surfaces presented below and the following key result: If general, if g : E → E ′ is an isomorphism of smooth line bundles over a Riemann surface X ,then for any section and connection pair, ( ψ, a ) on E , ( gψ, a + ig − dg ) is a section and connectionpair on E ′ . inzburg–Landau equations on Riemann surfaces, April 23, 2019 Theorem 1.5.
Let a c ∈ A cc,n and assume the space Null ∂ ′′ a c is one-dimensional.Then the statement of Theorem 1.1 is true. Theorem 1.5, and therefore Theorem 1.1, could be readily generalized tothe case when the space Null ∂ ′′ a c is odd-dimensional.We now consider the dimension of the space Null ∂ ′′ a c . On the first step,we use the following result, which is a constructive version of known results(see Appendix E, Theorem E.1 (i)): Theorem 1.6.
There is a smooth, invertible map from U (1) bundles withunitary connections onto holomorphic bundles with complex connections andcompatible hermitian metrics, which maps the former connections into thelatter ones. This map is constructed explicitly. We give a hands on proof of this theorem in Appendix C.By Theorem 1.6, the space Null ∂ ′′ a c is isomorphic to the space of holo-morphic sections of the holomorphic bundle corresponding to a c . We saya holomorphic structure, or its corresponding connection, a c , is admissible if the kernel of ∂ ′′ a c is exactly one-dimensional. We will give a completeclassification of the space of admissible connections including1. determinaion of its degree range n (Proposition 1.7);2. its description as an analytic moduli space (Corollary 1.8);3. explicit formulas for the elements of Null ∂ ′′ a c in terms of normalizeddifferentials of the third kind (Appendix F).This detailed classification is based entirely on classical results, thoughtheir application here to the Ginzburg-Landau equations is novel. This willbe explained below. We now state our principal classification. Proposition 1.7.
The following statements give a complete classificationof admissible connections and the possible degrees of their associated linebundles.(i) For either n < or n > g , there are no admissible holomorphicstructures a c .(ii) For ≤ n ≤ g , a holomorphic structure a c ∈ A cc,n is admissible ifand only if it corresponds to a regular value of the Abel-Jacobi map Φ (see (1.14) below) restricted to X ( n ) .(iii) For n = 0 , only the trivial bundle supports admissible connectionsand φ in this case may be gauge transformed to a unitary constant, c with | c | = 1 . In the first statement, n < In the physics literature such connections are called non-degenerate . inzburg–Landau equations on Riemann surfaces, April 23, 2019 which is the same as the degree, n , of the bundle. Hence, one must have n ≥ n > g , Riemann’s inequality states that h ( X, E ) := dim H ( X, E )satisfies h ( X, E ) ≥ n − g + 1 > n ≤ g .The second statement follows from Abel’s theorem as discussed in Ap-pendix E. For the last statement, meromorphic sections of a degree zerobundle have the same number of zeroes as poles. Hence a global holomor-phic section has no zeroes and so the bundle must be trivializable.Let X ( n ) denote the n -fold symmetric product of X with itself (this is asmooth, compact complex manifold, in fact a projective algebraic variety, ofcomplex dimension n ). The main ingredient in the proof of Proposition 1.7is the Abel-Jacobi map , Φ, extended naturally to X ( n ) and defined there asΦ : X ( n ) → J ac ( X ) := C g / Λ ,P + · + P n → n X k =1 Z P k P ~ζ, (1.14)where ~ζ = ( ζ , . . . , ζ g ) is a basis for the space of holomorphic differentials on X , normalized with respect to a canonical homology basis, a , . . . , a g ; b , . . . , b g ,meaning H a j ζ k = δ jk , and Λ is the rank 2 g lattice generated by the periods, δ jk and τ ij = R b j ζ i , of ~ζ . The map depends on the choice of basepoint P ,but for various P ’s, differs only by a translation in J ac ( X ) under a changeof this basepoint.The set of admissible connections a c , of degree n , has the structure ofa complex manifold which can be described in terms of the Abel-Jacobiimage, W n := Φ( X ( n ) ) ⊂ J ac ( X ), which is an analytic (in fact algebraic)subvariety of J ac ( X ) (see Appendix E). Denote by W n, smooth the set of theregular values of Φ in W n . The singular points of W n correspond precisely tothe holomorphic structures on E for which h ( X, E ) >
1. Again this followsfrom Abel’s theorem [14], III.11.12. So we have
Corollary 1.8.
The space of holomorphic line bundles with the underlyingsmooth bundle E and having one-dimensional spaces of holomorphic sectionsis parametrized by the complex sub-manifold W n, smooth ⊂ J ac ( X ) . Since itis the set of regular values of Φ , by Sard’s theorem, this latter space is anopen dense submanifold of W n . Theorems 1.6 and 1.5 and Corollary 1.8 imply Theorem 1.1. (Theo-rem 1.6 connects the original U (1) − bundle considered in Theorem 1.5 to aholomorphic one from the space described in Corollary 1.8.) Remark 1.9.
There is a one-to-one correspondence between homomor-phisms of E and sections of line bundles: a homomorphism E → E ′ is a inzburg–Landau equations on Riemann surfaces, April 23, 2019 section of the line bundle E ′ ⊗ E − and a section, g , of a line bundle J defines the homomorphism g : E → E ′ := J ⊗ E .Theorem 1.1 can be formulated and proved entirely on the trivial bundle,˜ E := ˜ X × C , over the universal cover, ˜ X , of X . While the approach taken inthis paper is more economical, its uniformized version is more explicit. Wepresent some details in Appendix A.The second reason in pursuing the uniformized approach is that it offersa much more general definition of the (dynamical) stability solutions of the(lifting of the) GLE (1.1), which disrupts the relation between the descrip-tion on X and its universal cover ˜ X (see Proposition A.2). This will beexplained in more detail elsewhere.Considering a family, h = rh ′ , r > , of Hermitian metrics, it is conve-nient to rescale the Ginzburg-Landau equations (1.1) to the fixed hermitianmetric, h ′ . Then (1.1) rescale to the form ( − ∆ a ψ = ( µ − κ | ψ | ) ψ,d ∗ da = Im( ψ ∇ a ψ ) , (1.15)where µ := κ r (see Appendix B.1). From now on, we consider (1.15), with µ an arbitrary positive number, and omitting the prime, the hermitian metric h . The paper is organized as follows. After a preliminary Sections 2 andSection 3, we prove Theorem 1.5, in Sections 4 and 5, with some technicalderivations given in Appendix D. In Section 6 we prove Theorem 1.2.In Appendix A we present an approach to the problem on the universalcover of X and in Appendices C and E and F, we prove Theorem 1.6 anddiscuss the proof of Proposition 1.7 and an explicit form of Corollary 1.8,respectively.We also have Appendix G, which gives some additional results not usedin the proofs and comments. We have the following well-known result (the magnetic flux quantization orthe Chern-Weil relation):
Theorem 2.1. If E is a line bundle on X , with a connection a , then π Z X F a = deg( E ) ∈ Z , (2.1) where deg( E ) is the degree of E . inzburg–Landau equations on Riemann surfaces, April 23, 2019 c ( E ) := π R X F a is called sometimes the first Chern number. It can bedefined in terms of the automorphy factor of X , see [20].Now, suppose ∇ a is a connection of constant curvature on E . Then, by(2.1), we have b = 1 | X | Z X F a = 2 πc ( E ) / | X | = 2 πn/ | X | . (2.2)where | X | is the area of X . For the hyperbolic metric the Gauss-Bonnetformula gives that | X | = 2 π (2 g − b = n/ (2 g − . (2.3) For reference reason we present the next well-known fact (see e. g. [23],Proposition 4.2.2 and [13], Sect 2.2.1, page 50) as
Lemma 3.1.
Any constant curvature connection, a , on E is of the form a ∗ + β, where a ∗ is a fixed constant curvature connection on E and β is aclosed one-form, dβ = 0 . Next, the important statement (1.6) is implied by the following
Lemma 3.2. a is an constant curvature connection on E iff it satisfies the(Maxwell) equation d ∗ da = 0 . (3.1) Consequently, the triple (0 , a, µ ) solves the Ginzburg-Landau equations (1.15) for every constant curvature connection a and every µ > , which proves (1.6) .Proof. Let ∗ be the Hodge star operator (the properties ∗ are listed in (B.2)).Using the relation d ∗ = ( − k ∗ d ∗ on k − forms, we find d ∗ da = − ∗ d ∗ da .If ∗ da = b for some constant b , then d ∗ da = − ∗ db = 0.Conversely, if d ∗ da = 0, then df = 0, where f := ∗ da is a function on X ,which implies that f is constant.Recall that we call the branch { (0 , a, µ ) : a is a constant curvature con-nection and µ > } of solutions of the Ginzburg-Landau equations (1.15),the normal (or constant curvature) branch. We will bifurcate from thisbranch. inzburg–Landau equations on Riemann surfaces, April 23, 2019 Let Ω c be the subspace of real (smooth or square integrable) closed one-forms on X with values in E ∗ n . Lemma 3.1 implies that the space, A cc,n , ofconstant curvature connections on E n is of the form A cc,n = a n + Ω c , (3.2)where a n is a fixed constant curvature connection on E n . By (2.1), da n =2 πnω . The first step in the bifurcation analysis is to investigate the linearized equa-tions.We linearize the Ginzburg-Landau equations (1.15) at the solution (0 , a n ),where a n is a constant curvature connection on E n (see Lemmas 3.2 and 3.1),to obtain the pair of equations − ∆ a n φ − µφ = 0 , d ∗ dα = 0 , (4.1)on X . Our goal is to solve these equations. We begin with the first equation.We decompose, as usual, the unitary connection ∇ a as ∇ a = ∂ ′ a + ∂ ′′ a ,where ∂ ′ a : Ω p,q → Ω p +1 ,q and ∂ ′′ a : Ω p,q → Ω p,q +1 (cf. Section A.3). We show Proposition 4.1.
Let a n be a connection of the constant curvature, F a = bω ( b = πn | X | ) . Then the operator − ∆ a n has purely discrete spectrum, isbounded below as − ∆ a n ≥ b and b is the smallest eigenvalue of − ∆ a n iff Null ∂ ′′ a n = ∅ . More precisely, Null( − ∆ a n − b ) = Null ∂ ′′ a n . (4.2)The value µ = b is a bifurcation point for our problem. More precisely,if Null ∂ ′′ a n = ∅ , then Null( − ∆ a n − b ) = ∅ , if µ < b , and Null( − ∆ a n − b ) = ∅ if µ = b .The statement that the operator − ∆ a n has purely discrete spectrum isa standard consequence of the condition that X is compact. The rest ofProposition 4.1 follows from the following Weitzenb¨ock-type formula or har-monic oscillator representation, which was proven in [34] and whose simpleproof is presented in Appendix D. Proposition 4.2. ∂ ′′ a ∗ ∂ ′′ a = 12 ( − ∆ a − ∗ F a ) . (4.3) inzburg–Landau equations on Riemann surfaces, April 23, 2019 Since ∗ F a n = b , this proposition implies Proposition 4.1.This finishes the first equation in (4.1). For the second equation in (4.1),let Ω be the subspace of real (smooth or square integrable) one-formson X with values in E ∗ n , which are harmonic, i.e. satisfying dβ = 0 , d ∗ β = 0 . (4.4)Then, by Lemma 3.2, the definition of ~ H and the integration by parts( h β, d ∗ dβ i = k dβ k ), we have Corollary 4.3.
Null d ∗ d (cid:12)(cid:12) ~ H = Ω . (4.5) Remark.
Let H dR ( H , R ) be the first de Rham cohomology group. Thenwe have the following standard result (which follows from the Hodge decom-position theorem)Ω ( X, R ) ≈ H dR ( X, R ) , (4.6)where ≈ stands for the isomorphism identifying elements of Ω ( X, R )with equivalence classes of H dR ( X, R ). We define the spaces, L n and ~ L , of L − sections of E n and (weakly) co-closed L − one-forms on X (with values in End ( E n )) and let H sn and ~ H sn be their natural Sobolev versions.We look for solutions of the Ginzburg–Landau equations, (1.15), in theform ( ψ, a n + α ), where ( ψ, α ) ∈ H n × ~ H n . Using (3.1) ( d ∗ da n = 0), werewrite (1.15) as ( − ∆ a n + α ψ + ( κ | ψ | − µ ) ψ = 0 ,d ∗ dα − Im( ψ ∇ a n + α ψ ) = 0 . (5.1)Our strategy is, following [36, 10], to solve first the following modified equa-tion F ( ψ, α, µ ) = 0 , F : H × ~ H × R + → L n × ~ L n,σ , (5.2) F ( ψ, α, µ ) = ( − ∆ a n + α ψ + ( κ | ψ | − µ ) ψ, d ∗ dα − P co − clo J ( ψ, α )) , (5.3)where J ( ψ, α ) := Im( ψ ∇ a n + α ψ ) and P co − clo is the orthogonal projection inthe space of real square integrable one-forms onto the subspace of real co-closed one-forms, Ω c ∗ . Then we go from (5.2) back to the original system(5.1) using the following proposition and its corollary: inzburg–Landau equations on Riemann surfaces, April 23, 2019 Proposition 5.1.
Assume ( ψ, a ) solves the first equation in (5.1) . Then J ( ψ, a ) is a co-closed one-form (i.e. d ∗ J = 0 ). Corollary 5.2.
A solution ( ψ, α, µ ) to (5.2) solves also (5.1) .Proof of Proposition 5.1. Given a base point z ∈ ˜ X and a closed form β on X , we define the family of maps˜ g s ( z ) ≡ ˜ g ( z ) s , ˜ g ( z ) ≡ ˜ g z ,β ( z ) := e i R zz ˜ β : ˜ X → U (1) , (5.4)where ˜ β is a lift of β to ˜ X and R zz ˜ β is the integral of ˜ β over a path, c z ,z , in˜ X from z to z . Since β is a closed one-form, these maps are independentof the choice of paths c z ,z from the same homotopy class, are differentiableand satisfy˜ g s ( γz ) = ˜ g s ( z ) σ ′ s ( γ ) , σ ′ s ( γ ) := e i s R γ β ∈ Hom (Γ , U (1)) . (5.5)Here R γ β is a period of β . (For the last equation, we have ˜ g s ( γz )˜ g s ( z ) − = e i s R γzz ˜ β = e i s R γ β . For more details, see [15], Theorem 10.13.)As a consequence of the last equality, the maps ˜ g s ( z ) descend to sections g s of a line bundle over X with fibers isomorphic to U (1), which satisfy g − s dg s = isβ .The sections g s generate the gauge transformations and the rescaledGinzburg-Landau energy functional E ( ψ, a, h ) = k∇ a ψ k + k da k + κ k ( | ψ | − µ/κ ) k . (5.6)is invariant under these transformations: E ( ψ, a ) = E ( g s ψ, a + ig − s dg s ). Wedifferentiate the last equation with respect to s at s = 0, to obtainRe h∇ a ψ, ∇ a (ln gψ ) i + h ( κ | ψ | − µ ) ψ, ln gψ i + h d ∗ da − J ( ψ, a ) , β i = 0 , (5.7)Assume β is exact (and therefore its periods vanish). Then we can write g = e if , where f is a real (single-valued) function on X and integrating byparts in the first term and using da = dα and h d ∗ dα, β i = h dα, dβ i = 0, wefind Re h− ∆ a ψ + ( κ | ψ | − µ ) ψ, if ψ i − h J ( ψ, a ) , β i = 0 . (5.8)This, together with the first equation in (5.1), implies h J ( ψ, a ) , β i = 0. Sincethe last equation holds for any exact form β , we conclude that J ( ψ, a ) isco-closed. inzburg–Landau equations on Riemann surfaces, April 23, 2019 Thus, by Corollary 5.2, it suffices to solve the equation F = 0, with F given in (5.3). Let u := ( ψ, α ). The map, F ( u, µ ), has the properties (a) F is C ; (b) F ( T gauge η u, µ ) = F ( u, µ ) and (c) F (0 , µ ) = 0 , ∀ µ , and therefore(0 , µ ) is a trivial branch of solutions to F ( u, µ ) = 0. Clearly, dF (0 , µ ) = ( − ∆ a n − µ ) ⊕ d ∗ d. (5.9)By Proposition 4.1, Null( − ∆ a n − b ) = Null ∂ ′′ a n and by (4.5), we haveNull ~ H d ∗ d = Ω . Hence the null space, K µ , of dF (0 , b ) is K µ = ( Null ∂ ′′ a n × Ω if µ = b, { } × Ω if µ < b. (5.10)We define P to be the projection on K b and let P ⊥ = 1 − P . By (5.10) andthe definition of P ⊥ , d w P ⊥ F (cid:12)(cid:12) w =0 = P ⊥ d w F (cid:12)(cid:12) w =0 is invertible on Ran P ⊥ .Now, we perform Lyapunov-Schmidt reduction: P F ( v + w, µ ) = 0 , (5.11) P ⊥ F ( v + w, µ ) = 0 , (5.12)where v := P ( ψ, α ) and w := P ⊥ ( ψ, α ). Since d w P ⊥ F (cid:12)(cid:12) w =0 is invertible onRan P ⊥ , provided v and µ − b are sufficiently small, we can use the implicitfunction theorem to find a unique solution, w = w ( v, µ ), for (5.12) (againprovided v and µ − b are sufficiently small).By the assumption of Theorem 1.5, ∂ ′′ a n has a non trivial one-dimensionalkernel. Let φ be the unique (modulo gauge transformations) solution to theequation ∂ ′′ a n φ = 0. We assume φ is normalized, i.e. k φ k = 1. Observe that v ≡ v ( s, µ ) := P ( ψ, α ) = ( sφ, β ) , (5.13)where s = h φ, ψ i and β is the projection of α onto Ω . So that the solutionto (5.12) can be written as w = w ( s, β, µ ), with ( s, β, µ ) in a neighbourhoodof (0 , , b ).To estimate w ( s, µ ), we decompose P ⊥ F into the linear and nonlinearcomponents, P ⊥ F ( v + w, µ ) = L ⊥ w + N ( v, w ), and rewrite (5.12) as the fixedpoint problem w = − ( L ⊥ ) − N ( v, w ). Then using estimates on N ( v, w ) andits derivatives, it is not hard to show that, for ( s, µ ) in a neighbourhood of(0 , b ), ∂ ν w ( s, β, µ ) = ( O ( s ) , O ( s )) , ∀ ν, (5.14) In fact, there is a larger trivial branch of the form ( ψ = 0 , β, µ ), where β ∈ Ω , but wewill avoid using the latter. inzburg–Landau equations on Riemann surfaces, April 23, 2019 where ∂ ν is a derivative of the order ν in s, β, µ (recall that β ∈ Ω , afinite dimensional vector space) and O ( s k ) stands for a remainder, which isuniformly bounded, together with its derivatives in µ , by Cs k .Substituting the solution w = w ( s, β, µ ) to (5.12) into equation (5.11),we arrive at the equation P F ( s, β, µ ) = 0 , F ( s, β, µ ) := F ( v ( s, β, µ ) + w ( s, β, µ ) , µ ) . (5.15)This equation has the U (1) symmetry in s : for α constant, we have P F ( e iα s, β, µ ) = e iα P F ( s, β, µ ).Let F = ( F , F ) and P harm be the the second component of the projec-tion P , i.e. the orthogonal projection onto Ω . By the definition of P , theequation P F ( s, β, µ ) = 0 is equivalent to the equations h φ, F ( s, β, µ ) i = 0and P harm F ( s, β, µ ) = 0. By (5.13) and (5.14), ψ = sφ + O ( s ) and α = O ( s ). Using this, we expand F i ( s, β, µ ) in s to the leading order: F ( s, β, µ ) = s ( − ∆ a n + β − µ I ) φ + O ( s ) , (5.16) F ( s, β, µ ) = | s | [ − Im( ¯ φ ∇ a n φ ) + | φ | β ] + O ( s ) . (5.17)In the next proposition, we show that the leading term on the r.h.s. of (5.17)drops out. Recall the decomposition ∇ a = ∂ ′ a + ∂ ′′ a . Proposition 5.3.
Let a n be a constant curvature connection on E n and φ is a solution to ∂ ′′ a n φ = 0 . Then J ( φ, a n ) is co-exact and consequently P harm Im( ¯ φ ∇ a n φ ) = 0 . (5.18) Proof.
Using that ∇ a = ∂ ′ a + ∂ ′′ a and that φ is a solution to ∂ ′′ a n φ = 0, wefind Im( φ ∇ a n φ ) = Im( φ [ ∂ ′ a n + ∂ ′′ a n ] φ ) (5.19)= Im( φ∂ ′ a n φ ) . (5.20)Next, we can proceed either locally or by lifting to the universal cover andusing ∇ A = ∂ ′ A c + ∂ ′′ A c , where A c := ( A − iA ) dz, ∂ ′ A c = ∂ + iA c and ∂ ′′ A c = ∂ + i ¯ A c = − ∂ ∗ A c (see Appendix A.1 and equations (A.8) and (A.9)).We take the former route.Using that φ∂φ = ∂ | φ | − φ∂φ , we find φ∂ ′ a n φ = ∂ | φ | − φ ( ∂ − ia n ) φ = ∂ | φ | . Next, Im ∂f = ∗ df (to see this we recall that locally or on theuniversal cover, we have ∂ := ∂∂z ⊗ dz , where, as usual, ∂∂z := ∂ z := ( ∂ x − i∂ x ) /
2, and ∗ = − , ∗ ( dx ) = dx and ∗ ( dx ) = − dx ). The last tworelations giveIm( φ∂ ′ a n φ ) = Im( ∂ | φ | ) = 12 ∗ d | φ | = − d ∗ ( ∗| φ | ) (5.21)Hence J ( φ, a n ) is co-exact. inzburg–Landau equations on Riemann surfaces, April 23, 2019 Hence using (5.16) - (5.17), (5.18), ∆ a n + β φ = ∆ a n φ + O ( sβ ) and ( − ∆ a n − b ) φ = 0, we find P F ( s, β, µ ) = (cid:0) O ( sβ )+ s ( µ − b ) φ + O ( s ) , (5.22) | s | P harm ( | φ | β ) + O ( s ) (cid:1) . (5.23)Recall that Ω is a 2 g − dimensional space. We fix an orthonormalbasis { ω i } gi =1 for Ω so that β = P i t i ω i and let t := ( t , . . . , t g ). Then P ( ψ, α ) = ( sφ, X i t i ω i ) . Then the solution for (5 .
12) can be written in the form of w ≡ w ( s, t, µ ), for s, t, µ in a neighbourhood of (0 , , b ).Let v ( s, t, µ ) := P ( ψ, α ) = ( sφ, P i t i ω i ). Substituting the solution w = w ( s, t, µ ) = ( ψ ⊥ ( s, t, µ ) , α ⊥ ( s, t, µ )) to (5.12) into equation (5.11), we arriveat the equation G ( s, t, µ ) = 0, where G ( s, t, µ ) := P F ( v ( s, t, µ ) + w ( s, t, µ ) , µ ) . (5.24)Let G = ( G , G ). The equations G ( s, t, µ ) = 0 and G ( s, t, µ ) = 0 areequivalent to the equations h φ, F ( s, t, µ ) i = 0 and h ω i , F ( s, t, µ ) i = 0 , i =1 , . . . , g, which, after using (5.22) - (5.23) and division by s and | s | , re-spectively, give( b − µ ) k φ k + O ( | t | ) + O ( s ) = 0 , X j B ij t i + O ( s ) = 0 , ∀ i, where B ij := h ω j , | φ | ω i i X and O ( s ) stand for a remainder uniformlybounded, together with its derivatives in t, µ , by Cs . Clearly, the ma-trix B ij is strictly positive and therefore is invertible. Hence, by the implicitfunction theorem, the latter system of equations has a unique solution for( µ, t ) of the form µ = b + O ( s ) , t = O ( s ) . (5.25)Thus, equation (5.15) (or G ( s, t, µ ) = 0) has a solution for µ and t as afunction of s . Together with the previous conclusion, this gives the solutionto the equation F ( u, µ ) = 0 for u := ( ψ, α ), µ and t in terms of s . Thus wehave proven Proposition 5.4.
For s sufficiently small, the equation F ( u, µ ) = 0 has afamily of solutions of the form u s = ( sφ + O ( s ) , O ( s )) , µ s = bκ + O ( s ) .By Corollary 5.2, u s solve (5.1) . inzburg–Landau equations on Riemann surfaces, April 23, 2019 Therefore (1.15) has a solution of the form (1.4) with the propertiesstated in Theorem 1.1, i.e. that the new solutions bifurcate from the trivialone. Next, we prove (1.5). It will follow from
Proposition 5.5.
Let b := 2 πn/ | X | . The bifurcating solutions, ( ψ s , a s , µ s ) ,constructed in Proposition 5.4, have the following expansions ψ s = sφ + O ( s ) , a s = a n + s a + O ( s ) , µ s = bκ + s µ + O ( s ) , (5.26) where the first and the second remainder are understood in the sense of thenorms in H n and ~ H n , respectively (with the first and second derivatives ofthe remainders obeying similar estimates). Moreover, φ , a and µ satisfythe equations − ∆ a n φ = bφ and da = 12 ∗ (1 − | φ | ) , (5.27) and µ = (cid:20)
12 + (cid:18) κ − (cid:19) β ( a n ) (cid:21) , (5.28) where β ( a n ) is the Abrikosov function given by (1.9) .Proof. The Lyapunov-Schmidt arguments of the main proof (more precisely,the equations w = − ( L ⊥ ) − N ( v, w ) , N = ( N , N ) and N = − J ( ψ, α ))give (5.26). Plugging (5.26) into (1.15) and taking s → d ∗ da = Im( ¯ φ ∇ a n φ ) . (5.29)This together with (5.21) gives d ∗ ( da + ∗ | φ | ) = 0. Hence da = h − ∗ | φ | , (5.30)where h is a two form obeying d ∗ h = 0 and therefore h is constant. Since R X da = 0, we have h = ∗ , which gives the second equation in (5.27).Now we prove (5.28). First we note that by the self-adjointness of theoperator − ∆ a , expansions (5.26) and ( − ∆ a n − b ) φ = 0 (see (5.27). h φ, ( − ∆ a − µ ) ψ i = s [ h φ, ia · ∇ a n ψ i − µ h φ, ψ i ] + O ( s ) . Next, we multiply the first equation in (1.15) scalarly (in L ( X )) by φ , usethe above relation, substitute the expansion for ψ in (5.26) and take s = 0,to obtain h φ, ia · ∇ a n φ i − µ h φ, φ i + κ h φ, | φ | φ i = 0 . (5.31) inzburg–Landau equations on Riemann surfaces, April 23, 2019 This expression implies that the imaginary part of the first term on the lefthand side of (5.31) is zero. (We arrive at the same conclusion by integratingby parts - using the (gauge-) periodicity of ψ and a - and using that a ,like α , is co-closed, i.e. d ∗ a = 0.) Therefore h φ, ia · ∇ a n φ i = 2 i Z X ¯ φa ∧ ∗∇ a n φ = − Z X a ∧ ∗ Im( ¯ φ ∇ a n φ ) . Using the second equation in (1.15) in the last term and integrating byparts, we obtain h φ, ia · ∇ a n φ i = − h a , d ∗ da i = − h da , da i . Next,using (5.27) gives furthermore h da , da i = h da , h| φ | i − | φ | i = −h da , | φ | i = −h h| φ | i − | φ | , | φ | i Thus we conclude that1 | X | k da k = − h| φ | i + 14 h| φ | i . (5.32)The last two relations imply1 | X | h φ, ia · ∇ a n φ i = 12 h| φ | i − h| φ | i . This equation together with (5.31), the relations h φ, φ i = | X |h| φ | i and κ h φ, | φ | φ i = | X |h| φ | i and the definition (1.9) gives (5.28).Eqn (5.28) fixes the parameter s uniquely up to the normalization of ψ .Indeed, we observe that the third equation in (5.26) implies s = µ − bκ µ + O (( µ − b ) ), which, together with (5.28) and the normalization h| φ | i = 1,yields s = µ − bκ ( κ − ) β ( a n ) + + O (( µ − bκ ) ) . (5.33)This implies (1.5). This proves Theorem 1.5. (cid:3) Corollary 5.6.
The unrescaled bifurcating solutions, ( ψ s , a s , r s ) , con-structed in Theorem 1.1 have the following expansions ψ s = sφ + O ( s ) , a s = a n + s a + O ( s ) , r s = b + s r + O ( s ) , (5.34) where φ and a satisfy the equations in (5.27) and r = µ /κ , with µ givenby (5.28) . inzburg–Landau equations on Riemann surfaces, April 23, 2019 Multiplying the first equation in (1.1) scalarly by ψ and integrating by partsgives h∇ a ψ, ∇ a ψ i = κ Z X (cid:0) | ψ | − | ψ | (cid:1) . Substituting this into the expression, (5.6), for the energy, we find E ( ψ, a ) = − κ k| ψ | k + k da k + κ | X | . (6.1)Using the expansions (5.34) and the facts that da = 2 πω and h da i = 0gives k da k = k da n k + 2 s h da n , da i + s k da k + O ( s ) (6.2)= (2 π ) | X | + s k da k + O ( s ) . (6.3)The last two relations, together with the first equation in (5.26) and equation(5.32), and the normalization h| φ | i = 1 give1 | X | E ( ψ s , a s ) = κ π ) − s [( κ −
12 ) β ( a n ) + 12 ] + O ( s ) . (6.4)This together with (5.33) implies (1.10). (cid:3) Acknowledgements
The first and fourth authors’ research is supported in part by NSERC GrantNo. NA7901. During the work on the paper, the fourth author enjoyed thesupport of the NCCR SwissMAP. The second author’s work is supportedby NSF Grant No. DMS 1615921. The third author acknowledges theUniversity of Saskatchewan for a New Faculty Recruitment Grant.
A Analysis on the universal cover
A.1 Generalities
To lift (1.1) to the universal cover, ˜ X , of X , let π ( X ) act on ˜ X by decktransformations, whose group is denoted by Γ, and let ρ be an automorphymap, i.e. ρ : Γ × ˜ X → C ∗ and satisfies the co-cycle relation ρ ( γ · γ ′ , z ) = ρ ( γ, γ ′ z ) ρ ( γ ′ , z ) . (A.1) inzburg–Landau equations on Riemann surfaces, April 23, 2019 Then any holomorphic line bundle, E over X is isomorphic to one of theform E ρ = ˜ E/ρ , where ˜
E/ρ is a factor space according to the action of Γ( z, ψ ) → ( γz, ρ ( γ, z ) ψ ) , ∀ γ ∈ Γ . (A.2)The Chern class, c ( ρ ), of ρ equals the degree of E ρ (see [20], Theorem 2afor the definition of c ).Pulling back the section ψ and connection form a to ˜ E , we arrive at thesection Ψ and connection form A on ˜ E , which satisfy the relations γ ∗ Ψ = ρ γ Ψ , γ ∗ A = A − iρ − γ dρ γ , ∀ γ ∈ π ( X ) , (A.3)where γ ∗ is the pull back of sections and connections by γ and where wehave written ρ ( γ, z ) ≡ ρ γ ( z ). Conversely, if Ψ and A are a section anda connection form in ˜ E = ˜ X × C , then Ψ and A project to a section ψ and a connection form a on E if and only if they satisfy (A.3), with theautomorphy factor ρ corresponding to E . Moreover, F A on ˜ E descends to F a on E .We say that a pair (Ψ , A ) is ρ − equivariant (or gauge π − invariant ) iff itsatisfies (A.3) for some automorphy factor ρ .For convenience of references we summarize a part of the above discussionas Proposition A.1.
There is a one to one correspondence between sectionsand connections on E and ρ − equivariant sections and connections on ˜ E =˜ X × C (i.e ρ − equivariant functions and one-forms on ˜ X ). It is convenient to formulate the next property of the correspondence( ψ, a ) ⇔ (Ψ , A ) as an elementary proposition: Proposition A.2.
The form of equations (1.1) does not change when liftedto the universal cover. ( ψ, a ) solve (1 . on E iff its lift (Ψ , A ) to ˜ E solves (1 . on ˜ E . Dealing with ρ − equivariant functions and one-forms on ˜ X , rather thanwith sections and connections on the bundle E , is convenient because of theglobal coordinates on ˜ X . Using this we give in Appendix C a hands-on proofof Theorem 1.6.Recall that, since X is a Riemann surface of the genus g ≥
2, its universalcover, ˜ X , can be identified with the Poincar´e upper complex half plane, H ,and π ( X ) with a Fuchsian group Γ (acting on H ). Using the standard co-ordinate z on H , we see that π ( X ) is a subgroup of P GL (2 , R ) acting on H by M¨obius transforms, γz = az + bcz + d , γ = (cid:0) a bc d (cid:1) ∈ Γ.Recall also that
P GL (2 , R ) is a group of isometries of H with the standardhyperbolic metrics ˜ h = (Im z ) − | dz | . The latter generates the hyperbolicvolume form ω = i z ) − dz ∧ dz. inzburg–Landau equations on Riemann surfaces, April 23, 2019 Finally, recall that the character of Γ is a homomorphism σ : Γ → U (1). Wehave the following results proven in Sections A.2 and A.3. Theorem A.3.
For any n ∈ Z , the map ρ n : P GL (2 , R ) × H → U (1) , givenby ρ n ( γ, z ) = (cid:20) cz + dcz + d (cid:21) − n g − , for γ = (cid:0) a bc d (cid:1) ∈ P GL (2 , R ) , (A.4) is an automorphy factor. Consequently, for any Fuchsian group Γ and anycharacter σ : Γ → U (1) , the map ρ n,σ : Γ × H → U (1) given by ρ n,σ ( γ, z ) = σ ( γ ) ρ n ( γ, z ) , for γ ∈ Γ , (A.5) is also an automorphy factor. The Chern class of ρ n,σ is c ( ρ n,σ ) = n . Theorem A.4.
For any n ∈ Z , the connection, A n , on the trivial bundle ˜ E := H × C , given by A n = n g − y − dx, (A.6) (a) has a constant curvature with respect to the standard hyperbolic volumeform on H ( F A n = n g − ω ); (b) is equivariant with respect to the automorphyfactor (A.5) for any Fuchsian group Γ and any character σ : Γ → U (1) ; (c)is unique up to proper, i.e. with g = e if , f : X → R , gauge transformations. The projection of A n to E n,σ gives the distinguished connection a n,σ on E n,σ . Remark A.5.
As mentioned above one may associate to an automorphy ρ the bundle E ρ . In fact this extends to the non-uniformized setting. Theingredients for this are the characters σ on Γ and the degree n which wemay think of as a character on the abelian group Z . In the general settingthe automorphy factors of the uniformized case become unitary characterson the product S = Z × Γ. There is a 1:1 correspondence between unitarycharacters and holomorphic line bundles on X which gives one yet anotherway to describe the Picard group: P ic ( n ) ( X ) = group of unitary characters ρ on S restricted to c ( ρ ) = n .Hence, it is natural to label holomorphic line bundles as E n,σ where σ isa unitary character of Γ.Because these characters take their values in an abelian group, U (1), theyonly depend, on the abelianization of π ( X ) which we will denote by Γ abel .As with π ( X ) itself, Γ abel depends on the basepoint for the paths. This ap-pears in the explicit realizations of these characters as, for instance, in (A.4)which depends on z , unless n = 0. However, abstractly, the abelianizationis isomorphic to the first homology group: Γ abel ≃ H ( X, Z ). inzburg–Landau equations on Riemann surfaces, April 23, 2019 Remark A.6.
In the more general setting of vector bundles, charactersare replaced by representations and when c = 0 one must consider repre-sentations of central extensions of π ( X ) [2]. However, in the case of linebundles where the gauge group is abelian, the central extension splits into adirect product of Z with π ( X ); the main remnant of this extension in thecharacter is the degree n . Remark A.7.
1) The description in the above theorem does not depend atall on the complex structure of the underlying Riemann surface and thereforeserves as a very useful gauge for analyzing this problem.2) A section g of the line bundle over X with fibers Aut ( E x ) defines theisomorphism gE ρ = E g ∗ ρ , g ∗ ρ γ := ( γ ∗ ˜ g ) ρ γ ˜ g − , (A.7)where ˜ g is a lift of a section g and we used the notation ρ γ ( z ) = ρ ( γ, z ). Thisgives the ‘gauge’ transformations of the automorphy factors. Thus, changinga connection using gauge equivalence changes the bundle automorphy factor,so, in general, it is impossible to adjust both simultaneously.Consider a special case of (A.7):If g is such that its lifting ˜ g to ˜ X ( g ◦ π = ˜ g ) satisfies˜ g ( γz ) = ˜ g ( z ) σ ′ ( γ ), σ ′ ∈ Hom ( π ( X ) , U (1)) , then g ∗ ρ n,σ = ρ n,σ ′ σ . In particular, if g : X → U (1) (i.e. it is a section of atrivial bundle, so that ˜ g satisfies ˜ g ( γx ) = ˜ g ( x ) , ∀ γ ∈ π ( X )), then g ∗ ρ = ρ .2) The gauge invariance implies that we can consider GLEs on a fixedbundle E n,σ , n ∈ Z , σ ∈ Hom ( π ( X ) , U (1)). A.2 Automorphy Factors
In this section, we give the calculation proving Theorem A.3. Recall H = { z ∈ C | Im( z ) > } with the group P GL (2 , R ) acting on H by (cid:0) a bc d (cid:1) · z = az + bcz + d . Proof of Theorem A.3.
Let β = n g − and s = (cid:0) a bc d (cid:1) , t = (cid:16) e fg h (cid:17) ∈ Γ. Using(A.4), we compute ρ n ( s · t, z ) = (cid:20) ( ce + dg ) z + ( cf + dh )( ce + dg ) z + ( cf + dh ) (cid:21) − β = (cid:20) c ( ez + f ) + d ( gz + h )( c ( ez + f ) + d ( gz + h ) (cid:21) − β = (cid:20) c ez + fgz + h + dc ez + fgz + h + d (cid:21) − β (cid:20) gz + hgz + h (cid:21) − β = ρ n ( s, t · z ) ρ n ( t, z ) . inzburg–Landau equations on Riemann surfaces, April 23, 2019 Using the formula for the Chern class, c ( ρ ), of a co-cycle ρ (see [20], Theo-rem 2a), we compute c ( ρ ) = n . A.3 Uniform constant curvature connection andits holomorphic structure
In this section we prove Theorem A.4. We begin with some preliminaryconstructions.We consider the trivial bundle, ˜ E := H × C with the standard complexstructure on H associated to the hyperbolic metric ˜ h = (Im z ) − | dz | .Since we work here on a global product space, it is natural to take thefiber metric to be induced from the metric on the base. So we take themetric on the fiber C over the point z ∈ H to be k z = (Im z ) − | dw | where w is the coordinate on the fiber C z .Let the connection A be given by A := A dx + A dx . We decomposethe covariant derivative ∇ A into (1 ,
0) and (0 ,
1) parts as ∇ A = ∂ ′ A + ∂ ′′ A ,where ∂ ′ A := ∂ + iA c , ∂ ′′ A := ∂ + i ¯ A c . (A.8)Here ∂ := ∂∂z ⊗ dz and ∂ := ∂∂z ⊗ dz , where, as usual, ∂∂z := ∂ z := ( ∂ x − i∂ x ) / ∂∂z ≡ ∂ ¯ z := ( ∂ x + i∂ x ) / A c := 12 ( A − iA ) ⊗ dz, ¯ A c := 12 ( A + iA ) ⊗ d ¯ z. (A.9)We call the complex one-form A c the complexification of the real connection A . In the reverse direction, we have A = 2 Re A c .In terms of A c , the curvature is given by F A = 2 Re ¯ ∂A c . Moreover,if A c satisfies the equivariance relation s ∗ ¯ A c = ¯ A c − i∂ ˜ f s , then A satisfies s ∗ A = A + df s , with f s satisfying df s := 2 Im ∂ ˜ f s .According to (A.9), the complexification of the connection A n given inthe theorem is A nc = n g − z ) dz . Proof of Theorem A.4.
We omit the superindex n in A n and A nc and use thenotation b = n g − . The operators ∂ ′ A and ∂ ′′ A are defined directly as follows. Let J be the natural almost complexstructure on X (a linear endomorphism of T ∗ ˜ X satisfying J = − ), generated by J ( dx ) = dx and J ( dx ) = − dx . Then Π ± : T ∗ ˜ X → T ∗ , X /T ∗ , X via v ( v ± iJ ( v )) / ∂ ′ A = Π + ( ∇ A ) and ∂ ′′ A = Π − ( ∇ A ) which iscomputed explicitly to be as in (A.8). inzburg–Landau equations on Riemann surfaces, April 23, 2019 Proof of constant curvature.
Using that ω = i Im( z ) − dz ∧ dz , wefind ¯ ∂A c = ∂∂ ¯ z ibz − z d ¯ z ∧ dz = − ib ( z − z ) dz ∧ dz = ib z ) dz ∧ dz = b ω. Since F A = 2 Re ¯ ∂A c , this gives the desired result. Proof of uniqueness. If A and B , satisfy dB = dA then d ( A − B ) = 0.It follows from the simple connectedness of H that A − B = df for somefunction f : H → R and f is unique up to an additive constant. So we canmap B to A through a suitable retrivialization. This completes the proof. Proof of equivariance.
For a generic isometry s ( z ) = αz + βγz + δ , we have ∂s ( z ) ∂z = ( γz + δ ) − and Im( s ( z )) = Im( z ) | γz + δ | , which gives s ∗ ¯ A c = b s ( z )) ∂s ( z ) ∂z dz = k | γz + δ | z ) dz ( γz + δ ) = b z ) γz + δγz + δ dz = b z ) dz + b ( γz − γz )2 Im( z )( γz + δ ) dz = ¯ A c + ibγγz + δ dz =: ¯ A c + ∂ ˜ f s , where ˜ f s is the function defined by the last relation, i.e. ∂ ˜ f s = ibγγz + δ dz. Solving this equation, we find˜ f s = b ln( γz + δ ) + c s . Now, we define f s := 2 Re ˜ f s and use that Re( ¯ ∂ ˜ f s ) = d (Re ˜ f s ) (as can bechecked by the direct computation: b Re( ¯ ∂ ˜ f s ) = − γ x | γz + δ | dx + γ ( γx + δ ) | γz + δ | dx )to obtain s ∗ A = A + df s , with f s = 2 Re( ˜ f s ) = ib ln (cid:20) γz + δγz + δ (cid:21) + c s ,ρ ( s, z ) = e if s ( z ) = e ic s (cid:20) γz + δγz + δ (cid:21) − b . (A.10)Since H is the upper half plane, the complex logarithm is well defined and γz + d is always non zero. Remark A.8.
1) The function ˜ f s ( z ) appearing above gives the character˜ ρ ( z ) = e ˜ f s ( z ) , which is now C ∗ valued instead of U (1) valued.2) A n is R - linear while A c is naturally C - linear. It is natural toask what the action of i on A nc does when we map back to A n . A simplecalculation shows that iA nc = i b z ) dz maps to by − dy which is flat! Itturns out that the complex action of i induces a rotation into the space offlat connections. inzburg–Landau equations on Riemann surfaces, April 23, 2019 Since ( ∂ ′′ A n ) = 0, the partial connection ∂ ′′ A n gives a holomorphic struc-ture on E n , unique up to a complex gauge transformation, which in turncorresponds to the character (A.5) (see Remark A.5). Proposition A.9. [22] For b ∈ Z the holomorphic sections of H ( X, E ⊗ F ) are modular forms of weight − b for Γ . (See Appendix G for details.) Remark A.10.
For non-integral b , the sections of the proposition lift tomodular forms of weight − n gcd( n, g − on a g − n, g − cover of X ([14] III.9.13). B Variation of the hermitian metric
Given a complex structure, the Riemannian metric can be specified in localcomplex coordinate z by a positive smooth function λ ( z ) as ds = λ ( z ) | dz | and the corresponding volume form by ω = λ ( z ) dz ∧ d ¯ z . Solutions to (1.1)depend parametrically on h only through the positive, smooth function λ ( z ).In this appendix we first consider how our variational equations change if λ ( z ) is simply rescaled. Then we describe what happens under more generaldeformaitons. B.1 Rescaled equations
Here we show that the Ginzburg-Landau equations, (1.1), with the hermitianmetric rh are equivalent to (1.15), with µ = rκ . We distinguish the quan-tities related to the family of Hermitian metrics ds = rλ ( z ) | dz | by tildes. Itsuffices to show that˜∆ A = 1 r ∆ A , ˜ M = 1 r M, (B.1)where, recall, M := d ∗ d . We note that − ∆ A = − ∗ ( − d + iA ) ∗ ( d + iA ),where ∗ is the Hodge star operator and that the scaling of the metric arisesonly in the application of the Hodge star. To prove the first relation, we usethat, since g ij is conformally flat and therefore dx and dx are orthogonal,we have that ∗ dx = dx , ∗ dx = − dx , ∗ λdx ∧ dx , ∗ dx ∧ dx = 1 /λ. (B.2)Indeed, ∗ dx = || dx |||| dx || dx = √ g √ g dx = √ /λ √ /λ dx = dx and similarly for theother relations. This gives ˜∆ A = ˜ ∗ ( d + iA )˜ ∗ ( d + iA ) = r ( d + iA ) ∗ ( d + iA ) = r ∆ A . We note that taking ˜ ψ = √ rψ produces the rescaled equations, whichafter omitting the tilde read (1.15) with µ := κ r (cf. [6]). inzburg–Landau equations on Riemann surfaces, April 23, 2019 B.2 Gauged Deformations of the metric
We fix a base volume form λ ( z ) dz ∧ d ¯ z which determines a fixed base *operator. Then any other * operator gotten from the base volume form, orequivalently the base * operator, by multiplying with the factor λ ( z ) /λ ( z ).To ensure the non-vanishing of the volume form we set λ = e h ( z ) and vary λ ( z ) by varying h . Now we make some convenient definitions: λ ( z ) = e h ( z ) λ ( z ) /λ ( z ) = e h ( z ) − h ( z ) g ( z ) = e − ( h ( z ) − h ( z )) Finally we let b ∗ denote the * operator associated to λ while ∗ denotes the *operator associated to λ . Now note how the three terms in the energy (5.6)scale with λ ( z ). The first term is independent, the second term scales as λ − ( z ) and the third term scales as λ ( z ). Now, integrating by parts to getthe variational equations a piece of the first term combines with the thirdterm to get the first variational equation in (1.1) while the remaining piece ofthe first term combines with the second term to yield the second variational.It is this mixing of λ scalings between terms that determines how varying themetric affects the variationa equations. Indeed, the following calculationsare straightforward. d b ∗ da = g (cid:0) ( d + g − dg ) ∗ da (cid:1) = Im( ψ ∇ a ψ ) − b ∆ a ψ = − g ∆ a ψ = ( κ | ψ | − ψ. Now if one makes the gauged scaling ψ → √ gψ one sees that varying h by h − h amounts to transforming the original variational equations to( d + g − dg ) ∗ da = Im( ψ ∇ a + g − dg ψ ) − ∆ a + g − dg ψ = ( κ | ψ | − g − ) ψ. We note that this is effectively a gauge transformation; however, one whichis real-valued and non-unitary. As a consequence the unit 1 in the nonlinearterm of the final equations must replaced by g − . This is completely analo-gous to, and indeed is an extension of, the constant rescaling carried out inAppendix B.1. We may view this gauging as a smoothly varying pointwisedilation which is completely consistent with the geometric significance of λ in our description of surface metrics below. C Bundle Holomorphisation
In this section we prove Theorem 1.6. Let E be a U (1) bundle over H with anautomorphy factor ρ and a unitary, ρ − equivariant connection ∇ A = ∇ + iA .The following result is a consequence of the Dolbeault lemma (see [15],Theorem 13.2): inzburg–Landau equations on Riemann surfaces, April 23, 2019 Proposition C.1.
There exists a complex valued gauge transformation g that solves g − ∂ ′′ A g = ∂ and it is unique up to a holomorphic term. Inparticular, g satisfies ∂g = igA c . (C.1) g inherits an equivariance property from A .Proof. Define g = e κ , then ∂κ = iA c . Passing from forms to functionsand using the Dolbeault lemma (see [15], Theorem 13.2), we arrive at thestatement of the propositions. Proposition C.2.
Viewing g as a multiplicative map from ˜ E to a newbundle ˜ E ′ , the induced automorphy factors ρ ′ ( γ ) := γ ∗ ( g − ) ρ ( γ ) g on thenew bundle are holomorphic.Proof. We start with two key components about how holomorphic functionscompose with regular complex valued functions. If w : U → V is holomor-phic in a neighbourhood, x : R → U and g : C → V are smooth then dw ◦ xdt = dwdz ( x ( t )) dxdt∂ ( g ◦ w ) = ( ∂g )( w ( z )) dwdz which can be checked by regarding maps on C as maps on R . With thesetwo formulas we can perform the computation: ∂ ( γ ∗ ( g − ) ρ ( γ ) g ) (C.2)= ∂ ( γ ∗ ( g − )) ρ ( γ ) g + γ ∗ ( g − ) ∂ ( ρ ( γ )) g + γ ∗ ( g − ) ρ ( γ ) ∂g (C.3)= iγ ∗ ( g − ) ρ ( γ ) g ( A c + ∂f γ ) + ∂ (1 /z ◦ g ◦ γ ) ρ ( γ ) f (C.4)= iγ ∗ ( g − ) ρ ( γ ) g ( A c + ∂f γ ) (C.5) − /z ◦ g ◦ γ · ∂g ◦ γ · ∂γ (C.6)= iγ ∗ ( g − ) ρ ( γ ) g ( A c + ∂f γ ) − iγ ∗ ( g − ) ρ ( γ ) gγ ∗ A c (C.7)= iγ ∗ ( g − ) ρ ( γ ) g ( A c + ∂f γ − γ ∗ A c ) (C.8)=0 . (C.9)The first equality is the Leibniz rule, the second equality follows from( C. A .Now we derive Theorem 1.6 from these propositions. Let ˜ E be the uni-formization of the unitary bundle E in Theorem 1.6 and let ∇ A be the lift ofthe connection, ∇ a , on that bundle. Hence E = ˜ E/ρ for some automorphyfactor ρ . Furthermore, let ˜ E ′ , ρ ′ and ∇ A ′ be the bundle, the automprphy inzburg–Landau equations on Riemann surfaces, April 23, 2019 factor and the connection constructed in Propositions C.1 and C.2 and let E ′ = ˜ E ′ /ρ ′ .Since the function g in Proposition C.1 is equivariant, it descends to X as a section of a line bundle over X , which induces the map of E into E ′ .This map takes also the connection ∇ a on E into the connection ∇ a ′ on E ′ .By Propositions C.1 and C.2, ∇ a ′ comes from d on ˜ E ′ via a holomorphicprojection. Hence the connection ∇ a ′ is holomorphic. This proves Theorem1.6. (cid:3) For connection (A.6) (with the standard metric on H ), we have an explicitform the transformation g , which we now denote ˜ g . Namely, we have Proposition C.3.
Let b := πn | X | = n g − . Then ˜ g = y b solves ˜ g − ∂ ′′ A n ˜ g = ∂ . ˜ g transforms under s = (cid:0) a bc d (cid:1) ∈ P SL (2 , R ) as s ∗ ˜ g = ˜ g ˜ ρ, where ˜ ρ : P SL (2 , R ) × H → R , ˜ ρ ( s, z ) := | cz + d | − b . (C.10) Proof.
We omit the superindex n , let ˜ g = e α and solve: e − α ∂ ′′ A e α = ∂ ⇐⇒ e − α ( ∂ + i ¯ A c ) e α = ∂ ⇐⇒ ∂α = i ¯ A c Since A c = n g − z ) dz , it then follows that α = b ln (cid:18) z − z i (cid:19) solves ∂α = i ¯ A c , which gives that ˜ g = y b . We compute the automorphy of˜ g to find (C.10).Let F be the line bundle over X = H / Γ, defined by the automorphy map˜ ρ − , where ˜ ρ is given in (C.10), and, recall, that E is the line bundle over X , with the automorphy map ρ . Then ˜ g − decends to a section, g , of F . Itfollows that E ⊗ F is a holomorphic line bundle and the equation ∂ ′′ a n φ = 0is equivalent to g − φ being a holomorphic section of E ⊗ F . Hence, we have Corollary C.4.
Let b := πn | X | = n g − and, as above, s be the section of theline bundle F coming from the equivariant function ˜ g = y b on H . We have Null ∂ ′′ a n = sH ( X, E ⊗ F ) , (C.11) where, as usual, H ( X, L ) is the space of holomorphic sections on a bundle L . inzburg–Landau equations on Riemann surfaces, April 23, 2019 D Weitzenb¨ock-type formula
We prove Proposition 4.2 by passing to the universal cover, H , and provingthere an equivalent relation. We use the complex covariant derivatives as inthe beginning of Section A.3. Proposition D.1. ∂ ′′ A ∗ ∂ ′′ A = 12 ( − ∆ A − ∗ F A ) . (D.1) Proof.
To prove this we need ∗ dz = ∗ ( dx − idx ) = dx + idx = idz and dz ∧ dz = − idx ∧ dx . Furthermore we compute the adjoint of ∂ ′′ A to be ∂ ′′ A ∗ = ∗ ( − ∂ z ⊗ dz − ( A + iA ) ⊗ dz ) ∗ . Now, we have ∂ ′′ A ∗ ∂ ′′ A = i ∗ ( − ∂ z ⊗ dz − ( A + iA ) ⊗ dz )(2 ∂ z ⊗ dz − ( A − iA ) ⊗ dz )= i ∗ ( − ∂ z ∂ z + A + A − A + iA ) ∂ z + 2 ∂ z ( A − iA )) dz ∧ dz = i ∗ ( − ∆ + A + A − ddx A + ddx A − iA · ∇ − i ∇ · A )) dz ∧ dz = 12 λ ( − ∆ + A + A − ddx A + ddx A − iA · ∇ − i ∇ · A ) , which gives (D.1). Corollary D.2.
Let A n be a connection of the constant curvature, i.e. ∗ F A n = b := πn | X | = n g − . Then Null( − ∆ a n − b ) = Null ∂ ′′ a n (D.2) Hence, b is an eigenvalue of − ∆ A n iff Null ∂ ′′ A n = { } and if b is an eigen-value of − ∆ A n , then it is the smallest eigenvalue. E Admissible connections
The purpose of this appendix is to outline relevant parts of the theory ofholomorphic line bundles over Riemann surfaces - specifically, the existenceof holomorphic sections of such bundles - to help the motivated reader, withthe background elsewhere, to navigate the main body of this paper.First, recall that holomorphic structures on E n are typically specified interms of transition functions for E n that are holomorphic with respect tothe complex structure on X .For our purpose it is natural to use another description of holomorphicbundles that is phrased directly in terms of objects we use, namely, deriva-tions (connections), ∂ ′′ E , of type (0 ,
1) on E ; i.e., operators satisfying ∂ ′′ E : E ( X, E ) → E (0 , ( X, E ) (E.1) ∂ ′′ E f ξ = ¯ ∂f · ξ + f ∂ ′′ E ξ (E.2) inzburg–Landau equations on Riemann surfaces, April 23, 2019 where E ( X, E ) and E (0 , ( X, E ) are the spaces of sections and (0 , − formson X with values in E , respectively, f is a function on X and ξ is a sectionof E . Two derivations are equivalent if and only if they are conjugate to oneanother under a complex-valued gauge transformation.Let A cc denote the space of constant curvature unitary (with respect to k ) connections on E and let G denote the group of gauge transformationswhich preserve k . Theorem E.1. (i) The space of holomorphic structures on E is in a 1:1correspondence with the space, C , of derivations, ∂ ′′ E of type (0 , on E . Thiscorrespondence descends to the corresponding gauge-equivalence classes.(ii) Let ∂ ′′ a denote the type (0 , -component of ∇ a and G c denote thegroup of complex-valued gauge transformations. Then A cc / G ≃ C / G c , (E.3)[ ∇ a ] → (cid:2) ∂ ′′ a (cid:3) (E.4) where the brackets denote the corresponding gauge equivalence class in eachcase. For the proof of the first statement, see [26], Propositions 1.3.5, 1.3.7 and1.4.17. We comment on it in Remark E.2. The second one is a special caseof a more general result for vector bundles due to Narasimhan and Seshadrithat can be found in [11] and Appendix by O. Garc´ıa-Prada in [37] (seeTheorem 2.7).Thus equivalent derivations correspond to equivalent holomorphic struc-tures. Given this we will henceforth refer to C as the space of holomorphicstructures on E . Remark E.2.
Given a holomorphic structure on a line bundle E , in thesense of holomorphic transition functions on the bundle, and a hermitianmetric, k ′ on E there is a canonical connection, D k ′ , on E which is compat-ible with the metric k ′ and whose (0 , P ic ( n ) ( X ) denotes the moduli space(i.e. the space of complex gauge equivalence classes ) of holomorphic linebundles of fixed degree n . For each n , P ic ( n ) ( X ) is isomorphic to J ac ( X ).This isomorphism is effectively determined by the Abel-Jacobi map (see(1.14)).As we mentioned above, we are interested in existence of holomorphicsections of holomorphic line bundles. Though every holomorphic bundle hasa meromorphic section, not every such bundle has a holomorphic one. LetΣ ( n ) ⊂ P ic ( n ) ( X ) denote the subset of degree n holomorphic line bundleswhich have a holomorphic section. inzburg–Landau equations on Riemann surfaces, April 23, 2019 Recall that the divisor, D , on a Riemann surface X is a finite collectionof points, P i ∈ X , with associated integers, n i , written as D := P n i P i . Thenumber deg( D ) := P n i is called the degree of D .With every meromorphic section, ϕ , one can associate the divisor, de-noted as ( ϕ ), which is the collection of its zeros and poles, together withtheir positive and negative orders (multiplicities), respectively. Differentsections of a holomorphic line bundle have linearly equivalent divisors: Twodivisors are said to be linearly equivalent if their difference is a divisor of ameromorphic function on X . In this way with each line bundle we associatea collection of linearly equivalent divisors. Conversely, a divisor uniquelydetermines an equivalence class of holomorphic line bundles; i.e., a point in P ic ( X ). ([18]).It is a consequence of the definition of the degree of a line bundle interms of zeros of its sections that the degree of the divisor is equal to thedegree of the bundle.By this description, the divisor of a holomorphic section has only positiveinteger coefficients (the value of the coefficient corresponds to the multipic-ity of the divisor at that point). Such divisors are said to be effective .An effective divisor corresponds to a unique point in the n -fold symmetricproduct, X ( n ) , which parametrizes unordered n -tuples of points on X . (Itis straightforward to check that X ( n ) is a smooth manifold [18].)Let ˜Σ ( n ) denote the set of pairs ( E, D ) where E is a holomorphic linebundle of degree n , which has a holomorphic section, and D is the divisorof a holomorphic section of E . Then projection onto the second factor,( E, D ) → D , defines a 1:1 map π : ˜Σ ( n ) → X ( n ) , (E.5)which is in fact onto, since one can construct a bundle and its holomorphicsection directly from an effective divisor [18]. One then makes use of aspecial case of Abel’s theorem : Theorem E.3. [18] Two effective divisors are the zeroes of two holomorphicsections of the same bundle if and only if they are mapped to the same pointin W n := Φ( X ( n ) ) ⊂ J ac ( X ) , (E.6) Moreover, if
D, D ′ ∈ Φ − ( z ) for z ∈ W n , then D and D ′ are linearlyequivalent, i.e. there exists a meromorphic function, f on X such that ( f ) = D − D ′ . It also follows from the first statement in Abel’s theorem that the admis-sible bundles, i.e. the bundles with a dimension one space of holomorphicsections, of degree n correspond to those points z ∈ W n for which Φ − ( z ) isa unique point in X ( n ) . inzburg–Landau equations on Riemann surfaces, April 23, 2019 We can now more properly define the map I : X ( n ) → A cc,n by I := I ◦ π ◦ π − , where I : C / G c → A cc / G is the reverse direction of the Narasimahn-Seshadriisomorphism (E.4), π : ˜Σ ( n ) → P ic ( n ) ( X ) is given by projection onto thefirst factor ( E, D ) → E in ˜Σ ( n ) and π is given in (E.5). (In defining thecomposition with I here we are using the isomorphism, P ic ( n ) ( X ) ≃ C / G c ,which is due to Theorem E.1 (i).In fact, one can say a bit more here: under I the admissible connectionsin I (Σ ( n ) ) correspond to the regular values in W n of Φ (see [14], III.11.11).By Sard’s theorem, this is an open dense submanifold of W n .Equivalently, by the implicit function theorem, these are the smoothpoints in the variety W n . So we denote this set by W n, smooth . Then S ( n ) :=Φ − ( W n, smooth ) must also be an open dense submanifold of X ( n ) . Explicit form of W n, smooth . For certain degrees one can give explicitequations that determine W n, smooth . These are expressed in terms of Rie-mann’s theta function, θ ( ~z, τ ) = X N ∈ Z g exp 2 πi (cid:18) t N τ N + t N~z (cid:19) (E.7)where τ is a g × g matrix with entries τ ij = R b j ζ i where ζ j are normalizedholomorphic differentials and ~z ∈ C g . Theorem E.4.
There is a unique constant vector − κ , known as Riemann’sconstant, [14] VI.3.6, depending on P such that(i) For n = g , W g, smooth = J ac ( X ) \ ( { ~z ∈ J ac ( X ) : θ ( ~z, τ ) = 0 } − κ ) . (ii) For n = g − ,W g − , smooth = W g − \ ( { ~z ∈ J ac ( X ) : θ ( ~z, τ ) = 0 , θ z i ( ~z, τ ) = 0 ∀ i } − κ ) . (iii) for n = 1 , W , smooth = W ; i.e., W is entirely smooth. The first two statements are a direct consequence of Riemann’s vanish-ing theorem, [14], VI.3.7. We give a self-contained proof here of the laststatement, which will also serve to illustrate the general result.For all points on W to be regular means that Φ restricted to just X isan embedding; i.e., that Φ is both 1:1 and an immersion on X . Φ on X canonly fail to be 1:1 if the there are two distinct points, p and p , that mapto the same point z ∈ W . If that were the case, by the second statement inzburg–Landau equations on Riemann surfaces, April 23, 2019 of Theorem E.3, there must exist a meromorphic function, f on X with asingle zero at p and a single pole at p .Any meromorphic function, f , may be viewed as a non-constant holo-morphic map from X to the Riemann sphere S in which the poles of f map to the point at infinity on S . By the open mapping theorem this mapmust be onto S . A submersive mapping between two compact manifoldsof the same dimension has a well-defined finite degree equal to the numberof points, counted with multiplicity, in the inverse image, f − ( a ), indepen-dent of whatever point a one chooses. But for a = 0 we already know that f − (0) = { p } and so the degree of f is 1. In other words f is a 1:1 holomor-phic map of X onto S . But a 1:1 holomorphic map is a homeomorphism,implying that the genus of X equals the genus of S which is zero. But thiscontradicts our assumption that g ( X ) > ~ζ . So this fails to be an immersion if and only ifthere exists a point p ∈ X at which every holomorphic differential vanishes.Suppose there was such a point and let L p be the line bundle associatedto it [21]. Recall that, by duality, H ( X, L p ) is isomorphic to the spaceof holomorphic differentials vanishing at p which, by our assumption, hasdimension g . So by the Riemann-Roch theorem one has dim H ( X, L p ) =1 − g +1+ g = 2. But then again, by the argument in the previous paragraph,this would imply that X has genus 0. Hence Φ is also an immersion on X and therefore all points in W are regular. (cid:3) F Explicit representation of Null ∂ ′′ a c We briefly recall the description of multivalued functions, f , on X that aremultiplicative with respect to the lattice Λ generated by the periods of ~ζ as described in Section 1. (This lattice is isomorphic to the first homologygroup H ( X, Z ).) More precisely, multiplicativity means that, for elements γ ∈ Λ, f ( P + γ ) = χ ( γ ) f ( P ) χ ( γ ) ∈ C ∗ , (F.1) χ ( γ + γ ) = χ ( γ ) · χ ( γ ) (F.2)Maps, χ , from Λ to C ∗ satisfying the multiplicativity property (F.2) arecalled characters of Λ. If f is a function for which (F.1, F.2) are bothsatisfied, then one says that χ is the character of the multi-valued function f . One says that a character is normalized if it takes its values in U (1) andthat it is inessential if it is the character of a non-vanishing, holomorphicmulti-valued function. inzburg–Landau equations on Riemann surfaces, April 23, 2019 Now we fix, once and for all,1. a point Q ∈ X ;2. the associated one-point line bundle L Q (see Theorem E.4 (iii));3. a holomorphic section, s ( P ), of L Q , unique up to an overall constant.We also make use of the following constructive result: Theorem F.1 ([14] III.9.10) . Every divisor D of degree zero is the divisor ofa unique (up to a multiplicative constant) multiplicative multivalued functionbelonging to a unique normalized character.
This leads to yet another description of line bundles associated to divisorsof degree zero, [14] III.9.16:
J ac ( X ) ≃ { characters on Λ } / { inessential characters } (F.3) ≃ { unitary characters on on Λ } which is given by association of the Abel image Φ( D ) ∈ J ac ( X ) to theunique normalized character χ specified in the theorem.For our case, the associated multiplicative function that the theoremspecifies is explicitly constructed as follows: consider D = P + · · · + P n − nQ where P + · · · + P n is an effective divisor corresponding to an admissibleconnection under I . Then the multiplicative function belonging to character χ , associated to D , is explicitly given by f ( P ) = exp n X j =1 Z PP τ P j ,Q (F.4)where τ P,Q is the normalized differential which is holomorphic except for twosimple pole of residues -1 and +1 at P and Q , respectively. By normalizedhere one means that R a j τ P,Q = 0, where the a j belong to the canonicalhomology basis chosen in (1.14). (Such differentials are called differentialsof the third kind and are unique once P and Q are specified.) This is adirect consequence of the Riemann-Roch Theorem, as we will now sketch.The differential τ P,Q is a section of the bundle L = K ⊗ L P ⊗ L Q where K is the canonical bundle (see Proposition G.1). This bundle has degree 2 g and, by duality dim H ( X, L ) = 0. Hence by the Riemann-Roch Theoremwe know that dim H ( X, L ) = 2 g − g +1 = g +1. Normalization of the differ-ential imposes g independent conditions and so the normalized differentialsof this type are unique up to a constant multiple. But since the residuesat P and Q must be − inzburg–Landau equations on Riemann surfaces, April 23, 2019 The holomorphic section, unique up to an overall constant multiple, ofthe holomorphic bundle associated to this divisor P + · · · + P n is thenexplicitly represented asˆ φ ( P ) = f ( P ) s n ( P ) . (F.5)where s n ( P ) is the n -fold product of the section s ( P ), fixed above, withitself. Let a c correspond to the divisor P + · · · + P n under the map I . ByTheorem 1.6, there is a gauge transformation g that conjugates ∂ ′′ a c to the¯ ∂ − operator on the holomorphic bundle associated to P + · · · + P n . It followsthat the element of the kernel of ∂ ′′ a c that corresponds to ˆ φ is given by φ ( P ) = g ( P ) f ( P ) s n ( P ) . (F.6)This representation has the form of a Baker-Akhiezer section [27].
Remark F.2. i) The transformation from (F.5) to (F.6) can be viewed ascorresponding to a change of bundle inner product. Indeed, the isomorphism(E.3) is mediated by gauge equivalences in the symmetric space G c / G . Thisgauged space corresponds to changes of hermitian inner product on thebundle. Indeed constructions on each side of the isomorphism are made byfixing a bundle inner product.ii) When n = g , the Baker-Akhiezer section (F.6) may be re-expressedin terms of Riemann’s theta function (E.7) as φ ( P ) = g ( P ) θ (Φ( P ) − Φ( D ) − κ ) . iii) It is interesting to inquire how the character of the explidit represen-tation (F.6) is related to the automorphy of the uniformized connections weconsidered in Section A.3. Because τ P,Q is normalized, the correspondingcharacter, χ , of f is unitary and corresponds precisely to σ in (A.5) and Q corresponds to the base point z in ρ n,σ . The holomorphic structure on E n and corresponding constant curvature connection under (E.3) are deter-mined by s n ( P ). In this way all degrees of freedom are accounted for. G The bundle E ⊗ F Proposition G.1.
Let E and F the bundles have the automorphies (A.4) and (C.10) . The holomorphic line bundle E ⊗ F is isomorphic to K ⊗ b ,where K is the holomorphic cotangent bundle of the Riemann surface X (the canonical bundle) and b := n g − . Moreover, E ⊗ F has degree n .Proof. Since the bundles E and F have the automorphies ρ and ˜ ρ − (see(A.4) and (C.10)), respectively, the bundle E ⊗ F has the automorphy ρ ( s, z )˜ ρ ( s, z ) − = (cid:20) cz + dcz + d (cid:21) − b | cz + d | b = ( cz + d ) b , inzburg–Landau equations on Riemann surfaces, April 23, 2019 where again b := πn | X | = n g − . So E ⊗ F is a holomorphic line bundle withautomorphy factor ( cz + d ) b . Since the holomorphic cotangent bundle, K ∼ = T ∗ X , of the Riemann surface X has the automorphy factor ( cz + d ) ,we conclude that E ⊗ F ∼ = K b .Since ( E ⊗ F ) ⊗ (2 g − = K n , K has degree 2 g −
2, and the degree in atensor product is additive, we have that (2 g −
2) deg( E ⊗ F ) = deg K n = n (2 g − . Hence the bundle E ⊗ F has degree n .Proposition G.1 implies, in particular that the line bundle F has degree0. References [1] L. V. Ahlfors,
Complex analysis . McGraw-Hill, New York, 1979.[2] Atiyah, M. F. and R. Bott, The Yang-Mills equations over Riemannsurfaces. Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505,523–615.[3] M. F. Atiyah, “Riemann surfaces and spin structures”,
Annales sci-entifiques de l’ ´E.N.S. , 4 e s´erie, tome 4, n o A New Proof of a Theorem of Narasimhan andSeshadri , 1982., J. Diff. Geom. 18 (1983) 269–277.[12] S. K. Donaldson,
Riemann Surfaces , Oxford University Press, 2011. inzburg–Landau equations on Riemann surfaces, April 23, 2019 [13] S. K. Donaldson and P. B. Kronheimer, Geometry of Four-Manifolds ,Oxford University Press, 1990.[14] H. M. Farkas, I. Kra, Riemann surfaces. Springer-Verlag, New York,1992.[15] O. Forster.
Lectures on Riemann surfaces , Graduate Texts in Math-ematics Vol. 81, Springer-Verlag, New York, 1991.[16] W. Fulton,
Algebraic Topology: A First Course . Springer, 1995.[17] O. Garc´ıa-Prada, A direct existence proof for the vortex equationsover a compact Riemann surface. Bull. London Math. Soc. 26 (1994),no. 1, 88–96.[18] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley,1994.[19] R. C. Gunning, The structure of factors of automorphy, AmericanJournal of Mathematics, Vol. 78, No. 2 (Apr., 1956), pp. 357–382.[20] R. C. Gunning,
Riemann Surfaces and Generalized Theta Functions ,Springer, 1976.[21] R. C. Gunning, Lectures on Riemann Surfaces, Princeton UniversityPress, 1967.[22] R.C. Gunning,
Lectures on Modular Forms , Princeton UniversityPress, 1962.[23] D. Huybrechts, Complex Geometry, Springer, 2005.[24] A. Jaffe and C.Taubes, Vortices and Monopoles, Birkh¨ausen, 1981.[25] J. Jost, Compact Riemann Surfaces, Third Edition, 2006.[26] S. Kobayshi,
Differential Geometry of Complex Vector Bundles ,Princeton University Press, New Jersey, 1987.[27] I.M. Krichever, Methods of Algebraic Geometry in the Theory ofNon-linear Equations, Russian Math Surveys, 32 (1977), 185-213.[28] N.S. Manton and N. A. Rink, Vortices on hyperbolic surfaces. J.Phys. A 43 (2010), no. 43, 434024[29] N.S. Manton and P. Sutcliffe,
Topological solitons . Cambridge Uni-versity Press, Cambridge, 2004.[30] A. Nagy, Irreducible Ginzburg–Landau fields in dimension 2,arXiv:1607.00232v2.[31] Jie Qing, Renormalized energy for Ginzburg-Landau vortices onclosed surfaces, Math. Z. 225, 1–34 (1997).[32] I. M. Sigal and T. Tzaneteas. Stability of Abrikosov lattices undergauge-periodic perturbations,
Nonlinearity
25 (2012) 1–24, arXiv. inzburg–Landau equations on Riemann surfaces, April 23, 2019 [33] I. M. Sigal, Magnetic vortices, Abrikosov lattices and automorphicfunctions, in Mathematical and Computational Modelling (With Ap-plications in Natural and Social Sciences, Engineering and the Arts),Wiley, (2015), arXiv:1308.5440.[34] C. Tejero Prieto, Holomorphic spectral geometry of magneticSchr¨odinger operators on Riemann surfaces, Diff Geom and its Ap-plications 24 (2006) 288 -310.[35] C. Teleman, Riemann Surfaces , https://math.berkeley.edu/ tele-man/math/Riemann.pdf[36] T. Tzaneteas and I. M. Sigal, Abrikosov lattice solutions of theGinzburg-Landau equations.
Contemporary Mathematics , 195– 213, 2011.[37] R. O. Wells, Jr.,
Differential Analysis on Complex Manifolds , Grad-uate Texts in Mathematics 65, Third Edition, Springer, 2000.[38] E. Witten, From superconductors and four-manifolds to weak inter-actions, Bulletin AMS44