A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS OVER A COMPACT RIEMANN SURFACE


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1 A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS OVER A COMPACT RIEMANN SURFACE OSCAR GARCIAPRADA ABSTRACT We give a direct proof of an existence theorem for the vortex equations over a compact Riemann surface, exploiting the interpretation of these equations in terms of moment maps. 1. The vortex equations In this paper we shall describe a direct existence proof for the vortex equations over a compact Riemann surface. These equations are a straightforward generalization of the vortex equations on U 2 which were introduced in 1950 by Ginzburg and Landau [10] in the theory of superconductivity. From a geometric point of view, they correspond to the absolute minima of the YangMillsHiggs functional. This is defined for a unitary connection A on a hermitian line bundle over R 2 and a smooth section 0 of that bundle as YMH(A,<f>) = where F A is the curvature of A and d A <f> is the covariant derivative of <}). If we regard IR 2 as the complex plane, we may decompose with respect to the complex structure to obtain d A = d' A + d" A. Then, by integration by parts, we can show that YMH is bounded below by 2nd, where d is an integer called the vortex number, and that this minimum is attained if and only if These equations are invariant under gauge transformations, and the moduli space of solutions is described by the basic existence theorem of Jaffe and Taubes [14]. They proved that given d points x t e U 2 (possibly with multiplicities), there exists a solution to the vortex equations, unique up to gauge equivalence, with <f>(x t ) = 0. This means that the moduli space of vortices is the space of unordered ^/tuples S d C. But an element of this space can be thought of as the set of zeros of a monic polynomial Hence the moduli space is just the vector space C d parametrizing all such polynomials. We shall study the following more general situation. Let Xbe a compact Riemann surface endowed with a metric having Kahler form co. Let L be a smooth complex line bundle over X, with a fixed hermitian metric h. Denote by sf the space of unitary connections on (L, h), and by Q (L) the space of smooth sections of L. Received 5 February 1992; revised 22 October Mathematics Subject Classification 53C07. Bull. London Math. Soc. 26 (1994) 8896
2 As in the defined by A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS 89 2 case, the YangMillsHiggs functional YMH: rf x Q (L) * U is YMH(A,</>) = \\F A f+\\d A </>\\ 2 +\\\Ml\\ 2, (1) where denotes the L 2 norm, F A eq 2 x is the curvature of A, d A <f>eq}(l) is the covariant derivative of <f>, and 0 A is the norm of <p with respect to h. This functional is invariant under the standard action of the gauge group ^ of unitary transformations of (L,h), so it defines a functional on the space The integration by parts in the IR 2 case is replaced here by the use of the Kahler identities (see [2, 7] for details) to rewrite the functional as + 2nd, where d" A is the (0,1) part of the connection, AF A eq. x is the contraction of F A with the Kahler form, and the integer d is the degree of L, that is, its first Chern class. We then conclude that the functional YMH is bounded below by 2nd, and that this lower bound is attained at (A, 0)e stf x Q (L) if and only if Note that the first equation simply says that 0 is holomorphic with respect to the holomorphic structure on L induced by Aesf. Due to the compactness of X, we may integrate the second equation to find an obstruction to its solution, given by the condition d< VolCT) An However, we may overcome this obstruction by introducing a positive parameter T in the YangMillsHiggs functional. The modified functional YMH T is obtained by substituting for the third term in (1). Our new functional can be rewritten as I 2 I A nrd, showing that 2nxd is a bound attained if and only if (2) These equations are called the xvortex equations. A necessary condition for the existence of solutions is d < xvo\{x)/an\ we shall show below that this condition is also sufficient. More precisely, we have the following.
3 90 OSCAR GARCIAPRADA THEOREM. Let X be a compact Riemann surface equipped with a metric. Let L be a C line bundle of degree d > 0 with a fixed hermitian metric h. Let D =? =1 x t be an effective divisor of degree d, and consider x > 0. Then there exists a smooth solution, unique up to gauge equivalence, of equations (2) if and only if </=deg(l)< ^ L. (3) Moreover, this solution is such that (L, d'2) = [D], the holomorphic bundle determined by D, and the set of zeros of <f> is the divisor D. This theorem is completely analogous to the result of Jaffe and Taubes [14], although the parameter x has no equivalent there. There are already several proofs of this theorem, in fact of a generalization of it to any compact Kahler manifold. Two of them are due to Bradlow. In [2], he gives a proof by reducing the vortex equations to a differential equation, already studied by Kazdan and Warner [15], of the type for a real function u on X, where h is a smooth function on X, not identically zero, and c is a real constant. His second proof is a particular case of a more general existence theorem for the vortex equations on a vector bundle of arbitrary rank [3]. He uses here methods analogous to those used by Simpson [17] in dealing with similar equations on Higgs bundles. In [7], the author has given another proof based on dimensional reduction arguments: namely, we have shown that the vortex equations are a dimensional reduction of the HermitianEinstein equation for a connection on a rank two bundle over XxCP 1. The existence of solutions is then related to the notion of stability for a holomorphic bundle via the theorem of Donaldson, Uhlenbeck and Yau [5, 6, 20]. 2. The proof We shall model our proof on that of Hitchin for the selfduality equations [13], which in turn is modelled on Donaldson's proof of the theorem of Narasimhan and Seshadri [4]. The key fact that we shall exploit is that the equation can be regarded as a moment map equation. So before starting our proof, let us pause to recall the basic definitions of moment maps. Let (M,co) be a symplectic manifold (for example, a Kahler manifold with its Kahler form) and let G be a Lie group acting on M symplectically, that is, preserving the symplectic form. Then if X is a vector field generated by the action, the Lie derivative L x co vanishes. Now for co, as for any differential form, + d(i(x)co); hence d(i(x) co) = 0. So, if H\M, U) = 0, there exists a function n x : M > R such that
4 A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS 91 As X ranges over the set of vector fields generated by the elements of the Lie algebra g of G, these functions can be chosen, under quite general conditions, to fit together to give a map to the dual of the Lie algebra, //: M * g*, defined by where A is the vector field generated by A e g. There is a natural action of G on both sides, and an ambiguity of a constant in the choice of /i x. If this can be adjusted so that n is Gequivariant, that is, compatible with both actions, then n is called a moment map for the action of G on M. The only remaining ambiguity in the choice of fi is the addition of a constant abelian character in g*. A valuable feature of the moment map is that it gives rise to new symplectic manifolds. More precisely, suppose that G acts freely and discontinuously; then is a symplectic manifold of dimension dim M 2 dim G. This is the MarsdenWeinstein quotient of a symplectic manifold by a group (see [12, 16], for instance). We shall be concerned with moment maps defined on infinitedimensional spaces. The first of these was constructed by Atiyah and Bott [1]. Let L be a C 00 complex line bundle over a compact Riemann surface X. Fix a hermitian metric h on L. The space of unitary connections can be identified with #, the space of holomorphic structures on L. On # there is an inner product <<*,/?> = f (<X J x for <x, e 7^# s Q \X). This inner product makes < a Kahler manifold with Kahler form The standard action of the unitary gauge group ^ preserves this Kahler form, and the moment map for the action is given, up to addition of a constant element of the centre, by Now consider the space of sections Q (L). This is also a Kahler manifold, with metric given by r y/,rj) h for \j/,neq 0 (L). The symplectic form that we shall consider on Q (L) is i The action of the gauge group & is symplectic, and it is easy to see that the moment map is given by <fi\* \i\<f>\l. We may combine the moment maps on sf and Cl (L) to obtain the moment map for the symplectic action of ^ on s& x Q\L).
5 92 OSCAR GARCIAPRADA We shall wish in future to restrict this moment map to the Kahler submanifold Jf = {(A,</>)erfxQ 0 (L)\<f> 0 and d" A <f> = 0}, since pairs (A,<p) in JV satisfy the first equation of our system of vortex equations. In order to introduce the parameter z in the moment map, consider the symplectic action on Jf of the subgroup U(l) <= ^ of constant unitary transformations, acting trivially on A and by multiplication on 0. The moment map for this action, jx: Jf * u(l), is given by The range of the moment map fi is u(l) = iu~, where M~ = ( oo,0). So if ceiu~, we can consider the symplectic quotient Jf = fir^c)/u(l). The group $ = acts symplectically on Jf y and the moment map /i for this action is given by 27HV/ We shall be able to solve this equation for any ceiu~. Writing / 2nid then c e iu~ is equivalent to d < z Vol (X)/4n, and The second equation of our system of vortex equations is then equivalent to The complex gauge group ^c acts on $0 by the action obtained by identifying $0 with the space of holomorphic structures on L and on Q (L) by multiplication. This induces an action on Jf. We can identify the quotient space Jf'/^c with the space of effective divisors of degree d, that is, the dfold symmetric product of the Riemann surface S d X. This follows from the very standard facts that a holomorphic line bundle is the line bundle of an effective divisor if and only if it has a nontrivial holomorphic section, and that, moreover, the divisor is given by the zeros of this holomorphic section (see [11], for example). However, we clearly can also identify the space of effective divisors with the quotient./p/# c, where # c = ^C/C*. The action of e# c on Jf is given by choosing a lifting ge^c that leaves the L 2 norm of the Higgsfieldfixed, that is, g0 ^ = 0 ^. Clearly, this extends the action of #. We shall solve the equation fi(a, (j>) = 0 by considering the orbit of a representative of the divisor D in Jf under the complex group # c = ^C/C*. We shall find a minimum for /i(y4,0) ^ on this orbit. Since # c acts freely, we shall produce a
6 A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS 93 solution to the equation n{a, 0) = 0. To see this, suppose that the symplectic manifold M considered at the beginning of our discussion on moment maps is a Kahler manifold, and that the group G acts by isometries as well as symplectically. We choose an invariant positive definite inner product on g, allowing us to identify g* with g. Suppose that G has a complexification G c with Lie algebra g c = g R C. Then the action of G can be extended to an action of G c. This action preserves the complex structure of M, but not necessarily the metric or symplectic structure. Now consider the function/: M> U defined by f{x) = //(x) 2. The gradient vector field grad/at a point xem satisfies (4) where / denotes the complex structure on TM X, and //(x) is the vector field generated by fi{x) evaluated at x. Hence grad z /=2/// / (x). By (4), the gradient lines are contained in the orbits of G c. Let T be such an orbit; then the critical points of the restriction of/to T are also critical points of/on M. If x is a critical point, then //(x) is zero. Also, if the isotropy group under the action of G is trivial (or finite), then fi(x) must be zero. Proof of Theorem. The necessity of (3) has been shown already. We shall show that it is sufficient. As in [4, 13], we shall be working with generalized connections of class L\, that is, connections which differ from a smooth connection by an element of the Sobolev space L\. We shall also use gauge transformations in L\. Since, as shown in [1], every L\ orbit in the L\ space of connections contains a C 00 connection, there is no loss of generality as far as A is concerned. Also, since 0 satisfies the elliptic equation d' A '(j) = 0, we deduce by elliptic regularity that 0 is C 00. As explained in [1, 14], the group action and properties of curvature that we use extend without substantial change in particular, L\ <= C so the topology of the line bundle is preserved. We observe that the functional ^(/4,0) 2 2 on Jf is essentially the Yang MillsHiggs functional, Indeed, as shown above, if (A, fye^v, YMH r (A,</>)= AF A + \cf>\lt + 2nrdeg(L) (5) The YangMillsHiggs functional extends to a smooth functional for A and 0 in the L\ spaces. Notice that 06 L 4 since, as a particular case of the Sobolev inequalities, the inclusion L\ c L 4 is compact. So given DeS d X, choose a smooth representative (A o, 0 o )e./k, and consider the restriction of \\/i(a, 0) «to the orbit of (A o, 0 O ) under (# c ) 2, the group of L\ complex gauge transformations modulo C*. Take a minimizing sequence (A n, 0 n ) for // ^ in this orbit. Then for some constant C, This, together with equation (5), gives an L 2 bound on F A. The main ingredient in the proofs of Donaldson and Hitchin referred to above is the weak compactness theorem
7 94 OSCAR GARCIAPRADA of Uhlenbeck [19]. This states that if A n is a sequence of L\ connections for which F A is bounded in L 2, then there are unitary gauge transformations u n for which u n (A n ) has a weakly convergent subsequence. In our abelian situation, this is an easy consequence of the ellipticity of the Coulomb gauge. We have then a subsequence A n. and L\ unitary gauge transformations u n. such that u n.{a n ) converges weakly in L\. Rename A n = u n (A n ) and 0 n = u n ((j) n ). We shall now find L\ uniform bounds for <f> n, so that by the weak compactness of L\, the sequence 0 n will have a weakly convergent subsequence in L\. To do this, consider the elliptic estimate We have d" A <f) n = 0; on the other hand, the constants K n can be uniformly bounded since the d" A converge. We need only find uniform bounds for H^JI^. First note that we have uniform bounds for 0J L 4 as a consequence of (A n,<f> n ) being a minimizing sequence for //(;4,0) 2 2, and the equality (5). Now Holder's inequality gives us the uniform L 2 bounds for 0. We conclude that (after possibly renaming again) (A n, 0J converges weakly in L\ to {A,(j>). We need to show that (A,<f>) is in the same orbit as (A o, 0 O ). The (A n,<f> n ) are related to (A 0,<f> 0 ) by elements g n e(^c) 2 : satisfying II0JL=IIS.0OIL«= II0OIL« ( 6 ) We shall prove first that g n has a subsequence that converges to a holomorphic isomorphism g from (L,d A ) to (L,d A ). From this we may conclude that (A,<j>) is in the same orbit as (A o, 0 O ) and is a solution to the vortex equations. The operators d' A ' n and d" A are related by d" A d" A = a n, where a n is an element in the L\ completion of the space of (0, l)forms on X. The element a n defines a class in H 0> \X), since by elliptic regularity the cohomology of the Dolbeault complex of L 2 forms is isomorphic to the cohomology of the ordinary Dolbeault complex. This class can be regarded as an element of H\X, 0) via the Dolbeault isomorphism. Consider the standard short exact sequence of sheaves defined by the exponential map 0 >Z >Q >0* >1, and the corresponding long exact sequence in cohomology 0 > H\X, Z) > H\X, 0) > H\X, (9*). Now, since a n = g^d"g n, the image in H\X, &*) of the class defined by <x n is the trivial element, thus this class comes from an element in H\X, Z). The convergence of the a n implies the convergence of the classes that they define, as can be seen by identifying these classes with their harmonic representatives. So, for n large enough, the integral class defined by a n must be constant, thus by applying a fixed unitary gauge transformation we can assume that this class is zero. Hence there exists h n such that a = d"h«.
8 A DIRECT EXISTENCE PROOF FOR THE VORTEX EQUATIONS 95 The L\ convergence of A n implies the L\ convergence of <x n. Now using the elliptic estimate \\h n c n \\ Ll^C\\d"h n \\ Ll, where c n = j x h n and C is a constant, we deduce that the L\ convergence of a n = d"h n gives the L\ convergence of h n c n. To see this, recall that in general one has the estimate and the term / L s can be omitted if/is L 2 orthogonal to the kernel of d". But the kernel of d" consists of the constant functions, and f h n c n is certainly L 2  orthogonal to those. Since L\ <= C, we obtain a uniform bound on h n c n, that is, there is a constant M such that \h n c n \<M. (7) Now, g n = K n exp(h n c n ) (8) for some nonzero constant K n. From (7) and (8) we obtain \K n \e~ M ^ \g n \ ^ \K n \e M, and then (6) gives e~ M ^ \K n \ ^ e M. This shows that we can choose a subsequence of g n which converges uniformly to a nonzero gauge transformation. As mentioned above, the methods of our proof have already been used in dealing with other gaugetheoretical equations, like the antiselfdual equations on a fourmanifold, or Hitchin's selfduality equations on a Riemann surface [13]. The case of the vortex equations is, however, slightly different to the other cases. The difference is due to the fact that the relevant group for the vortex equations is U(«), while in the other cases it is typically SU(«). We have dealt here with the abelian case U(l), but it is conceivable that the same methods could work in the nonabelian situation studied in [3, 8, 9]. Our approach to solving the vortex equations is also useful in the study of the moduli space of vortices, that is, the space of all solutions modulo gauge equivalence. We have realized this space as the symplectic quotient On the other hand, the existence theorem establishes a bijection 11 ) = 0}/#< > Jfl& 9* S d X. Under this identification, S"*^ inherits a symplectic structure which can be seen to be compatible with the complex structure. It defines then a Kahler structure whose study would be interesting to pursue. ACKNOWLEDGEMENTS. It is a pleasure to thank my thesis supervisors, Professors Simon Donaldson and Nigel Hitchin, for many ideas and good advice. I also wish to thank Jorgen Andersen for discussions, and the referee for valuable comments. Thanks are due to the Spanish Consejo Superior de Investigaciones Cientificas and
9 96 OSCAR GARCIAPRADA The British Council for financial support. Finally, I am very grateful to the IHES, where the final version of this paper has been written, for their hospitality and financial support. References 1. M. F. ATIYAH and R. BOTT, 'The YangMills equations over Riemann surfaces', Philos. Trans. Roy. Soc. London Ser. A 308 (1982) S. B. BRADLOW, 'Vortices in holomorphic line bundles over closed Kahler manifolds', Comm. Math. Phys. 135 (1990) S. B. BRADLOW, 'Special metrics and stability for holomorphic bundles with global sections', J. Differential Geom. 33 (1991) S. K. DONALDSON, 'A new proof of a theorem of Narasimhan and Seshadri', /. Differential Geom. 18 (1983) S. K. DONALDSON, 'Antiselfdual YangMills connections over complex algebraic surfaces and stable bundles', Proc. London Math. Soc. (3) 50 (1985) S. K. DONALDSON, 'Infinite determinants, stable bundles and curvature', Duke Math. J. 54 (1987) O. GARCIAPRADA, 'Invariant connections and vortices', Comm. Math. Phys., to appear. 8. O. GARCIAPRADA, 'Dimensional reduction of stable bundles, vortices and stable pairs', Internal. J. Math., to appear. 9. O. GARCIAPRADA, 'The geometry of the vortex equation', D.Phil. Thesis, Oxford, V. L. GINZBURG and L. D. LANDAU, Zh. Eksper. Teoret. Fiz. 20 (1950) P. GRIFFITHS and J. HARRIS, Principles of algebraic geometry (Wiley, New York, 1978). 12. V. GUILLEMIN and S. STERNBERG, Symplectic techniques in physics (Cambridge University Press, 1990). 13. N. J. HITCHIN, 'The selfduality equations on a Riemann surface', Proc. London Math. Soc. 55 (1987) A. JAFFE and C. H. TAUBES, Vortices and monopoles, Progr. Phys. 2 (Birkhauser, Boston, 1980). 15. J. L. KAZDAN and F. W. WARNER, 'Curvature functions for compact 2manifolds', Ann. of Math. (2) 99 (1978) F. C. KIRWAN, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes 31 (Princeton University Press, Princeton, NJ, 1984). 17. C. T. SIMPSON, ' Constructing variations of Hodge structure using YangMills theory and applications to uniformization', J. Amer. Math. Soc. 1 (1988) C. H. TAUBES, 'Arbitrary ivvortex solutions to the first order GinzburgLandau equations', Comm. Math. Phys. 72 (1980) K. K. UHLENBECK, 'Connections with L p bounds on curvature', Comm. Math. Phys. 83 (1983) K. K. UHLENBECK and S. T. YAU, 'On the existence of HermitianYangMills connections in stable vector bundles', Comm. Pure Appl. Math. 39S (1986) Institut des Hautes Etudes Scientifiques 35 Route de Chartres BuressurYvette France
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