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## The self-duality equations on a Riemann surface (1987)

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Venue: | Proc. Lond. Math. Soc., III. Ser |

Citations: | 501 - 6 self |

### Citations

479 |
Differential Geometry and Symmetric Spaces
- Helgason
- 1962
(Show Context)
Citation Context ...inite-dimensional vector space, we see that the weakly convergent sequence becomes a sequence converging to (A,<&). Hence y{sn)->m in the topology of M. Standard differential geometric arguments (see =-=[14]-=-) show this to be a contradiction. The natural metric on M is a very special one—it is a hyperkdhler metric. Recall that a hyperkahler metric on a 4n-dimensional manifold is a Riemannian metric which ... |

416 |
Cohomology of quotients in symplectic and algebraic geometry
- Kirwan
- 1984
(Show Context)
Citation Context ...-2 and so d - \ = deg L - \ deg A2V ^ g - 1; hence d^g-1. (iii) The non-degenerateness of the critical submanifolds follows from their description as the fixed point set of a circle action. Also (see =-=[5, 21]-=-), the index of the critical submanifold Y is equal to the real rank of the subbundle N~ of the normal bundle of Y on which the holomorphic circle action acts with negative weights. In the case of /^(... |

293 | Vector Bundles over an Elliptic Curve
- Atiyah
- 1957
(Show Context)
Citation Context ...(m) is O-invariant. However, deg O(m) = m&i(m + n) = l deg(A2V). (iv) On an elliptic curve, every indecomposable bundle is, after tensoring with a line bundle, equivalent to the non-trivial extension =-=[3]-=- defined by H\M;0) = Cor where 0(1) is a line bundle of degree 1, defined by HX(M ; 0 ( - l ) ) = C. Since the canonical bundle is trivial, the Higgs fields O are endomorphisms of V, but in the first ... |

292 |
Hyperkahler Metrics and Supersymmetry
- Hitchin, Karlhede, et al.
- 1987
(Show Context)
Citation Context ... fact is basically a consequence of the structure of the self-duality equations themselves and is due to the fact that, formally speaking, the moduli space M is a hyperkahler quotient in the sense of =-=[18]-=-. A hyperkahler metric is one which is Kahlerian with respect to complex structures /, / , and K which satisfy the algebraic relations of the quaternions i, ;, k. Its existence means that M has many c... |

286 |
Self-duality in four-dimensional Riemannian geometry
- Atiyah, Hitchin, et al.
- 1978
(Show Context)
Citation Context ...onsider the space of all solutions on a fixed principal bundle P, modulo the group of gauge transformations. This is a setting which is by now familiar in the context of gauge theories on 4-manifolds =-=[4, 7, 30]-=- or stable bundles on Riemann surfaces [5]. The basic idea is to linearize the equations and consider the elliptic complex which arises from this. An application of the Atiyah-Singer index theorem and... |

272 |
Space of unitary vector bundles on a compact Riemann surface
- Seshadri
- 1967
(Show Context)
Citation Context ...Let <J> = 0; then the self-duality equations are simply F(A) = 0 and we are reduced to the consideration of flat unitary connections. This, via the 66 N. J. HITCHIN theorem of Narasimhan and Seshadri =-=[25]-=-, is equivalent to the study of stable holomorphic bundles. EXAMPLE (1.5). Let M be given a Riemannian metric g = hdzdz compatible with the conformal structure. Then the Levi-Civita connection is a U(... |

227 |
functions on complete Riemannian manifolds, Comm. Pure and
- Yau
- 1975
(Show Context)
Citation Context ...rmonic exhaustion function for M with respect to the complex structure /. Thus M is a Stein manifold. (iii) Since a hyperkahler manifold has vanishing Ricci tensor, this follows from a theorem of Yau =-=[34]-=-. REMARKS. (1) From (9.10) the Kahler forms co2 and <w3 are cohomologous to zero. The form o)x is not, as it restricts to a Kahler form on No, the moduli space of stable bundles and from (7.7) the res... |

221 | The symplectic nature of fundamental groups of surfaces
- Goldman
- 1984
(Show Context)
Citation Context ...vita connection of /i dz dz. Hence in all cases, the flat SL(2, R) connection d'A + d\ + <]> + <I>* is equivalent to the canonical connection associated to the metric of constant curvature REMARK. In =-=[11]-=-, Goldman showed that every flat PSL(2, U) connection with Euler class 2g — 2 is the connection associated to a constant-curvature metric. This is also a corollary of Donaldson's theorem (9.19), for e... |

212 |
Harmonic spinors
- Hitchin
- 1974
(Show Context)
Citation Context ...onnection is a U(l) connection defined on the canonical bundle K. Let K% denote a holomorphic line bundle such that with the induced U(l) connection. (These are spinor bundles for the Riemann surface =-=[15]-=-.) Let P be the principal SU(2) bundle associated to the rank-2 vector bundle V = K? © K~3 and A the SU(2) connection (reducible to U(l)) defined by the Levi-Civita connection. With respect to this de... |

141 |
The Yang-Mills equations over Riemann surfaces, Philos
- Atiyah, Bott
- 1982
(Show Context)
Citation Context ...irs occur with more frequency, but there are still restrictions on the holomorphic structure of the underlying vector bundle V. We recall the various types of rank-2 bundles on a Riemann surface (see =-=[5]-=-): (i) V is stable if for each subbundle L, deg L < \ deg(A2V); (ii) V is semi-stable if for each subbundle L, deg L *s \ deg(A2F); (iii) if V is not semi-stable, there is a unique subbundle Lv with d... |

133 |
Quaternionic Kähler manifolds
- Salamon
- 1982
(Show Context)
Citation Context ... + x2J + x3K: x e S2} of the hyperkahler metric. The product manifold MxS2 has a natural complex structure / = (/x, Isi) which is integrable (this is the twistor space of the hyperkahlerian structure =-=[17,18, 31]-=-) and the circle action preserves the complex structure. The action projects under the holomorphic projection p2: MxS2-*S2 = CP1 to the standard rotation leaving ±7 fixed. If we can show that the holo... |

127 |
Twisted harmonic maps and the self-duality equations
- Donaldson
- 1987
(Show Context)
Citation Context ...y, and we may expect an analogous statement for the viewpoint considered here: every irreducible flat connection arises from a solution to the self-duality equations. This is the theorem of Donaldson =-=[8]-=-, proved in the paper following this one: 112 N. J. HITCHIN THEOREM (9.19) (S. K. Donaldson). Let P be a principal SO(3) bundle over a compact Riemann surface M. For any irreducible flat connection on... |

106 | On the existence of a connection with curvature zero
- MILNOR
- 1958
(Show Context)
Citation Context ...))/C/PSL(2, IR) is a smooth manifold of dimension (6g — 6) which is diffeomorphic to a complex vector bundle of rank (g — 1 + k) over the symmetric product S2g~2~kM. COROLLARY (10.9) (Milnor and Wood =-=[24, 33]-=-). The Euler class k of any flat PSL(2, IR) bundle satisfies \k\^2g-2. Proof By Donaldson's theorem (9.19) the flat connection arises from a 118 N. J. HITCHIN solution of the self-duality equations in... |

106 |
Moduli of vector bundles on a compact Riemann surface
- Narasimhan, Ramanan
- 1969
(Show Context)
Citation Context ...hic line bundle such that In Case (i), of course, {V, <£) is stable for any <J> in the 3-dimensional space H°(M ; End0 V <8> K). The stable bundle V itself is, from a result of Narasimhan and Ramanan =-=[26]-=-, determined by the subbundles of deg - 1 it contains. This is a divisor of the system 20 in the Jacobian Jl{M), and so each stable bundle is determined by a point in the 3-dimensional projective spac... |

102 |
A fibre bundle description of Teichmüller theory
- Earle, Eells
- 1969
(Show Context)
Citation Context ...that g0 is t n e constant-curvature metric compatible with the complex structure of M, and g another metric with the same constant curvature. We use the Earle and Eells approach to Teichmuller theory =-=[9]-=- next. Since (M, g) has negative curvature, it follows from the Eells-Sampson existence theorem that there is a unique harmonic diffeomorphism which is homotopic to the identity and minimizes the ener... |

99 |
Connections with Lp bounds on curvature
- Uhlenbeck
- 1982
(Show Context)
Citation Context ...roof is modelled on Donaldson's proof of this theorem [6]. It is an analytical proof which makes use of one non-trivial and highly effective tool in gauge theory: Uhlenbeck's weak compactness theorem =-=[32]-=-. To motivate the proof, we consider the structure of the equation F + [<!>, <P*] = 0 (4.1) which forms part of (1.3), in terms of moment maps. Recall that if N is a Kahler manifold with Kahler form c... |

75 |
Symmetric products of an algebraic curve’, Topology 1
- Macdonald
- 1962
(Show Context)
Citation Context ...be (1 + f\2g _ ,2g/i ,sf\2g (78) The other critical submanifolds are coverings of symmetric products of the Riemann surface M. The Poincar6 polynomial of a symmetric product was computed by Macdonald =-=[23]-=-: (1 + xt)2g R(SnM)is equal to the coefficient of x" in ,sWH —57. (7.9) vs'sMs(l~x)(l~xt2) Consider now the 22g-fold covering S"M of SnM, induced by the map This covering is a principal Z2g bundle ove... |

66 |
Fixed points and torsion on Kähler manifolds
- Frankel
- 1959
(Show Context)
Citation Context ...e of stable pairs (V, O), where V is a rank-2 vector bundle of odd degree and fixed determinant. We shall investigate the algebraic topology of M using the Morse function The method is due to Frankel =-=[10]-=- (see also [5]), and uses the fact that, since by (6.10), d\i = -2i(X)(ou the critical points of /x are the fixed points of the circle action generated by the vector field X. PROPOSITION (7.1). The fu... |

61 |
Bundles with totally disconnected structure group
- Wood
(Show Context)
Citation Context ...))/C/PSL(2, IR) is a smooth manifold of dimension (6g — 6) which is diffeomorphic to a complex vector bundle of rank (g — 1 + k) over the symmetric product S2g~2~kM. COROLLARY (10.9) (Milnor and Wood =-=[24, 33]-=-). The Euler class k of any flat PSL(2, IR) bundle satisfies \k\^2g-2. Proof By Donaldson's theorem (9.19) the flat connection arises from a 118 N. J. HITCHIN solution of the self-duality equations in... |

40 |
Lectures on vector bundles over Riemann surfaces
- Gunning
- 1967
(Show Context)
Citation Context ...lf-duality equations which reduce to U(l). (ii) If 0 = 0, then any subbundle is O-invariant. The condition deg (L) < |deg(A2V) for a bundle of rank 2 over a Riemann surface is the notion of stability =-=[13]-=-. In the next section we shall examine the algebro-geometric idea of stability for a pair of objects—a rank-2 holomorphic vector bundle V, and a holomorphic section of End V <S> K—which Theorem (2.1) ... |

28 |
Characteristic classes of stable bundles of rank 2 over an algebraic curve
- Newstead
- 1972
(Show Context)
Citation Context ...= 6g — 6 and dimK Nd = Ag — Ad — 2, it is clear that P,(M) is a polynomial of degree 6g — 6 and hence bi = 0 if i > 6g - 6. To compute the Betti numbers more explicitly, we use the result of Newstead =-=[29]-=- or Atiyah and Bott [5] for No the modulo space of stable bundles of rank-2, odd degree, and fixed determinant. The Poincar6 polynomial is there shown to be (1 + f\2g _ ,2g/i ,sf\2g (78) The other cri... |

25 |
Univalency of harmonic mappings between surfaces
- Jost
- 1981
(Show Context)
Citation Context ...function to compute the Betti numbers of the moduli space restricts to be simply the energy of the harmonic map. It is interesting to note that a proof of Tromba that Teichmuller space is a ball (see =-=[20]-=-) uses E(<f>) as a Morse function with just one critical point. (2) Our version of Teichmuller space ZT is C3g~3 as a complex manifold and has a complete Kahler metric with a circle action. None of th... |

24 |
An application of gauge theory to the topology of 4-manifolds
- Donaldson
- 1983
(Show Context)
Citation Context ...onsider the space of all solutions on a fixed principal bundle P, modulo the group of gauge transformations. This is a setting which is by now familiar in the context of gauge theories on 4-manifolds =-=[4, 7, 30]-=- or stable bundles on Riemann surfaces [5]. The basic idea is to linearize the equations and consider the elliptic complex which arises from this. An application of the Atiyah-Singer index theorem and... |

22 |
Representations of fundamental groups of surfaces. Geometry and topology
- Goldman
- 1985
(Show Context)
Citation Context ...r o, and hence in particular giving a holomorphic vector bundle V = La © L2. By Remark (3.12), the stability condition for the pair (V, O) gives deg Lx - deg L2 ^ (2g - 2). COROLLARY (10.10) (Goldman =-=[12]-=-). Ifg-2andk = l, then , PSL(2, R))7PSL(2, R ) s j | f x U\ Proof. From (10.6), w^E) = 0, so as a real vector bundle, E is trivial. In the case where k = 2g - 2, this moduli space is simply a vector s... |

19 |
Geometry of SU(2) gauge fields
- NARASIMHAN, RAMADAS
- 1979
(Show Context)
Citation Context ...s a stable pair, then there exists an automorphism of V of determinant 1, unique modulo SO(3) gauge transformations, which takes (A, 4>) to a solution of the equation F(A) + [O, O*] = 0. Proof. As in =-=[27, 6]-=- we shall work with connections which differ from a smooth connection by an element of the Sobolev space L\, and use automorphisms which lie in L\. Since (from [5]) every L\ orbit in the L\ space of c... |

17 |
Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7
- Newstead
- 1968
(Show Context)
Citation Context ...ith ° 2 In Case (i), (V, <1>) is again stable for any O in the 3-dimensional space H°(M ; Endo V®K). The stable bundles considered here are parametrized by the intersection of two quadrics in P5 (see =-=[26, 28]-=-). Each point y eM, considered as a double covering of the projective line, parametrizes a quadric Qy of the pencil together with a choice of ar-plane or /S-plane. Fixing a point q in the intersection... |

16 |
A new proof of a theorem of Narasimhan and Seshadri
- Donaldson
- 1983
(Show Context)
Citation Context ...n g(d) is not the identity for d^lkn, and so the second equation implies that A is reducible to a U(l) connection. This means that the associated vector bundle V is decomposable: V = L® L*A2V. Since g=-=(6)-=- is diagonal with respect to this decomposition, from (7.2), <£ must be lower triangular with respect to one ordering of the decomposition. Let <D = ( V where tf>eQ°(M;L-2/«:<8>AV).\(p 0/ By the self-... |

14 | Gauge theories on four-dimensional Riemannian manifolds
- PARKER
- 1982
(Show Context)
Citation Context ...onsider the space of all solutions on a fixed principal bundle P, modulo the group of gauge transformations. This is a setting which is by now familiar in the context of gauge theories on 4-manifolds =-=[4, 7, 30]-=- or stable bundles on Riemann surfaces [5]. The basic idea is to linearize the equations and consider the elliptic complex which arises from this. An application of the Atiyah-Singer index theorem and... |

13 |
Metrics on moduli spaces
- Hitchin
- 1984
(Show Context)
Citation Context ...ent maps with respect to ^, then the self-duality equations are given by lii{A, 3>) = 0, where 1 ^ i ̂s3,lii , 3) 0, and the moduli space M is the quotient (6.6) It is a theorem, in finite dimensions =-=[17, 18]-=-, that the natural metric on the quotient in this sense (a generalization of the Marsden-Weinstein quotient in symplectic geometry) of a hyperkahler manifold is again hyperkahler. We shall adapt the p... |

5 |
Differential manifolds (Addison-Wesley
- LANG
- 1972
(Show Context)
Citation Context ...equations. This is a principal •^-bundle over M. The curve y(s) may be lifted to a horizontal curve y(s) in M. This follows from the existence theorem for differential equations in Banach spaces (see =-=[22]-=-). By definition of the metric, the length of y from s = s0 to s = sn is the same as the length of y, that is, \sn —so\, which is bounded above. This length is greater than or equal to the straight li... |

3 |
Singularities of ray systems
- ARNOL'D
- 1983
(Show Context)
Citation Context ...s not a local diffeomorphism. The projection onto Jf gives a subspace which is called a caustic. The singularities of projections of Lagrangian submanifolds have been investigated in depth by Arnol'd =-=[2]-=-. For our purposes we merely wish to determine the locus of points in the Prym variety at which the polarization has a tangential part. This is equivalent to seeking a tangent vector X to M, tangent t... |

1 |
Geometry of algebraic curves 1
- ARBARELLO, CORNALBA, et al.
- 1985
(Show Context)
Citation Context ...; L2K 0 A2V*) = 0 since deg(L2£ 0 A2V*) >2g-2, so ch{p,(U-lK2)) = ch(E) where E=pifV~xK2 and therefore, since the tangent bundle of the Jacobian is trivial, ch(E) =p*(ch(U-'K2)td(M)). (10.4) Now (see =-=[1]-=-), c^U) = (2g — 2d — l)rj + y where r\ is the pull-back of the class of a point on M and y e Hl{M, Z) 0 //1(Jac(M), Z) the canonical element. Hence p*{ch{U-xK2)td{M)) = (g + 2d-2)-d, where 6 e H2(Jac(... |

1 |
Monopoles and geodesies
- HITCHIN
- 1982
(Show Context)
Citation Context ...ism L2 = o*Ll restricted to the fixed points of o, and 6D: H\D;L*2Lx)-+H\M;L2®p*{A2V*)) is the coboundary map for this sequence. (An analogous situation occurs in the twistor description of monopoles =-=[16]-=-.) It follows that d(s) = 0 if and only if dD(o*s) = 0. Now o*s is a section of L*L2 ®p*(A2V) on D and this maps to zero if and only if it restricts from a section of the same bundle on M. However, si... |