3096:
3387:
The moduli space is usually a manifold. For generic metrics, after gauge fixing, the equations cut out the solution space transversely and so define a smooth manifold. The residual U(1) "gauge fixed" gauge group U(1) acts freely except at reducible monopoles i.e. solutions with
3993:
2902:
2008:
2749:
2447:
4152:
1718:
4523:. The value of the invariant on a spin structure is easiest to define when the moduli space is zero-dimensional (for a generic metric). In this case the value is the number of elements of the moduli space counted with signs.
822:
3514:
1157:
215:
2913:
1586:
439:
2284:
2222:
3849:
1256:
897:
2760:
1795:
1400:
70:
and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than
Donaldson invariants; for example,
3689:
718:
492:
4488:
4371:
it has two connected components (chambers). The moduli space can be given a natural orientation from an orientation on the space of positive harmonic 2 forms, and the first cohomology.
253:
3612:
1348:
3183:
1880:
957:
539:
2622:
4418:
3338:
3216:
2126:
636:
4186:
2545:
1839:
2165:
1076:
4316:
4263:
3137:
2311:
1447:
585:
313:
5110:
1302:
1026:
2343:
2581:
1875:
4438:
4369:
4283:
4206:
3840:
3820:
3800:
3780:
3709:
3566:
3540:
3519:
which for generic metrics is the actual dimension away from the reducibles. It means that the moduli space is generically empty if the virtual dimension is negative.
3412:
3292:
2607:
2521:
2348:
4060:
3740:
3272:
2501:
2474:
2087:
2060:
1184:
2037:
1631:
4052:
4336:
4226:
3760:
3632:
3370:
3298:
bounded in
Sobolev norms of arbitrary regularity, which shows all solutions are smooth, and that the space of all solutions up to gauge equivalence is compact.
3240:
1815:
1738:
1626:
1606:
1467:
1420:
1046:
997:
977:
842:
738:
353:
283:
743:
3424:
1081:
127:
3091:{\displaystyle \Delta |\phi |^{2}+|\nabla ^{A}\phi |^{2}+{\tfrac {1}{4}}|\phi |^{4}=(-s)|\phi |^{2}-{\tfrac {1}{2}}h(\phi ,\gamma (\omega )\phi )}
4734:
4703:
1472:
327:
is a reduction of the structure group to Spin, i.e. a lift of the SO(4) structure on the tangent bundle to the group Spin. By a theorem of
365:
5164:
5186:
5092:
4901:
4871:
4810:
4766:
3988:{\displaystyle =F_{0}^{\mathrm {harm} }=i(\omega ^{\mathrm {harm} }+\alpha ^{\mathrm {harm} })\in H^{2}(M,\mathbb {R} )}
4983:
4928:
3294:. After adding the gauge fixing condition, elliptic regularity of the Dirac equation shows that solutions are in fact
2897:{\displaystyle \Delta _{g}|\phi |_{h}^{2}=2h({\nabla ^{A}}^{*}\nabla ^{A}\phi ,\phi )-2|\nabla ^{A}\phi |_{g\otimes h}}
2230:
2171:
4926:(1994a), "Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory",
78:
tends to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in
Donaldson theory.
1197:
847:
1743:
1353:
4629:
on which the
Seiberg–Witten invariant is 1. In particular it cannot be split as a connected sum of manifolds with
4855:
3637:
3415:
645:
444:
60:
4443:
2003:{\displaystyle H^{1}(M,\mathbb {R} )^{\mathrm {harm} }/H^{1}(M,\mathbb {Z} )\oplus d^{*}A_{\mathbb {R} }^{+}(M)}
223:
5087:, First International Press Lecture Series, vol. 2, Somerville, MA: International Press, pp. vi+401,
3571:
1307:
4828:
3142:
4893:
4838:
4548:
if the
Seiberg–Witten invariant vanishes whenever the expected dimension of the moduli space is nonzero. The
2744:{\displaystyle {\nabla ^{A}}^{*}\nabla ^{A}\phi =(D^{A})^{2}\phi -({\tfrac {1}{2}}\gamma (F_{A}^{+})+s)\phi }
902:
497:
4787:, Lecture Notes in Mathematics, vol. 1629 (2nd ed.), Berlin: Springer-Verlag, pp. viii+121,
4833:
4748:
4385:
3304:
3188:
2092:
595:
102:
4160:
4758:
20:
4653:
Hirzebruch, F.; Hopf, H. (1958). "Felder von Flächenelementen in 4-dimensionalen
Mannigfaltigkeiten".
3380:
The space of solutions is acted on by the gauge group, and the quotient by this action is called the
360:
40:
4793:
2526:
1820:
4740:
2134:
1051:
4288:
4235:
3105:
2289:
1425:
1259:
548:
291:
2442:{\displaystyle \phi \mapsto \left(\phi h(\phi ,-)-{\tfrac {1}{2}}h(\phi ,\phi )1_{W^{+}}\right)}
1265:
1002:
356:
4788:
4147:{\displaystyle \omega ^{\mathrm {harm} }\in 2\pi K+{\mathcal {H}}^{-}\in H^{2}(X,\mathbb {R} )}
2316:
320:
286:
2550:
1844:
4655:
4423:
4341:
4268:
4265:, the moduli space is a (possibly empty) compact manifold for generic metrics and admissible
4191:
3825:
3805:
3785:
3765:
3694:
3545:
3525:
3391:
3277:
2586:
2506:
4567:) ≥ 2 then the manifold is of simple type. This is true for symplectic manifolds.
1713:{\displaystyle D^{A}=\gamma \otimes 1\circ \nabla ^{A}=\gamma (dx^{\mu })\nabla _{\mu }^{A}}
5181:
5152:
5132:
5102:
5065:
5045:
5012:
4992:
4967:
4947:
4911:
4881:
4820:
4776:
4726:
3718:
3245:
2479:
2452:
2065:
2045:
1162:
328:
2013:
8:
3712:
67:
5136:
5049:
4996:
4951:
3998:
5156:
5122:
5069:
5035:
4971:
4937:
4681:
4321:
4211:
3843:
3745:
3617:
3355:
3349:
3225:
1800:
1723:
1611:
1591:
1452:
1405:
1031:
982:
962:
827:
723:
338:
268:
5026:(1994b), "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD",
4754:
The
Seiberg–Witten equations and applications to the topology of smooth four-manifolds
5088:
5057:
5004:
4959:
4897:
4867:
4806:
4762:
4685:
4676:
263:
5160:
5073:
4975:
5140:
5053:
5000:
4955:
4859:
4858:, vol. 28, Providence, RI: American Mathematical Society, pp. xviii+484,
4798:
4712:
4671:
4663:
4717:
5148:
5098:
5080:
5061:
5008:
4963:
4907:
4877:
4816:
4772:
4722:
4698:
3345:
2523:
is a real selfdual two form, often taken to be zero or harmonic. The gauge group
817:{\displaystyle \gamma :\mathrm {Cliff} (M,g)\to {\mathcal {E}}{\mathit {nd}}(W)}
5144:
5019:
4919:
4490:, and hence only finitely many Spin structures, with a non empty moduli space.
2610:
1877:, leaving an effective parametrisation of the space of all such connections of
542:
319:
to SO(4) and is harmless from a homotopical point of view. A Spin-structure or
44:
4752:
3509:{\displaystyle (K^{2}-2\chi _{\mathrm {top} }(M)-3\operatorname {sign} (M))/4}
1152:{\displaystyle \wedge ^{+}M\cong {\mathcal {E}}{\mathit {nd}}_{0}^{sh}(W^{+})}
5175:
5023:
4923:
1159:
of the selfdual two forms with the traceless skew
Hermitian endomorphisms of
590:
210:{\displaystyle (U(1)\times \mathrm {Spin} (4))/(\mathbb {Z} /2\mathbb {Z} ).}
48:
32:
355:
admits a Spin structure. The existence of a Spin structure is equivalent to
4846:
2547:
acts on the space of solutions. After adding the gauge fixing condition
642:
representation of Spin(4) on which U(1) acts by multiplication. We have
494:
Conversely such a lift determines the Spin structure up to 2 torsion in
255:
acts as a sign on both factors. The group has a natural homomorphism to
5127:
5040:
4942:
4802:
4667:
2609:. For technical reasons, the equations are in fact defined in suitable
1581:{\displaystyle \nabla _{X}^{A}(\gamma (a)):==\gamma (\nabla _{X}^{g}a)}
332:
28:
4863:
740:
comes with a graded
Clifford algebra bundle representation i.e. a map
2583:
the residual U(1) acts freely, except for "reducible solutions" with
315:. This reduces the structure group from the connected component GL(4)
4054:
the necessary and sufficient condition for a reducible solution is
4232:
met and solutions are necessarily irreducible. In particular, for
3418:
the moduli space is finite dimensional and has "virtual dimension"
4701:(1996), "The Seiberg-Witten equations and 4-manifold topology.",
4537:) = 1, but then it depends on the choice of a chamber.
1078:
by anti-symmetrising. In particular this gives an isomorphism of
5085:
Seiberg Witten and Gromov invariants for symplectic 4-manifolds
639:
434:{\displaystyle w_{2}(M)\in H^{2}(M,\mathbb {Z} /2\mathbb {Z} )}
3222:
bounded with the bound depending only on the scalar curvature
3742:
is closed, the only obstruction to solving this equation for
256:
2062:
for a spinor field of positive chirality, i.e. a section of
638:
coming from the 2 complex dimensional positive and negative
101:, Chapter 10). For the relation to symplectic manifolds and
81:
For detailed descriptions of
Seiberg–Witten invariants see (
4188:
is the space of harmonic anti-selfdual 2-forms. A two form
4596:
is the connected sum of two manifolds both of which have
1114:
800:
1628:. The Clifford connection then defines a Dirac operator
1797:
acts as a gauge group on the set of all connections on
4585:) ≥ 2 then all Seiberg–Witten invariants of
4526:
The Seiberg–Witten invariant can also be defined when
4006:
3047:
2977:
2694:
2388:
4603: ≥ 1 then all Seiberg–Witten invariants of
4513:) ≥ 2 is a map from the spin structures on
4446:
4426:
4388:
4344:
4324:
4291:
4271:
4238:
4214:
4194:
4163:
4063:
4001:
3852:
3828:
3808:
3788:
3768:
3748:
3721:
3697:
3640:
3620:
3574:
3548:
3528:
3427:
3394:
3358:
3307:
3280:
3248:
3228:
3191:
3145:
3108:
2916:
2763:
2625:
2589:
2553:
2529:
2509:
2482:
2455:
2351:
2319:
2292:
2233:
2174:
2137:
2095:
2068:
2048:
2016:
1883:
1847:
1823:
1803:
1746:
1726:
1634:
1614:
1594:
1475:
1455:
1428:
1408:
1356:
1310:
1268:
1200:
1165:
1084:
1054:
1034:
1005:
985:
965:
905:
850:
830:
746:
726:
648:
598:
589:
A Spin structure determines (and is determined by) a
551:
500:
447:
368:
341:
294:
271:
262:
Given a compact oriented 4 manifold, choose a smooth
226:
130:
3185:, so this shows that for any solution, the sup norm
4757:, Mathematical Notes, vol. 44, Princeton, NJ:
2503:identified with an imaginary self-dual 2-form, and
4482:
4432:
4412:
4363:
4330:
4310:
4277:
4257:
4220:
4200:
4180:
4146:
4046:
3987:
3834:
3814:
3794:
3774:
3754:
3734:
3703:
3683:
3626:
3606:
3560:
3534:
3508:
3406:
3364:
3332:
3286:
3266:
3234:
3210:
3177:
3131:
3090:
2896:
2743:
2601:
2575:
2539:
2515:
2495:
2468:
2441:
2337:
2305:
2278:
2216:
2159:
2120:
2081:
2054:
2031:
2002:
1869:
1833:
1809:
1789:
1732:
1712:
1620:
1600:
1580:
1461:
1441:
1414:
1394:
1342:
1296:
1250:
1178:
1151:
1070:
1040:
1020:
991:
971:
951:
891:
836:
816:
732:
712:
630:
579:
533:
486:
433:
347:
307:
277:
247:
209:
71:
2345:is its self-dual part, and σ is the squaring map
2279:{\displaystyle F^{A}\in iA_{\mathbb {R} }^{2}(M)}
2217:{\displaystyle F_{A}^{+}=\sigma (\phi )+i\omega }
5173:
4498:The Seiberg–Witten invariant of a four-manifold
3375:
2907:to solutions of the equations gives an equality
1229:
1207:
4338:-admissible two forms is connected, whereas if
1251:{\displaystyle L=\det(W^{+})\equiv \det(W^{-})}
892:{\displaystyle \gamma (a):W^{\pm }\to W^{\mp }}
5018:
4918:
4652:
4574:has a metric of positive scalar curvature and
3842:, and the harmonic part, or equivalently, the
1790:{\displaystyle {\mathcal {G}}=\{u:M\to U(1)\}}
56:
52:
4704:Bulletin of the American Mathematical Society
4625:) ≥ 2 then it has a spin structure
2476:to the a traceless Hermitian endomorphism of
1395:{\displaystyle A\in iA_{\mathbb {R} }^{1}(M)}
75:
4493:
3199:
3192:
1784:
1757:
3684:{\displaystyle F_{0}+dA=i(\alpha +\omega )}
3340:of the Seiberg–Witten equations are called
1841:can be "gauge fixed" e.g. by the condition
1189:
713:{\displaystyle K=c_{1}(W^{+})=c_{1}(W^{-})}
487:{\displaystyle K\in H^{2}(M,\mathbb {Z} ).}
335:, every smooth oriented compact 4-manifold
4844:
4483:{\displaystyle K\in H^{2}(M,\mathbb {Z} )}
2616:An application of the Weitzenböck formula
1048:. It gives an induced action of the forms
248:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
94:
27:are invariants of compact smooth oriented
5126:
5039:
4941:
4792:
4716:
4697:
4675:
4473:
4137:
3978:
3607:{\displaystyle \nabla _{A}=\nabla _{0}+A}
2256:
1980:
1954:
1904:
1372:
1343:{\displaystyle \nabla _{A}=\nabla _{0}+A}
521:
474:
424:
411:
241:
228:
197:
184:
82:
66:Seiberg–Witten invariants are similar to
3178:{\displaystyle \Delta |\phi |^{2}\geq 0}
4887:
4732:
4614:is simply connected and symplectic and
3568:, and so are determined by connections
110:
98:
5174:
5108:
5079:
4747:
1422:, there is a unique spinor connection
952:{\displaystyle \gamma (a)^{2}=-g(a,a)}
534:{\displaystyle H^{2}(M,\mathbb {Z} ).}
106:
90:
36:
4785:Lectures on Seiberg-Witten invariants
4782:
959:. There is a unique hermitian metric
545:proper requires the more restrictive
86:
4826:
4413:{\displaystyle F^{\mathrm {harm} }}
4378:bound on the solutions, also gives
3333:{\displaystyle (\phi ,\nabla ^{A})}
3211:{\displaystyle \|\phi \|_{\infty }}
2121:{\displaystyle (\phi ,\nabla ^{A})}
2089:. The Seiberg–Witten equations for
1028:is skew Hermitian for real 1 forms
631:{\displaystyle W=W^{+}\oplus W^{-}}
121:The Spin group (in dimension 4) is
116:
13:
5083:(2000), Wentworth, Richard (ed.),
4404:
4401:
4398:
4395:
4181:{\displaystyle {\mathcal {H}}^{-}}
4167:
4103:
4079:
4076:
4073:
4070:
3946:
3943:
3940:
3937:
3922:
3919:
3916:
3913:
3892:
3889:
3886:
3883:
3589:
3576:
3459:
3456:
3453:
3318:
3203:
3146:
2949:
2917:
2864:
2831:
2814:
2765:
2646:
2629:
2532:
2294:
2286:is the closed curvature 2-form of
2106:
1923:
1920:
1917:
1914:
1826:
1749:
1696:
1661:
1558:
1516:
1477:
1430:
1325:
1312:
1111:
1103:
797:
790:
766:
763:
760:
757:
754:
296:
159:
156:
153:
150:
14:
5198:
4420:. There are therefore (for fixed
4228:-admissible if this condition is
2613:of sufficiently high regularity.
59:) during their investigations of
5111:"Monopoles and four-manifolds."
4856:Graduate Studies in Mathematics
3542:, the reducible solutions have
19:In mathematics, and especially
5187:Partial differential equations
4848:Notes on Seiberg-Witten theory
4646:
4477:
4463:
4141:
4127:
4035:
4002:
3982:
3968:
3952:
3904:
3866:
3853:
3691:for some anti selfdual 2-form
3678:
3666:
3495:
3492:
3486:
3471:
3465:
3428:
3327:
3308:
3261:
3249:
3159:
3150:
3119:
3110:
3085:
3079:
3073:
3061:
3033:
3024:
3020:
3011:
2998:
2989:
2963:
2944:
2930:
2921:
2878:
2859:
2849:
2808:
2784:
2775:
2735:
2726:
2708:
2690:
2675:
2661:
2540:{\displaystyle {\mathcal {G}}}
2414:
2402:
2381:
2369:
2355:
2273:
2267:
2202:
2196:
2115:
2096:
2026:
2020:
1997:
1991:
1958:
1944:
1909:
1894:
1834:{\displaystyle {\mathcal {G}}}
1781:
1775:
1769:
1692:
1676:
1575:
1554:
1545:
1542:
1536:
1512:
1506:
1503:
1497:
1491:
1389:
1383:
1285:
1279:
1245:
1232:
1223:
1210:
1146:
1133:
1015:
1009:
946:
934:
916:
909:
876:
860:
854:
811:
805:
785:
782:
770:
707:
694:
678:
665:
568:
562:
525:
511:
478:
464:
428:
401:
385:
379:
201:
180:
172:
169:
163:
143:
137:
131:
109:). For the early history see (
72:the moduli spaces of solutions
1:
5115:Mathematical Research Letters
4894:American Mathematical Society
4890:The wild world of 4-manifolds
4845:Nicolaescu, Liviu I. (2000),
4718:10.1090/S0273-0979-96-00625-8
4639:
3846:of the curvature form i.e.
3376:The moduli space of solutions
3344:, as these equations are the
1469:i.e. a connection such that
5058:10.1016/0550-3213(94)90214-3
5005:10.1016/0550-3213(94)00449-8
4960:10.1016/0550-3213(94)90124-4
4783:Moore, John Douglas (2001),
2160:{\displaystyle D^{A}\phi =0}
1071:{\displaystyle \wedge ^{*}M}
7:
4888:Scorpan, Alexandru (2005),
4834:Encyclopedia of Mathematics
4736:A revolution in mathematics
4311:{\displaystyle b_{+}\geq 2}
4258:{\displaystyle b^{+}\geq 1}
3416:Atiyah–Singer index theorem
3132:{\displaystyle |\phi |^{2}}
2306:{\displaystyle \nabla ^{A}}
1442:{\displaystyle \nabla ^{A}}
1186:which are then identified.
580:{\displaystyle w_{2}(M)=0.}
308:{\displaystyle \nabla ^{g}}
61:Seiberg–Witten gauge theory
10:
5203:
5145:10.4310/MRL.1994.v1.n6.a13
4829:"Seiberg-Witten equations"
4759:Princeton University Press
3844:(de Rham) cohomology class
3802:, is the harmonic part of
1297:{\displaystyle c_{1}(L)=K}
1021:{\displaystyle \gamma (a)}
824:such that for each 1 form
4677:21.11116/0000-0004-3A18-1
4494:Seiberg–Witten invariants
2338:{\displaystyle F_{A}^{+}}
25:Seiberg–Witten invariants
4556:is simply connected and
2576:{\displaystyle d^{*}A=0}
1870:{\displaystyle d^{*}A=0}
1304:. For every connection
1190:Seiberg–Witten equations
103:Gromov–Witten invariants
76:Seiberg–Witten equations
5109:Witten, Edward (1994),
4733:Jackson, Allyn (1995),
4433:{\displaystyle \omega }
4364:{\displaystyle b_{+}=1}
4278:{\displaystyle \omega }
4201:{\displaystyle \omega }
3835:{\displaystyle \omega }
3815:{\displaystyle \alpha }
3795:{\displaystyle \omega }
3775:{\displaystyle \alpha }
3704:{\displaystyle \alpha }
3561:{\displaystyle \phi =0}
3535:{\displaystyle \omega }
3522:For a self dual 2 form
3407:{\displaystyle \phi =0}
3287:{\displaystyle \omega }
3274:and the self dual form
2602:{\displaystyle \phi =0}
2516:{\displaystyle \omega }
1260:determinant line bundle
357:the existence of a lift
5081:Taubes, Clifford Henry
4550:simple type conjecture
4484:
4434:
4414:
4365:
4332:
4312:
4279:
4259:
4222:
4202:
4182:
4148:
4048:
3989:
3836:
3816:
3796:
3776:
3756:
3736:
3705:
3685:
3628:
3608:
3562:
3536:
3510:
3408:
3366:
3334:
3288:
3268:
3236:
3212:
3179:
3133:
3092:
2898:
2745:
2603:
2577:
2541:
2517:
2497:
2470:
2443:
2339:
2307:
2280:
2218:
2161:
2122:
2083:
2056:
2033:
2004:
1871:
1835:
1811:
1791:
1734:
1714:
1622:
1602:
1582:
1463:
1443:
1416:
1396:
1344:
1298:
1252:
1180:
1153:
1072:
1042:
1022:
993:
973:
953:
893:
838:
818:
734:
714:
632:
581:
535:
488:
435:
349:
321:complex spin structure
309:
287:Levi Civita connection
279:
249:
211:
4761:, pp. viii+128,
4485:
4440:) only finitely many
4435:
4415:
4366:
4333:
4313:
4280:
4260:
4223:
4203:
4183:
4149:
4049:
3990:
3837:
3817:
3797:
3777:
3757:
3737:
3735:{\displaystyle F_{0}}
3706:
3686:
3629:
3609:
3563:
3537:
3511:
3409:
3367:
3335:
3289:
3269:
3267:{\displaystyle (M,g)}
3237:
3213:
3180:
3134:
3093:
2899:
2746:
2604:
2578:
2542:
2518:
2498:
2496:{\displaystyle W^{+}}
2471:
2469:{\displaystyle W^{+}}
2444:
2340:
2308:
2281:
2219:
2162:
2123:
2084:
2082:{\displaystyle W^{+}}
2057:
2055:{\displaystyle \phi }
2039:gauge group action.
2034:
2005:
1872:
1836:
1812:
1792:
1735:
1715:
1623:
1603:
1583:
1464:
1444:
1417:
1397:
1345:
1299:
1253:
1181:
1179:{\displaystyle W^{+}}
1154:
1073:
1043:
1023:
994:
974:
954:
894:
839:
819:
735:
720:. The spinor bundle
715:
633:
582:
536:
489:
436:
361:Stiefel–Whitney class
350:
310:
280:
250:
212:
41:Seiberg–Witten theory
16:4-manifold invariants
4991:(2): 485–486, 1994,
4444:
4424:
4386:
4342:
4322:
4289:
4269:
4236:
4212:
4192:
4161:
4061:
3999:
3850:
3826:
3806:
3786:
3766:
3746:
3719:
3695:
3638:
3618:
3572:
3546:
3526:
3425:
3392:
3356:
3305:
3278:
3246:
3226:
3189:
3143:
3106:
2914:
2761:
2623:
2587:
2551:
2527:
2507:
2480:
2453:
2349:
2317:
2290:
2231:
2172:
2135:
2093:
2066:
2046:
2032:{\displaystyle U(1)}
2014:
1881:
1845:
1821:
1801:
1744:
1740:. The group of maps
1724:
1632:
1612:
1592:
1473:
1453:
1426:
1406:
1354:
1308:
1266:
1198:
1163:
1082:
1052:
1032:
1003:
983:
963:
903:
848:
828:
744:
724:
646:
596:
549:
498:
445:
366:
339:
292:
269:
224:
128:
68:Donaldson invariants
5137:1994MRLet...1..769W
5050:1994NuPhB.431..484S
4997:1994NuPhB.430..485.
4952:1994NuPhB.426...19S
4827:Nash, Ch. (2001) ,
4699:Donaldson, Simon K.
3897:
3713:Hodge decomposition
2798:
2725:
2334:
2266:
2189:
1990:
1709:
1571:
1529:
1490:
1382:
1132:
4803:10.1007/BFb0092948
4668:10.1007/BF01362296
4480:
4430:
4410:
4361:
4328:
4308:
4275:
4255:
4218:
4198:
4178:
4144:
4047:{\displaystyle =K}
4044:
4023:
3995:. Thus, since the
3985:
3872:
3832:
3812:
3792:
3772:
3752:
3732:
3701:
3681:
3624:
3604:
3558:
3532:
3506:
3404:
3362:
3350:magnetic monopoles
3330:
3284:
3264:
3232:
3208:
3175:
3129:
3088:
3056:
2986:
2894:
2782:
2741:
2711:
2703:
2599:
2573:
2537:
2513:
2493:
2466:
2439:
2397:
2335:
2320:
2303:
2276:
2250:
2214:
2175:
2157:
2118:
2079:
2052:
2029:
2000:
1974:
1867:
1831:
1807:
1787:
1730:
1710:
1695:
1618:
1598:
1578:
1557:
1515:
1476:
1459:
1439:
1412:
1392:
1366:
1340:
1294:
1248:
1176:
1149:
1108:
1068:
1038:
1018:
989:
969:
949:
889:
834:
814:
730:
710:
628:
577:
531:
484:
431:
345:
305:
275:
257:SO(4) = Spin(4)/±1
245:
207:
45:Nathan Seiberg
5094:978-1-57146-061-5
5028:Nuclear Physics B
4984:Nuclear Physics B
4929:Nuclear Physics B
4903:978-0-8218-3749-8
4873:978-0-8218-2145-9
4812:978-3-540-41221-2
4768:978-0-691-02597-1
4743:on April 26, 2010
4544:is said to be of
4331:{\displaystyle K}
4221:{\displaystyle K}
4022:
3755:{\displaystyle A}
3627:{\displaystyle L}
3365:{\displaystyle M}
3235:{\displaystyle s}
3055:
2985:
2754:and the identity
2702:
2396:
1810:{\displaystyle L}
1733:{\displaystyle W}
1621:{\displaystyle X}
1608:and vector field
1601:{\displaystyle a}
1588:for every 1-form
1462:{\displaystyle W}
1415:{\displaystyle L}
1041:{\displaystyle a}
992:{\displaystyle W}
972:{\displaystyle h}
837:{\displaystyle a}
733:{\displaystyle W}
348:{\displaystyle M}
278:{\displaystyle g}
264:Riemannian metric
33:Edward Witten
5194:
5168:
5163:, archived from
5130:
5105:
5076:
5043:
5015:
4978:
4945:
4914:
4884:
4853:
4841:
4823:
4796:
4779:
4744:
4739:, archived from
4729:
4720:
4690:
4689:
4679:
4650:
4636: ≥ 1.
4610:If the manifold
4592:If the manifold
4570:If the manifold
4489:
4487:
4486:
4481:
4476:
4462:
4461:
4439:
4437:
4436:
4431:
4419:
4417:
4416:
4411:
4409:
4408:
4407:
4370:
4368:
4367:
4362:
4354:
4353:
4337:
4335:
4334:
4329:
4317:
4315:
4314:
4309:
4301:
4300:
4285:. Note that, if
4284:
4282:
4281:
4276:
4264:
4262:
4261:
4256:
4248:
4247:
4227:
4225:
4224:
4219:
4207:
4205:
4204:
4199:
4187:
4185:
4184:
4179:
4177:
4176:
4171:
4170:
4153:
4151:
4150:
4145:
4140:
4126:
4125:
4113:
4112:
4107:
4106:
4084:
4083:
4082:
4053:
4051:
4050:
4045:
4034:
4033:
4024:
4021:
4007:
3994:
3992:
3991:
3986:
3981:
3967:
3966:
3951:
3950:
3949:
3927:
3926:
3925:
3896:
3895:
3880:
3865:
3864:
3841:
3839:
3838:
3833:
3821:
3819:
3818:
3813:
3801:
3799:
3798:
3793:
3781:
3779:
3778:
3773:
3761:
3759:
3758:
3753:
3741:
3739:
3738:
3733:
3731:
3730:
3710:
3708:
3707:
3702:
3690:
3688:
3687:
3682:
3650:
3649:
3633:
3631:
3630:
3625:
3613:
3611:
3610:
3605:
3597:
3596:
3584:
3583:
3567:
3565:
3564:
3559:
3541:
3539:
3538:
3533:
3515:
3513:
3512:
3507:
3502:
3464:
3463:
3462:
3440:
3439:
3413:
3411:
3410:
3405:
3371:
3369:
3368:
3363:
3352:on the manifold
3339:
3337:
3336:
3331:
3326:
3325:
3293:
3291:
3290:
3285:
3273:
3271:
3270:
3265:
3241:
3239:
3238:
3233:
3217:
3215:
3214:
3209:
3207:
3206:
3184:
3182:
3181:
3176:
3168:
3167:
3162:
3153:
3138:
3136:
3135:
3130:
3128:
3127:
3122:
3113:
3097:
3095:
3094:
3089:
3057:
3048:
3042:
3041:
3036:
3027:
3007:
3006:
3001:
2992:
2987:
2978:
2972:
2971:
2966:
2957:
2956:
2947:
2939:
2938:
2933:
2924:
2903:
2901:
2900:
2895:
2893:
2892:
2881:
2872:
2871:
2862:
2839:
2838:
2829:
2828:
2823:
2822:
2821:
2797:
2792:
2787:
2778:
2773:
2772:
2750:
2748:
2747:
2742:
2724:
2719:
2704:
2695:
2683:
2682:
2673:
2672:
2654:
2653:
2644:
2643:
2638:
2637:
2636:
2608:
2606:
2605:
2600:
2582:
2580:
2579:
2574:
2563:
2562:
2546:
2544:
2543:
2538:
2536:
2535:
2522:
2520:
2519:
2514:
2502:
2500:
2499:
2494:
2492:
2491:
2475:
2473:
2472:
2467:
2465:
2464:
2448:
2446:
2445:
2440:
2438:
2434:
2433:
2432:
2431:
2430:
2398:
2389:
2344:
2342:
2341:
2336:
2333:
2328:
2312:
2310:
2309:
2304:
2302:
2301:
2285:
2283:
2282:
2277:
2265:
2260:
2259:
2243:
2242:
2223:
2221:
2220:
2215:
2188:
2183:
2166:
2164:
2163:
2158:
2147:
2146:
2127:
2125:
2124:
2119:
2114:
2113:
2088:
2086:
2085:
2080:
2078:
2077:
2061:
2059:
2058:
2053:
2038:
2036:
2035:
2030:
2010:with a residual
2009:
2007:
2006:
2001:
1989:
1984:
1983:
1973:
1972:
1957:
1943:
1942:
1933:
1928:
1927:
1926:
1907:
1893:
1892:
1876:
1874:
1873:
1868:
1857:
1856:
1840:
1838:
1837:
1832:
1830:
1829:
1817:. The action of
1816:
1814:
1813:
1808:
1796:
1794:
1793:
1788:
1753:
1752:
1739:
1737:
1736:
1731:
1719:
1717:
1716:
1711:
1708:
1703:
1691:
1690:
1669:
1668:
1644:
1643:
1627:
1625:
1624:
1619:
1607:
1605:
1604:
1599:
1587:
1585:
1584:
1579:
1570:
1565:
1528:
1523:
1489:
1484:
1468:
1466:
1465:
1460:
1448:
1446:
1445:
1440:
1438:
1437:
1421:
1419:
1418:
1413:
1401:
1399:
1398:
1393:
1381:
1376:
1375:
1349:
1347:
1346:
1341:
1333:
1332:
1320:
1319:
1303:
1301:
1300:
1295:
1278:
1277:
1257:
1255:
1254:
1249:
1244:
1243:
1222:
1221:
1185:
1183:
1182:
1177:
1175:
1174:
1158:
1156:
1155:
1150:
1145:
1144:
1131:
1123:
1118:
1117:
1107:
1106:
1094:
1093:
1077:
1075:
1074:
1069:
1064:
1063:
1047:
1045:
1044:
1039:
1027:
1025:
1024:
1019:
998:
996:
995:
990:
978:
976:
975:
970:
958:
956:
955:
950:
924:
923:
898:
896:
895:
890:
888:
887:
875:
874:
843:
841:
840:
835:
823:
821:
820:
815:
804:
803:
794:
793:
769:
739:
737:
736:
731:
719:
717:
716:
711:
706:
705:
693:
692:
677:
676:
664:
663:
637:
635:
634:
629:
627:
626:
614:
613:
586:
584:
583:
578:
561:
560:
540:
538:
537:
532:
524:
510:
509:
493:
491:
490:
485:
477:
463:
462:
440:
438:
437:
432:
427:
419:
414:
400:
399:
378:
377:
354:
352:
351:
346:
314:
312:
311:
306:
304:
303:
284:
282:
281:
276:
254:
252:
251:
246:
244:
236:
231:
216:
214:
213:
208:
200:
192:
187:
179:
162:
5202:
5201:
5197:
5196:
5195:
5193:
5192:
5191:
5172:
5171:
5095:
4980:
4920:Seiberg, Nathan
4904:
4874:
4864:10.1090/gsm/028
4851:
4813:
4794:10.1.1.252.2658
4769:
4749:Morgan, John W.
4694:
4693:
4651:
4647:
4642:
4635:
4620:
4602:
4580:
4562:
4552:states that if
4532:
4508:
4496:
4472:
4457:
4453:
4445:
4442:
4441:
4425:
4422:
4421:
4394:
4393:
4389:
4387:
4384:
4383:
4349:
4345:
4343:
4340:
4339:
4323:
4320:
4319:
4296:
4292:
4290:
4287:
4286:
4270:
4267:
4266:
4243:
4239:
4237:
4234:
4233:
4213:
4210:
4209:
4193:
4190:
4189:
4172:
4166:
4165:
4164:
4162:
4159:
4158:
4136:
4121:
4117:
4108:
4102:
4101:
4100:
4069:
4068:
4064:
4062:
4059:
4058:
4029:
4025:
4011:
4005:
4000:
3997:
3996:
3977:
3962:
3958:
3936:
3935:
3931:
3912:
3911:
3907:
3882:
3881:
3876:
3860:
3856:
3851:
3848:
3847:
3827:
3824:
3823:
3807:
3804:
3803:
3787:
3784:
3783:
3767:
3764:
3763:
3747:
3744:
3743:
3726:
3722:
3720:
3717:
3716:
3696:
3693:
3692:
3645:
3641:
3639:
3636:
3635:
3619:
3616:
3615:
3592:
3588:
3579:
3575:
3573:
3570:
3569:
3547:
3544:
3543:
3527:
3524:
3523:
3498:
3452:
3451:
3447:
3435:
3431:
3426:
3423:
3422:
3393:
3390:
3389:
3378:
3357:
3354:
3353:
3346:field equations
3321:
3317:
3306:
3303:
3302:
3279:
3276:
3275:
3247:
3244:
3243:
3227:
3224:
3223:
3202:
3198:
3190:
3187:
3186:
3163:
3158:
3157:
3149:
3144:
3141:
3140:
3123:
3118:
3117:
3109:
3107:
3104:
3103:
3046:
3037:
3032:
3031:
3023:
3002:
2997:
2996:
2988:
2976:
2967:
2962:
2961:
2952:
2948:
2943:
2934:
2929:
2928:
2920:
2915:
2912:
2911:
2882:
2877:
2876:
2867:
2863:
2858:
2834:
2830:
2824:
2817:
2813:
2812:
2811:
2793:
2788:
2783:
2774:
2768:
2764:
2762:
2759:
2758:
2720:
2715:
2693:
2678:
2674:
2668:
2664:
2649:
2645:
2639:
2632:
2628:
2627:
2626:
2624:
2621:
2620:
2588:
2585:
2584:
2558:
2554:
2552:
2549:
2548:
2531:
2530:
2528:
2525:
2524:
2508:
2505:
2504:
2487:
2483:
2481:
2478:
2477:
2460:
2456:
2454:
2451:
2450:
2426:
2422:
2421:
2417:
2387:
2362:
2358:
2350:
2347:
2346:
2329:
2324:
2318:
2315:
2314:
2297:
2293:
2291:
2288:
2287:
2261:
2255:
2254:
2238:
2234:
2232:
2229:
2228:
2184:
2179:
2173:
2170:
2169:
2142:
2138:
2136:
2133:
2132:
2109:
2105:
2094:
2091:
2090:
2073:
2069:
2067:
2064:
2063:
2047:
2044:
2043:
2015:
2012:
2011:
1985:
1979:
1978:
1968:
1964:
1953:
1938:
1934:
1929:
1913:
1912:
1908:
1903:
1888:
1884:
1882:
1879:
1878:
1852:
1848:
1846:
1843:
1842:
1825:
1824:
1822:
1819:
1818:
1802:
1799:
1798:
1748:
1747:
1745:
1742:
1741:
1725:
1722:
1721:
1704:
1699:
1686:
1682:
1664:
1660:
1639:
1635:
1633:
1630:
1629:
1613:
1610:
1609:
1593:
1590:
1589:
1566:
1561:
1524:
1519:
1485:
1480:
1474:
1471:
1470:
1454:
1451:
1450:
1433:
1429:
1427:
1424:
1423:
1407:
1404:
1403:
1377:
1371:
1370:
1355:
1352:
1351:
1328:
1324:
1315:
1311:
1309:
1306:
1305:
1273:
1269:
1267:
1264:
1263:
1239:
1235:
1217:
1213:
1199:
1196:
1195:
1192:
1170:
1166:
1164:
1161:
1160:
1140:
1136:
1124:
1119:
1110:
1109:
1102:
1101:
1089:
1085:
1083:
1080:
1079:
1059:
1055:
1053:
1050:
1049:
1033:
1030:
1029:
1004:
1001:
1000:
984:
981:
980:
964:
961:
960:
919:
915:
904:
901:
900:
883:
879:
870:
866:
849:
846:
845:
829:
826:
825:
796:
795:
789:
788:
753:
745:
742:
741:
725:
722:
721:
701:
697:
688:
684:
672:
668:
659:
655:
647:
644:
643:
622:
618:
609:
605:
597:
594:
593:
556:
552:
550:
547:
546:
520:
505:
501:
499:
496:
495:
473:
458:
454:
446:
443:
442:
423:
415:
410:
395:
391:
373:
369:
367:
364:
363:
340:
337:
336:
318:
299:
295:
293:
290:
289:
270:
267:
266:
240:
232:
227:
225:
222:
221:
196:
188:
183:
175:
149:
129:
126:
125:
119:
117:Spin-structures
95:Nicolaescu 2000
17:
12:
11:
5:
5200:
5190:
5189:
5184:
5170:
5169:
5128:hep-th/9411102
5121:(6): 769–796,
5106:
5093:
5077:
5041:hep-th/9408099
5034:(3): 484–550,
5016:
4943:hep-th/9407087
4924:Witten, Edward
4916:
4902:
4885:
4872:
4842:
4824:
4811:
4780:
4767:
4745:
4730:
4692:
4691:
4662:(2): 156–172.
4644:
4643:
4641:
4638:
4633:
4618:
4600:
4578:
4560:
4530:
4506:
4495:
4492:
4479:
4475:
4471:
4468:
4465:
4460:
4456:
4452:
4449:
4429:
4406:
4403:
4400:
4397:
4392:
4360:
4357:
4352:
4348:
4327:
4307:
4304:
4299:
4295:
4274:
4254:
4251:
4246:
4242:
4217:
4197:
4175:
4169:
4155:
4154:
4143:
4139:
4135:
4132:
4129:
4124:
4120:
4116:
4111:
4105:
4099:
4096:
4093:
4090:
4087:
4081:
4078:
4075:
4072:
4067:
4043:
4040:
4037:
4032:
4028:
4020:
4017:
4014:
4010:
4004:
3984:
3980:
3976:
3973:
3970:
3965:
3961:
3957:
3954:
3948:
3945:
3942:
3939:
3934:
3930:
3924:
3921:
3918:
3915:
3910:
3906:
3903:
3900:
3894:
3891:
3888:
3885:
3879:
3875:
3871:
3868:
3863:
3859:
3855:
3831:
3811:
3791:
3771:
3751:
3729:
3725:
3700:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3648:
3644:
3623:
3603:
3600:
3595:
3591:
3587:
3582:
3578:
3557:
3554:
3551:
3531:
3517:
3516:
3505:
3501:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3461:
3458:
3455:
3450:
3446:
3443:
3438:
3434:
3430:
3403:
3400:
3397:
3384:of monopoles.
3377:
3374:
3361:
3329:
3324:
3320:
3316:
3313:
3310:
3301:The solutions
3283:
3263:
3260:
3257:
3254:
3251:
3231:
3205:
3201:
3197:
3194:
3174:
3171:
3166:
3161:
3156:
3152:
3148:
3126:
3121:
3116:
3112:
3100:
3099:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3054:
3051:
3045:
3040:
3035:
3030:
3026:
3022:
3019:
3016:
3013:
3010:
3005:
3000:
2995:
2991:
2984:
2981:
2975:
2970:
2965:
2960:
2955:
2951:
2946:
2942:
2937:
2932:
2927:
2923:
2919:
2905:
2904:
2891:
2888:
2885:
2880:
2875:
2870:
2866:
2861:
2857:
2854:
2851:
2848:
2845:
2842:
2837:
2833:
2827:
2820:
2816:
2810:
2807:
2804:
2801:
2796:
2791:
2786:
2781:
2777:
2771:
2767:
2752:
2751:
2740:
2737:
2734:
2731:
2728:
2723:
2718:
2714:
2710:
2707:
2701:
2698:
2692:
2689:
2686:
2681:
2677:
2671:
2667:
2663:
2660:
2657:
2652:
2648:
2642:
2635:
2631:
2611:Sobolev spaces
2598:
2595:
2592:
2572:
2569:
2566:
2561:
2557:
2534:
2512:
2490:
2486:
2463:
2459:
2437:
2429:
2425:
2420:
2416:
2413:
2410:
2407:
2404:
2401:
2395:
2392:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2361:
2357:
2354:
2332:
2327:
2323:
2300:
2296:
2275:
2272:
2269:
2264:
2258:
2253:
2249:
2246:
2241:
2237:
2225:
2224:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2187:
2182:
2178:
2167:
2156:
2153:
2150:
2145:
2141:
2117:
2112:
2108:
2104:
2101:
2098:
2076:
2072:
2051:
2028:
2025:
2022:
2019:
1999:
1996:
1993:
1988:
1982:
1977:
1971:
1967:
1963:
1960:
1956:
1952:
1949:
1946:
1941:
1937:
1932:
1925:
1922:
1919:
1916:
1911:
1906:
1902:
1899:
1896:
1891:
1887:
1866:
1863:
1860:
1855:
1851:
1828:
1806:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1751:
1729:
1707:
1702:
1698:
1694:
1689:
1685:
1681:
1678:
1675:
1672:
1667:
1663:
1659:
1656:
1653:
1650:
1647:
1642:
1638:
1617:
1597:
1577:
1574:
1569:
1564:
1560:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1527:
1522:
1518:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1488:
1483:
1479:
1458:
1436:
1432:
1411:
1391:
1388:
1385:
1380:
1374:
1369:
1365:
1362:
1359:
1339:
1336:
1331:
1327:
1323:
1318:
1314:
1293:
1290:
1287:
1284:
1281:
1276:
1272:
1247:
1242:
1238:
1234:
1231:
1228:
1225:
1220:
1216:
1212:
1209:
1206:
1203:
1191:
1188:
1173:
1169:
1148:
1143:
1139:
1135:
1130:
1127:
1122:
1116:
1113:
1105:
1100:
1097:
1092:
1088:
1067:
1062:
1058:
1037:
1017:
1014:
1011:
1008:
988:
968:
948:
945:
942:
939:
936:
933:
930:
927:
922:
918:
914:
911:
908:
886:
882:
878:
873:
869:
865:
862:
859:
856:
853:
833:
813:
810:
807:
802:
799:
792:
787:
784:
781:
778:
775:
772:
768:
765:
762:
759:
756:
752:
749:
729:
709:
704:
700:
696:
691:
687:
683:
680:
675:
671:
667:
662:
658:
654:
651:
625:
621:
617:
612:
608:
604:
601:
576:
573:
570:
567:
564:
559:
555:
543:spin structure
530:
527:
523:
519:
516:
513:
508:
504:
483:
480:
476:
472:
469:
466:
461:
457:
453:
450:
430:
426:
422:
418:
413:
409:
406:
403:
398:
394:
390:
387:
384:
381:
376:
372:
359:of the second
344:
316:
302:
298:
274:
243:
239:
235:
230:
218:
217:
206:
203:
199:
195:
191:
186:
182:
178:
174:
171:
168:
165:
161:
158:
155:
152:
148:
145:
142:
139:
136:
133:
118:
115:
83:Donaldson 1996
31:introduced by
15:
9:
6:
4:
3:
2:
5199:
5188:
5185:
5183:
5180:
5179:
5177:
5167:on 2013-06-29
5166:
5162:
5158:
5154:
5150:
5146:
5142:
5138:
5134:
5129:
5124:
5120:
5116:
5112:
5107:
5104:
5100:
5096:
5090:
5086:
5082:
5078:
5075:
5071:
5067:
5063:
5059:
5055:
5051:
5047:
5042:
5037:
5033:
5029:
5025:
5021:
5017:
5014:
5010:
5006:
5002:
4998:
4994:
4990:
4986:
4985:
4977:
4973:
4969:
4965:
4961:
4957:
4953:
4949:
4944:
4939:
4935:
4931:
4930:
4925:
4921:
4917:
4913:
4909:
4905:
4899:
4895:
4891:
4886:
4883:
4879:
4875:
4869:
4865:
4861:
4857:
4850:
4849:
4843:
4840:
4836:
4835:
4830:
4825:
4822:
4818:
4814:
4808:
4804:
4800:
4795:
4790:
4786:
4781:
4778:
4774:
4770:
4764:
4760:
4756:
4755:
4750:
4746:
4742:
4738:
4737:
4731:
4728:
4724:
4719:
4714:
4710:
4706:
4705:
4700:
4696:
4695:
4687:
4683:
4678:
4673:
4669:
4665:
4661:
4658:
4657:
4649:
4645:
4637:
4632:
4628:
4624:
4617:
4613:
4608:
4606:
4599:
4595:
4590:
4588:
4584:
4577:
4573:
4568:
4566:
4559:
4555:
4551:
4547:
4543:
4538:
4536:
4529:
4524:
4522:
4521:
4516:
4512:
4505:
4501:
4491:
4469:
4466:
4458:
4454:
4450:
4447:
4427:
4390:
4381:
4377:
4372:
4358:
4355:
4350:
4346:
4325:
4318:the space of
4305:
4302:
4297:
4293:
4272:
4252:
4249:
4244:
4240:
4231:
4215:
4195:
4173:
4133:
4130:
4122:
4118:
4114:
4109:
4097:
4094:
4091:
4088:
4085:
4065:
4057:
4056:
4055:
4041:
4038:
4030:
4026:
4018:
4015:
4012:
4008:
3974:
3971:
3963:
3959:
3955:
3932:
3928:
3908:
3901:
3898:
3877:
3873:
3869:
3861:
3857:
3845:
3829:
3809:
3789:
3769:
3749:
3727:
3723:
3714:
3698:
3675:
3672:
3669:
3663:
3660:
3657:
3654:
3651:
3646:
3642:
3621:
3601:
3598:
3593:
3585:
3580:
3555:
3552:
3549:
3529:
3520:
3503:
3499:
3489:
3483:
3480:
3477:
3474:
3468:
3448:
3444:
3441:
3436:
3432:
3421:
3420:
3419:
3417:
3401:
3398:
3395:
3385:
3383:
3373:
3359:
3351:
3347:
3343:
3322:
3314:
3311:
3299:
3297:
3281:
3258:
3255:
3252:
3229:
3221:
3195:
3172:
3169:
3164:
3154:
3124:
3114:
3082:
3076:
3070:
3067:
3064:
3058:
3052:
3049:
3043:
3038:
3028:
3017:
3014:
3008:
3003:
2993:
2982:
2979:
2973:
2968:
2958:
2953:
2940:
2935:
2925:
2910:
2909:
2908:
2889:
2886:
2883:
2873:
2868:
2855:
2852:
2846:
2843:
2840:
2835:
2825:
2818:
2805:
2802:
2799:
2794:
2789:
2779:
2769:
2757:
2756:
2755:
2738:
2732:
2729:
2721:
2716:
2712:
2705:
2699:
2696:
2687:
2684:
2679:
2669:
2665:
2658:
2655:
2650:
2640:
2633:
2619:
2618:
2617:
2614:
2612:
2596:
2593:
2590:
2570:
2567:
2564:
2559:
2555:
2510:
2488:
2484:
2461:
2457:
2435:
2427:
2423:
2418:
2411:
2408:
2405:
2399:
2393:
2390:
2384:
2378:
2375:
2372:
2366:
2363:
2359:
2352:
2330:
2325:
2321:
2298:
2270:
2262:
2251:
2247:
2244:
2239:
2235:
2211:
2208:
2205:
2199:
2193:
2190:
2185:
2180:
2176:
2168:
2154:
2151:
2148:
2143:
2139:
2131:
2130:
2129:
2110:
2102:
2099:
2074:
2070:
2049:
2040:
2023:
2017:
1994:
1986:
1975:
1969:
1965:
1961:
1950:
1947:
1939:
1935:
1930:
1900:
1897:
1889:
1885:
1864:
1861:
1858:
1853:
1849:
1804:
1778:
1772:
1766:
1763:
1760:
1754:
1727:
1705:
1700:
1687:
1683:
1679:
1673:
1670:
1665:
1657:
1654:
1651:
1648:
1645:
1640:
1636:
1615:
1595:
1572:
1567:
1562:
1551:
1548:
1539:
1533:
1530:
1525:
1520:
1509:
1500:
1494:
1486:
1481:
1456:
1434:
1409:
1386:
1378:
1367:
1363:
1360:
1357:
1337:
1334:
1329:
1321:
1316:
1291:
1288:
1282:
1274:
1270:
1261:
1240:
1236:
1226:
1218:
1214:
1204:
1201:
1187:
1171:
1167:
1141:
1137:
1128:
1125:
1120:
1098:
1095:
1090:
1086:
1065:
1060:
1056:
1035:
1012:
1006:
986:
966:
943:
940:
937:
931:
928:
925:
920:
912:
906:
884:
880:
871:
867:
863:
857:
851:
831:
808:
779:
776:
773:
750:
747:
727:
702:
698:
689:
685:
681:
673:
669:
660:
656:
652:
649:
641:
623:
619:
615:
610:
606:
602:
599:
592:
591:spinor bundle
587:
574:
571:
565:
557:
553:
544:
528:
517:
514:
506:
502:
481:
470:
467:
459:
455:
451:
448:
420:
416:
407:
404:
396:
392:
388:
382:
374:
370:
362:
358:
342:
334:
330:
326:
322:
300:
288:
272:
265:
260:
258:
237:
233:
204:
193:
189:
176:
166:
146:
140:
134:
124:
123:
122:
114:
112:
108:
104:
100:
96:
92:
88:
84:
79:
77:
73:
69:
64:
62:
58:
54:
50:
47: and
46:
42:
39:), using the
38:
34:
30:
26:
22:
5165:the original
5118:
5114:
5084:
5031:
5027:
4988:
4982:
4936:(1): 19–52,
4933:
4927:
4889:
4847:
4832:
4784:
4753:
4741:the original
4735:
4711:(1): 45–70,
4708:
4702:
4659:
4654:
4648:
4630:
4626:
4622:
4615:
4611:
4609:
4604:
4597:
4593:
4591:
4586:
4582:
4575:
4571:
4569:
4564:
4557:
4553:
4549:
4545:
4541:
4539:
4534:
4527:
4525:
4519:
4518:
4514:
4510:
4503:
4499:
4497:
4379:
4375:
4373:
4229:
4156:
3521:
3518:
3386:
3382:moduli space
3381:
3379:
3348:of massless
3341:
3300:
3295:
3219:
3101:
2906:
2753:
2615:
2226:
2041:
1193:
588:
324:
261:
219:
120:
111:Jackson 1995
99:Scorpan 2005
80:
65:
24:
21:gauge theory
18:
5182:4-manifolds
5020:Seiberg, N.
4981:"Erratum",
4546:simple type
4540:A manifold
3139:is maximal
441:to a class
107:Taubes 2000
91:Morgan 1996
43:studied by
29:4-manifolds
5176:Categories
5024:Witten, E.
4707:, (N.S.),
4656:Math. Ann.
4640:References
4382:bounds on
3634:such that
329:Hirzebruch
220:where the
87:Moore 2001
4839:EMS Press
4789:CiteSeerX
4686:120557396
4451:∈
4428:ω
4303:≥
4273:ω
4250:≥
4196:ω
4174:−
4115:∈
4110:−
4092:π
4086:∈
4066:ω
4016:π
3956:∈
3933:α
3909:ω
3830:ω
3810:α
3790:ω
3770:α
3711:. By the
3699:α
3676:ω
3670:α
3590:∇
3577:∇
3550:ϕ
3530:ω
3484:
3475:−
3449:χ
3442:−
3414:. By the
3396:ϕ
3342:monopoles
3319:∇
3312:ϕ
3282:ω
3204:∞
3200:‖
3196:ϕ
3193:‖
3170:≥
3155:ϕ
3147:Δ
3115:ϕ
3083:ϕ
3077:ω
3071:γ
3065:ϕ
3044:−
3029:ϕ
3015:−
2994:ϕ
2959:ϕ
2950:∇
2926:ϕ
2918:Δ
2887:⊗
2874:ϕ
2865:∇
2853:−
2847:ϕ
2841:ϕ
2832:∇
2826:∗
2815:∇
2780:ϕ
2766:Δ
2739:ϕ
2706:γ
2688:−
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2656:ϕ
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2641:∗
2630:∇
2591:ϕ
2560:∗
2511:ω
2412:ϕ
2406:ϕ
2385:−
2379:−
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2364:ϕ
2356:↦
2353:ϕ
2295:∇
2245:∈
2212:ω
2200:ϕ
2194:σ
2149:ϕ
2107:∇
2100:ϕ
2050:ϕ
1970:∗
1962:⊕
1854:∗
1770:→
1701:μ
1697:∇
1688:μ
1674:γ
1662:∇
1658:∘
1652:⊗
1649:γ
1559:∇
1552:γ
1534:γ
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1495:γ
1478:∇
1431:∇
1361:∈
1326:∇
1313:∇
1241:−
1227:≡
1099:≅
1087:∧
1061:∗
1057:∧
1007:γ
929:−
907:γ
885:∓
877:→
872:±
852:γ
786:→
748:γ
703:−
624:−
616:⊕
452:∈
389:∈
297:∇
147:×
5161:10611124
5074:17584951
4976:14361074
4751:(1996),
4607:vanish.
4589:vanish.
4380:a priori
4376:a priori
3715:, since
3296:a priori
3220:a priori
2128:are now
844:we have
5153:1306021
5133:Bibcode
5103:1798809
5066:1306869
5046:Bibcode
5013:1303306
4993:Bibcode
4968:1293681
4948:Bibcode
4912:2136212
4882:1787219
4821:1830497
4777:1367507
4727:1339810
1258:be the
74:of the
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4775:
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4725:
4684:
4157:where
3762:given
2042:Write
640:spinor
49:Witten
5157:S2CID
5123:arXiv
5070:S2CID
5036:arXiv
4972:S2CID
4938:arXiv
4852:(PDF)
4682:S2CID
4502:with
2449:from
2227:Here
1350:with
1262:with
999:s.t.
285:with
105:see (
57:1994b
53:1994a
5089:ISBN
4898:ISBN
4868:ISBN
4807:ISBN
4763:ISBN
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1194:Let
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333:Hopf
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5141:doi
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4672:hdl
4664:doi
4660:136
4517:to
4230:not
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1449:on
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323:on
113:).
5178::
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