3313:
42:
51:
2733:
3308:{\displaystyle {\begin{aligned}{\frac {d}{dt}}\left(\operatorname {YM} (A+ta)\right)_{t=0}&={\frac {d}{dt}}\left(\int _{X}\langle F_{A}+t\,d_{A}a+t^{2}a\wedge a,F_{A}+t\,d_{A}a+t^{2}a\wedge a\rangle \,d\mathrm {vol} _{g}\right)_{t=0}\\&={\frac {d}{dt}}\left(\int _{X}\|F_{A}\|^{2}+2t\langle F_{A},d_{A}a\rangle +2t^{2}\langle F_{A},a\wedge a\rangle +t^{4}\|a\wedge a\|^{2}\,d\mathrm {vol} _{g}\right)_{t=0}\\&=2\int _{X}\langle d_{A}^{*}F_{A},a\rangle \,d\mathrm {vol} _{g}.\end{aligned}}}
62:
71:
3644:
5558:
The moduli space of ASD instantons may be used to define further invariants of four-manifolds. Donaldson defined polynomials on the second homology group of a suitably restricted class of four-manifolds, arising from pairings of cohomology classes on the moduli space. This work has subsequently been
1025:
of this functional, either the absolute minima or local minima. That is to say, Yang–Mills connections are precisely those that minimize their curvature. In this sense they are the natural choice of connection on a principal or vector bundle over a manifold from a mathematical point of view.
4795:. For various choices of principal bundle, one obtains moduli spaces with interesting properties. These spaces are Hausdorff, even when allowing reducible connections, and are generically smooth. It was shown by Donaldson that the smooth part is orientable. By the
5554:
in two ways: once using that signature is a cobordism invariant, and another using a Hodge-theoretic interpretation of reducible connections. Interpreting these counts carefully, one can conclude that such a smooth manifold has diagonalisable intersection form.
622:
In addition to the physical origins of the theory, the Yang–Mills equations are of important geometric interest. There is in general no natural choice of connection on a vector bundle or principal bundle. In the special case where this bundle is the
3461:
1965:
2457:
544:
1738:
710:
5770:
The duality observed for these solutions is theorized to hold for arbitrary dual groups of symmetries of a four-manifold. Indeed there is a similar duality between instantons invariant under dual lattices inside
4988:
4793:
4288:
831:
1542:
771:. The first attempt at choosing a canonical connection might be to demand that these forms vanish. However, this is not possible unless the trivialisation is flat, in the sense that the transition functions
5759:, who first described how to construct monopoles from Nahm equation data. Hitchin showed the converse, and Donaldson proved that solutions to the Nahm equations could further be linked to moduli spaces of
3796:
6145:
Atiyah, M. F., & Bott, R. (1983). The Yang–Mills equations over riemann surfaces. Philosophical
Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 308(1505),
5290:
635:, but in general there is an infinite-dimensional space of possible choices. A Yang–Mills connection gives some kind of natural choice of a connection for a general fibre bundle, as we now describe.
4155:
2712:
2738:
2023:
2567:
5892:
5451:
The moduli space of Yang–Mills equations was used by
Donaldson to prove Donaldson's theorem about the intersection form of simply-connected four-manifolds. Using analytical results of
4659:
2335:
922:
378:
4833:
4715:
3453:
1574:
1484:
1323:
5131:
4618:
4587:
301:
6233:
Nahm, W. (1983). All self-dual multimonopoles for arbitrary gauge groups. In
Structural elements in particle physics and statistical mechanics (pp. 301–310). Springer, Boston, MA.
4472:
4424:
2202:
612:
4556:
4525:
4009:
3978:
3947:
5552:
5520:
2103:
5831:
5798:
5731:
5672:
5635:
5602:
5379:
5350:
743:
6018:
3868:
3844:
3820:
3726:
2497:
2349:
class of a differential form. The analogy being that a Yang–Mills connection is like a harmonic representative in the set of all possible connections on a principal bundle.
4034:, which was proved in this form relating Yang–Mills connections to holomorphic vector bundles by Donaldson. In this setting the moduli space has the structure of a compact
5753:
2283:
858:
4875:
5928:
of Ward. In this sense it is a 'master theory' for integrable systems, allowing many known systems to be recovered by picking appropriate parameters, such as choice of
2151:
5087:
5027:
3699:
1777:
5972:
5188:
4311:
2043:
769:
5431:
5321:
1859:
1828:
1641:
1393:
1264:
1019:
953:
448:
3639:{\displaystyle \operatorname {YM} (g\cdot A)=\int _{X}\|gF_{A}g^{-1}\|^{2}\,d\mathrm {vol} _{g}=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}=\operatorname {YM} (A)}
2596:
5922:
5571:
Through the process of dimensional reduction, the Yang–Mills equations may be used to derive other important equations in differential geometry and gauge theory.
5214:
5402:
4205:
5481:
5155:
5051:
4679:
4379:
4359:
4339:
4182:
4088:
4056:
3916:
3892:
3750:
3667:
3421:
3401:
3356:
3336:
2616:
2517:
1801:
1614:
1594:
1448:
1413:
1363:
1343:
1288:
1234:
1211:
1187:
1167:
1147:
1127:
1104:
1084:
1052:
973:
413:
328:
1867:
6061:
387:
The essential points of the work of Yang and Mills are as follows. One assumes that the fundamental description of a physical model is through the use of
2369:
456:
1649:
251:
developed (essentially independent of the mathematical literature) the theory of principal bundles and connections in order to explain the concept of
5933:
641:
6320:
6185:
Donaldson, S. K. (1986). Connections, cohomology and the intersection forms of 4-manifolds. Journal of
Differential Geometry, 24(3), 275–341.
833:
are constant functions. Not every bundle is flat, so this is not possible in general. Instead one might ask that the local connection forms
4883:
1214:
6056:
6066:
5844:
is that in fact all known integrable ODEs and PDEs come from symmetry reduction of ASDYM. For example reductions of SU(2) ASDYM give the
4724:
4213:
6176:
Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of
Differential Geometry, 18(2), 279–315.
774:
6251:
Donaldson, S. K. (1984). Nahm's equations and the classification of monopoles. Communications in
Mathematical Physics, 96(3), 387–408.
6206:
Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of
Differential Geometry, 17(1), 139–170.
1489:
6294:
Axelrod, S., Della Pietra, S., & Witten, E. (1991). Geometric quantization of Chern Simons gauge theory. representations, 34, 39.
6270:
5975:
2052:
Assuming the above set up, the Yang–Mills equations are a system of (in general non-linear) partial differential equations given by
143:(bottom right). The BPST instanton is a solution to the anti-self duality equations, and therefore of the Yang–Mills equations, on
3755:
395:(change of local trivialisation of principal bundle), these physical fields must transform in precisely the way that a connection
4796:
4031:
6155:
Donaldson, S. K. (1983). A new proof of a theorem of
Narasimhan and Seshadri. Journal of Differential Geometry, 18(2), 269–277.
6164:
Friedman, R., & Morgan, J. W. (1998). Gauge theory and the topology of four-manifolds (Vol. 4). American
Mathematical Soc.
5849:
5219:
6310:
6109:
Yang, C.N. and Mills, R.L., 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical review, 96(1), p.191.
5134:
4096:
133:
6285:
Hitchin, N. J. (1990). Flat connections and geometric quantization. Communications in mathematical physics, 131(2), 347–380.
6224:
Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in
Mathematical Physics, 83(1), 11–29.
5560:
2624:
5381:
may be extended across the point at infinity using Uhlenbeck's removable singularity theorem. More generally, for positive
6315:
6046:
1366:
1190:
182:
558:
2718:
1022:
198:
6215:
Uhlenbeck, K. K. (1982). Connections with L bounds on curvature. Communications in Mathematical Physics, 83(1), 31–42.
1974:
450:
of the connection, and the energy of the gauge field is given (up to a constant) by the Yang–Mills action functional
2526:
6041:
5855:
2286:
174:
41:
50:
30:
This article discusses the Yang–Mills equations from a mathematical perspective. For the physics perspective, see
3729:
550:
339:
6242:
Hitchin, N. J. (1983). On the construction of monopoles. Communications in Mathematical Physics, 89(2), 145–190.
4623:
The moduli space of ASD connections, or instantons, was most intensively studied by Donaldson in the case where
4058:
is four. Here the Yang–Mills equations admit a simplification from a second-order PDE to a first-order PDE, the
5488:
244:
186:
6118:
Pauli, W., 1941. Relativistic field theories of elementary particles. Reviews of Modern Physics, 13(3), p.203.
5979:
5949:
4626:
860:
are themselves constant. On a principal bundle the correct way to phrase this condition is that the curvature
307:
and others. The novelty of the work of Yang and Mills was to define gauge theories for an arbitrary choice of
2295:
863:
345:
5925:
5460:
4802:
4688:
3426:
1547:
1457:
1296:
5092:
4592:
4561:
274:
6136:
Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Oxford University Press.
6021:
5678:
5446:
4429:
1780:
1424:
632:
268:
227:
5677:
By requiring the self-duality equations to be invariant under translation in two directions, one obtains
5608:
By requiring the anti-self-duality equations to be invariant under translations in a single direction of
4384:
2161:
925:
571:
31:
4530:
4499:
3983:
3952:
3921:
170:
5525:
5493:
2060:
990:
The best one can hope for is then to ask that instead of vanishing curvature, the bundle has curvature
5807:
5774:
5707:
5648:
5611:
5578:
5355:
5326:
5845:
5841:
715:
5985:
5463:) the moduli space of ASD instantons on a smooth, compact, oriented, simply-connected four-manifold
3849:
3825:
3801:
3707:
2478:
380:
corresponding to electromagnetism, and the right framework to discuss such objects is the theory of
6197:
Donaldson, S. K. (1990). Polynomial invariants for smooth four-manifolds. Topology, 29(3), 257–315.
5937:
5638:
2046:
1416:
1063:
628:
554:
264:
5736:
5696:
By requiring the anti-self-duality equations to be invariant in three directions, one obtains the
4038:. Moduli of Yang–Mills connections have been most studied when the dimension of the base manifold
2245:
5895:
5642:
4014:
Moduli spaces of Yang–Mills connections have been intensively studied in specific circumstances.
2346:
984:
836:
4838:
4035:
3874:
or a smooth manifold. However, by restricting to irreducible connections, that is, connections
2124:
6266:
5837:
5801:
5056:
4996:
3678:
1746:
1267:
987:
to the existence of flat connections: not every principal bundle can have a flat connection.
61:
5957:
5160:
259:
as it applies to physical theories. The gauge theories Yang and Mills discovered, now called
128:(bottom left). A visual representation of the field strength of a BPST instanton with center
17:
6051:
4682:
4296:
2240:
2028:
1831:
1451:
748:
381:
194:
116:
coefficient (top right). These coefficients determine the restriction of the BPST instanton
5459:, Donaldson was able to show that in specific circumstances (when the intersection form is
5407:
5299:
1837:
1806:
1619:
1371:
1242:
1066:. The Yang–Mills equations can be phrased for a connection on a vector bundle or principal
997:
931:
426:
5764:
5456:
5452:
4023:
3871:
3704:
There is a moduli space of Yang–Mills connections modulo gauge transformations. Denote by
3373:
2572:
223:
148:
5901:
5193:
5384:
4187:
3338:
is a critical point of the Yang–Mills functional if and only if this vanishes for every
5697:
5690:
5466:
5293:
5140:
5036:
4664:
4364:
4344:
4324:
4167:
4073:
4041:
4015:
3901:
3877:
3735:
3652:
3406:
3386:
3341:
3321:
2601:
2502:
1786:
1599:
1579:
1433:
1398:
1348:
1328:
1291:
1273:
1237:
1219:
1196:
1172:
1152:
1132:
1112:
1089:
1069:
1037:
958:
624:
398:
313:
304:
248:
87:
6304:
6029:
6025:
5682:
1960:{\displaystyle \langle d_{A}s,t\rangle _{L^{2}}=\langle s,d_{A}^{*}t\rangle _{L^{2}}}
1420:
1059:
1055:
190:
5604:, and imposing that the solutions be invariant under a symmetry group. For example:
4090:
is four, a coincidence occurs: the Hodge star operator maps two-forms to two-forms,
3918:, one does obtain Hausdorff spaces. The space of irreducible connections is denoted
994:. The Yang–Mills action functional described above is precisely (the square of) the
5760:
5704:
There is a duality between solutions of the dimensionally reduced ASD equations on
5686:
5030:
4027:
3381:
2520:
2452:{\displaystyle \operatorname {YM} (A)=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}.}
2342:
539:{\displaystyle \operatorname {YM} (A)=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}.}
219:
106:
5575:
is the process of taking the Yang–Mills equations over a four-manifold, typically
4480:), the connection is a Yang–Mills connection. These connections are called either
1733:{\displaystyle \langle s,t\rangle _{L^{2}}=\int _{X}\langle s,t\rangle \,dvol_{g}}
151:'s removable singularity theorem to a topologically non-trivial ASD connection on
5929:
5756:
4718:
980:
166:
4162:
4019:
3798:
classifies all connections modulo gauge transformations, and the moduli space
705:{\displaystyle A_{\alpha }\in \Omega ^{1}(U_{\alpha },\operatorname {ad} (P))}
5487:
between a copy of the manifold itself, and a disjoint union of copies of the
5484:
1107:
308:
214:
70:
1743:
where inside the integral the fiber-wise inner product is being used, and
342:
for more details). This group could be non-Abelian as opposed to the case
4161:
The Hodge star operator squares to the identity in this case, and so has
3895:
5954:
The moduli space of Yang–Mills equations over a compact Riemann surface
4496:. The spaces of self-dual and anti-self-dual connections are denoted by
2341:
In this sense the search for Yang–Mills connections can be compared to
162:
5216:, the intersection form is trivial and the moduli space has dimension
1021:-norm of the curvature, and its Euler–Lagrange equations describe the
4983:{\displaystyle \dim {\mathcal {M}}_{k}^{-}=8k-3(1-b_{1}(X)+b_{+}(X))}
6086:
2285:, so Yang–Mills connections can be seen as a non-linear analogue of
2121:
Since the Hodge star is an isomorphism, by the explicit formula for
4788:{\displaystyle c_{2}(P)\in H^{4}(X,\mathbb {Z} )\cong \mathbb {Z} }
2475:
To derive the equations from the functional, recall that the space
5836:
Symmetry reductions of the ASD equations also lead to a number of
4283:{\displaystyle \Omega ^{2}(X)=\Omega _{+}(X)\oplus \Omega _{-}(X)}
1236:
on the total space of the principal bundle. This connection has a
4026:. There the moduli space obtains an alternative description as a
2357:
The Yang–Mills equations are the Euler–Lagrange equations of the
826:{\displaystyle g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G}
1537:{\displaystyle \operatorname {ad} (P)\otimes \Lambda ^{2}T^{*}X}
124:
to this slice. The corresponding field strength centered around
5932:
and symmetry reduction scheme. Other such master theories are
27:
Partial differential equations whose solutions are instantons
5232:
4896:
4809:
4599:
4568:
4537:
4506:
3990:
3959:
3928:
3855:
3831:
3807:
3791:{\displaystyle {\mathcal {B}}={\mathcal {A}}/{\mathcal {G}}}
3783:
3771:
3761:
3713:
2484:
243:
In their foundational paper on the topic of gauge theories,
4022:
studied the Yang–Mills equations for bundles over compact
3822:
of Yang–Mills connections is a subset. In general neither
979:), then the underlying principal bundle must have trivial
5285:{\displaystyle \dim {\mathcal {M}}_{1}^{-}(S^{4})=8-3=5}
1643:-inner product on the sections of this bundle. Namely,
205:. They have also found significant use in mathematics.
4150:{\displaystyle \star :\Omega ^{2}(X)\to \Omega ^{2}(X)}
561:
of this functional, which are the Yang–Mills equations
5089:
is the dimension of the positive-definite subspace of
2707:{\displaystyle F_{A+ta}=F_{A}+td_{A}a+t^{2}a\wedge a.}
6265:. Oxford: Oxford University Press. pp. 151–154.
5988:
5960:
5904:
5858:
5810:
5804:
can be thought of as a duality between instantons on
5777:
5739:
5710:
5651:
5614:
5581:
5528:
5496:
5469:
5410:
5387:
5358:
5329:
5302:
5222:
5196:
5163:
5143:
5095:
5059:
5039:
4999:
4886:
4841:
4805:
4727:
4691:
4667:
4629:
4595:
4564:
4533:
4502:
4432:
4387:
4367:
4347:
4327:
4299:
4216:
4190:
4170:
4099:
4076:
4044:
3986:
3955:
3924:
3904:
3880:
3852:
3828:
3804:
3758:
3738:
3710:
3681:
3655:
3464:
3429:
3409:
3389:
3344:
3324:
2736:
2627:
2604:
2575:
2529:
2505:
2481:
2372:
2298:
2248:
2164:
2153:
the Yang–Mills equations can equivalently be written
2127:
2063:
2031:
1977:
1870:
1840:
1809:
1789:
1749:
1652:
1622:
1602:
1582:
1550:
1492:
1460:
1436:
1401:
1395:, defined on the adjoint bundle. Additionally, since
1374:
1351:
1331:
1299:
1276:
1245:
1222:
1199:
1175:
1155:
1135:
1115:
1092:
1072:
1040:
1000:
961:
934:
866:
839:
777:
751:
718:
644:
574:
459:
429:
401:
348:
316:
277:
5800:, instantons on dual four-dimensional tori, and the
2618:
in this affine space, the curvatures are related by
1454:, and combined with the invariant inner product on
6012:
5966:
5916:
5886:
5825:
5792:
5747:
5725:
5666:
5629:
5596:
5546:
5514:
5475:
5425:
5396:
5373:
5344:
5323:up to a 5 parameter family defining its centre in
5315:
5284:
5208:
5182:
5149:
5125:
5081:
5045:
5021:
4982:
4869:
4827:
4787:
4709:
4673:
4653:
4612:
4581:
4550:
4519:
4466:
4418:
4373:
4353:
4333:
4305:
4282:
4199:
4176:
4149:
4082:
4050:
4003:
3972:
3941:
3910:
3886:
3862:
3838:
3814:
3790:
3744:
3720:
3693:
3661:
3638:
3455:is invariant, the Yang–Mills functional satisfies
3447:
3415:
3395:
3350:
3330:
3307:
2706:
2610:
2590:
2561:
2511:
2491:
2451:
2329:
2277:
2196:
2145:
2097:
2037:
2017:
1959:
1853:
1822:
1795:
1771:
1732:
1635:
1608:
1588:
1568:
1536:
1478:
1442:
1407:
1387:
1357:
1337:
1317:
1282:
1258:
1228:
1205:
1181:
1161:
1141:
1121:
1098:
1078:
1046:
1013:
967:
947:
916:
852:
825:
763:
737:
704:
606:
538:
442:
407:
372:
322:
295:
5685:. These equations naturally lead to the study of
4059:
1450:is Riemannian, there is an inner product on the
557:for this physical theory should be given by the
2345:, which seeks a harmonic representative in the
2018:{\displaystyle d_{A}^{*}=\pm \star d_{A}\star }
1129:. Here the latter convention is presented. Let
4293:into the positive and negative eigenspaces of
2562:{\displaystyle \Omega ^{1}(P;{\mathfrak {g}})}
271:, which had been phrased in the language of a
5887:{\displaystyle \mathrm {SL} (3,\mathbb {R} )}
5833:and dual algebraic data over a single point.
5567:Dimensional reduction and other moduli spaces
2239:Every connection automatically satisfies the
8:
3587:
3573:
3529:
3499:
3273:
3239:
3168:
3155:
3139:
3114:
3095:
3066:
3048:
3034:
2951:
2841:
2415:
2401:
1941:
1913:
1894:
1871:
1707:
1695:
1666:
1653:
732:
719:
502:
488:
4835:, the moduli space of ASD connections when
638:A connection is defined by its local forms
4207:. In particular, there is a decomposition
4028:moduli space of holomorphic vector bundles
6020:. In this case the moduli space admits a
5987:
5959:
5903:
5877:
5876:
5859:
5857:
5817:
5813:
5812:
5809:
5784:
5780:
5779:
5776:
5741:
5740:
5738:
5717:
5713:
5712:
5709:
5658:
5654:
5653:
5650:
5621:
5617:
5616:
5613:
5588:
5584:
5583:
5580:
5538:
5534:
5531:
5530:
5527:
5506:
5502:
5499:
5498:
5495:
5468:
5409:
5386:
5365:
5361:
5360:
5357:
5336:
5332:
5331:
5328:
5307:
5301:
5255:
5242:
5237:
5231:
5230:
5221:
5195:
5174:
5162:
5142:
5116:
5115:
5100:
5094:
5064:
5058:
5038:
5004:
4998:
4962:
4940:
4906:
4901:
4895:
4894:
4885:
4846:
4840:
4819:
4814:
4808:
4807:
4804:
4781:
4780:
4770:
4769:
4754:
4732:
4726:
4690:
4666:
4628:
4604:
4598:
4597:
4594:
4573:
4567:
4566:
4563:
4542:
4536:
4535:
4532:
4511:
4505:
4504:
4501:
4457:
4449:
4437:
4431:
4409:
4401:
4392:
4386:
4366:
4346:
4326:
4298:
4265:
4243:
4221:
4215:
4189:
4169:
4132:
4110:
4098:
4075:
4043:
3995:
3989:
3988:
3985:
3964:
3958:
3957:
3954:
3933:
3927:
3926:
3923:
3903:
3879:
3854:
3853:
3851:
3830:
3829:
3827:
3806:
3805:
3803:
3782:
3781:
3776:
3770:
3769:
3760:
3759:
3757:
3737:
3712:
3711:
3709:
3680:
3654:
3612:
3601:
3596:
3590:
3580:
3567:
3554:
3543:
3538:
3532:
3519:
3509:
3493:
3463:
3428:
3408:
3388:
3343:
3323:
3292:
3281:
3276:
3261:
3251:
3246:
3233:
3204:
3193:
3182:
3177:
3171:
3149:
3121:
3108:
3086:
3073:
3051:
3041:
3028:
3003:
2981:
2970:
2959:
2954:
2936:
2920:
2915:
2903:
2881:
2865:
2860:
2848:
2835:
2810:
2791:
2741:
2737:
2735:
2686:
2670:
2654:
2632:
2626:
2603:
2574:
2550:
2549:
2534:
2528:
2504:
2483:
2482:
2480:
2440:
2429:
2424:
2418:
2408:
2395:
2371:
2312:
2297:
2263:
2253:
2247:
2182:
2169:
2163:
2137:
2132:
2126:
2083:
2073:
2068:
2062:
2030:
2006:
1987:
1982:
1976:
1949:
1944:
1931:
1926:
1902:
1897:
1878:
1869:
1845:
1839:
1814:
1808:
1788:
1763:
1748:
1724:
1710:
1689:
1674:
1669:
1651:
1627:
1621:
1601:
1581:
1549:
1525:
1515:
1491:
1459:
1435:
1400:
1379:
1373:
1350:
1330:
1298:
1275:
1250:
1244:
1221:
1198:
1174:
1154:
1134:
1114:
1091:
1071:
1039:
1005:
999:
960:
939:
933:
889:
871:
865:
844:
838:
811:
798:
782:
776:
750:
726:
717:
675:
662:
649:
643:
592:
579:
573:
527:
516:
511:
505:
495:
482:
458:
434:
428:
400:
347:
315:
276:
4799:, one may compute that the dimension of
4654:{\displaystyle G=\operatorname {SU} (2)}
4070:When the dimension of the base manifold
1486:there is an inner product on the bundle
419:) on a principal bundle transforms. The
6102:
6085:For a proof of this fact, see the post
6078:
6062:Deformed Hermitian Yang–Mills equations
5522:. We can count the number of copies of
5296:, which is the unique ASD instanton on
3949:, and so the moduli spaces are denoted
2330:{\displaystyle d\omega =d^{*}\omega =0}
917:{\displaystyle F_{A}=dA+{\frac {1}{2}}}
373:{\displaystyle G=\operatorname {U} (1)}
6193:
6191:
6172:
6170:
4828:{\displaystyle {\mathcal {M}}_{k}^{-}}
4710:{\displaystyle \operatorname {SU} (2)}
3448:{\displaystyle \operatorname {ad} (P)}
3368:Moduli space of Yang–Mills connections
1569:{\displaystyle \operatorname {ad} (P)}
1479:{\displaystyle \operatorname {ad} (P)}
1318:{\displaystyle \operatorname {ad} (P)}
631:, there is such a natural choice, the
208:Solutions of the equations are called
5126:{\displaystyle H_{2}(X,\mathbb {R} )}
4613:{\displaystyle {\mathcal {M}}^{\pm }}
4582:{\displaystyle {\mathcal {B}}^{\pm }}
296:{\displaystyle \operatorname {U} (1)}
7:
6132:
6130:
6128:
6126:
6124:
5934:four-dimensional Chern–Simons theory
5292:. This agrees with existence of the
4717:-bundle is classified by its second
4467:{\displaystyle F_{A}=-{\star F_{A}}}
2363:
2155:
2054:
1215:Lie algebra-valued differential form
263:, generalised the classical work of
4419:{\displaystyle F_{A}={\star F_{A}}}
2551:
2197:{\displaystyle d_{A}\star F_{A}=0.}
607:{\displaystyle d_{A}\star F_{A}=0.}
147:. This solution can be extended by
6263:Solitons, instantons, and twistors
5989:
5961:
5863:
5860:
5352:and its scale. Such instantons on
4551:{\displaystyle {\mathcal {A}}^{-}}
4520:{\displaystyle {\mathcal {A}}^{+}}
4262:
4240:
4218:
4129:
4107:
4004:{\displaystyle {\mathcal {M}}^{*}}
3973:{\displaystyle {\mathcal {B}}^{*}}
3942:{\displaystyle {\mathcal {A}}^{*}}
3608:
3605:
3602:
3550:
3547:
3544:
3358:, and this occurs precisely when (
3288:
3285:
3282:
3189:
3186:
3183:
2966:
2963:
2960:
2531:
2436:
2433:
2430:
1512:
659:
523:
520:
517:
355:
278:
25:
6087:https://mathoverflow.net/a/265399
5755:called the Nahm transform, after
5547:{\displaystyle \mathbb {CP} ^{2}}
5515:{\displaystyle \mathbb {CP} ^{2}}
4685:. In this setting, the principal
4494:anti-self-duality (ASD) equations
3423:, and since the inner product on
2098:{\displaystyle d_{A}^{*}F_{A}=0.}
5898:, and a particular reduction to
5826:{\displaystyle \mathbb {R} ^{4}}
5793:{\displaystyle \mathbb {R} ^{4}}
5726:{\displaystyle \mathbb {R} ^{3}}
5667:{\displaystyle \mathbb {R} ^{3}}
5630:{\displaystyle \mathbb {R} ^{4}}
5597:{\displaystyle \mathbb {R} ^{4}}
5374:{\displaystyle \mathbb {R} ^{4}}
5345:{\displaystyle \mathbb {R} ^{4}}
69:
60:
49:
40:
5404:the moduli space has dimension
1345:. Associated to the connection
738:{\displaystyle \{U_{\alpha }\}}
197:. They arise in physics as the
6321:Partial differential equations
6057:Hermitian Yang–Mills equations
6024:, discovered independently by
6013:{\displaystyle \Sigma \times }
6007:
5995:
5881:
5867:
5261:
5248:
5120:
5106:
5076:
5070:
5016:
5010:
4977:
4974:
4968:
4952:
4946:
4927:
4858:
4852:
4774:
4760:
4744:
4738:
4704:
4698:
4648:
4642:
4277:
4271:
4255:
4249:
4233:
4227:
4144:
4138:
4125:
4122:
4116:
3863:{\displaystyle {\mathcal {M}}}
3839:{\displaystyle {\mathcal {B}}}
3815:{\displaystyle {\mathcal {M}}}
3721:{\displaystyle {\mathcal {G}}}
3633:
3627:
3483:
3471:
3442:
3436:
2783:
2768:
2556:
2540:
2492:{\displaystyle {\mathcal {A}}}
2385:
2379:
1563:
1557:
1505:
1499:
1473:
1467:
1312:
1306:
911:
899:
817:
755:
712:for a trivialising open cover
699:
696:
690:
668:
562:
472:
466:
367:
361:
290:
284:
183:partial differential equations
1:
6047:Connection (principal bundle)
4361:-bundle over a four-manifold
3372:The Yang–Mills equations are
2523:modelled on the vector space
1367:exterior covariant derivative
18:Atiyah–Hitchin–Singer theorem
5748:{\displaystyle \mathbb {R} }
2569:. Given a small deformation
2278:{\displaystyle d_{A}F_{A}=0}
1971:Explicitly this is given by
203:Yang–Mills action functional
4797:Atiyah–Singer index theorem
4490:self-duality (SD) equations
4476:
4321:two-forms. If a connection
4066:Anti-self-duality equations
4060:anti-self-duality equations
4032:Narasimhan–Seshadri theorem
3671:
3360:
2723:
2287:harmonic differential forms
2228:
2222:
1830:-inner product, the formal
1415:is compact, its associated
853:{\displaystyle A_{\alpha }}
391:, and derives that under a
6337:
6067:Yang–Mills–Higgs equations
6042:Connection (vector bundle)
5947:
5850:Korteweg–de Vries equation
5444:
4870:{\displaystyle c_{2}(P)=k}
4486:anti-self-dual connections
955:vanishes (that is to say,
553:dictates that the correct
393:local gauge transformation
222:of instantons was used by
29:
6261:Dunajski, Maciej (2010).
6028:and Axelrod–Della Pietra–
5561:Seiberg–Witten invariants
3898:group is given by all of
2220:A connection satisfying (
2146:{\displaystyle d_{A}^{*}}
1616:is oriented, there is an
551:principle of least action
340:Gauge group (mathematics)
5489:complex projective plane
5082:{\displaystyle b_{+}(X)}
5022:{\displaystyle b_{1}(X)}
4488:, and the equations the
3694:{\displaystyle g\cdot A}
3403:of the principal bundle
1772:{\displaystyle dvol_{g}}
559:Euler–Lagrange equations
199:Euler–Lagrange equations
5967:{\displaystyle \Sigma }
5926:integrable chiral model
5765:complex projective line
5440:
5183:{\displaystyle X=S^{4}}
985:topological obstruction
6022:geometric quantization
6014:
5968:
5918:
5888:
5827:
5794:
5749:
5727:
5681:first investigated by
5668:
5631:
5598:
5548:
5516:
5477:
5427:
5398:
5375:
5346:
5317:
5286:
5210:
5184:
5151:
5127:
5083:
5047:
5023:
4984:
4871:
4829:
4789:
4711:
4675:
4655:
4614:
4583:
4552:
4521:
4468:
4420:
4375:
4355:
4335:
4307:
4306:{\displaystyle \star }
4284:
4201:
4178:
4151:
4084:
4052:
4005:
3974:
3943:
3912:
3888:
3864:
3840:
3816:
3792:
3746:
3722:
3695:
3663:
3640:
3449:
3417:
3397:
3352:
3332:
3309:
2708:
2612:
2592:
2563:
2513:
2499:of all connections on
2493:
2453:
2331:
2279:
2198:
2147:
2099:
2039:
2038:{\displaystyle \star }
2019:
1961:
1855:
1824:
1797:
1781:Riemannian volume form
1773:
1734:
1637:
1610:
1590:
1570:
1538:
1480:
1444:
1425:adjoint representation
1409:
1389:
1359:
1339:
1319:
1284:
1260:
1230:
1213:may be specified by a
1207:
1183:
1163:
1143:
1123:
1100:
1080:
1048:
1015:
969:
949:
924:vanishes. However, by
918:
854:
827:
765:
764:{\displaystyle P\to X}
739:
706:
633:Levi-Civita connection
608:
540:
444:
409:
374:
324:
297:
210:Yang–Mills connections
6311:Differential geometry
6015:
5974:can be viewed as the
5969:
5924:dimensions gives the
5919:
5889:
5828:
5795:
5750:
5728:
5669:
5632:
5599:
5573:Dimensional reduction
5549:
5517:
5478:
5428:
5426:{\displaystyle 8k-3.}
5399:
5376:
5347:
5318:
5316:{\displaystyle S^{4}}
5287:
5211:
5185:
5152:
5128:
5084:
5048:
5024:
4985:
4872:
4830:
4790:
4712:
4676:
4656:
4615:
4584:
4553:
4522:
4482:self-dual connections
4469:
4421:
4376:
4356:
4336:
4308:
4285:
4202:
4179:
4152:
4085:
4053:
4006:
3975:
3944:
3913:
3889:
3865:
3841:
3817:
3793:
3747:
3723:
3696:
3664:
3641:
3450:
3418:
3398:
3353:
3333:
3310:
2709:
2613:
2593:
2564:
2514:
2494:
2454:
2359:Yang–Mills functional
2332:
2280:
2234:Yang–Mills connection
2199:
2148:
2100:
2049:acting on two-forms.
2040:
2020:
1962:
1856:
1854:{\displaystyle d_{A}}
1825:
1823:{\displaystyle L^{2}}
1798:
1774:
1735:
1638:
1636:{\displaystyle L^{2}}
1611:
1591:
1576:-valued two-forms on
1571:
1539:
1481:
1445:
1410:
1390:
1388:{\displaystyle d_{A}}
1360:
1340:
1320:
1285:
1261:
1259:{\displaystyle F_{A}}
1231:
1208:
1184:
1164:
1144:
1124:
1101:
1081:
1049:
1016:
1014:{\displaystyle L^{2}}
970:
950:
948:{\displaystyle F_{A}}
919:
855:
828:
766:
740:
707:
609:
541:
445:
443:{\displaystyle F_{A}}
410:
375:
325:
298:
171:differential geometry
6316:Mathematical physics
5986:
5958:
5902:
5856:
5808:
5775:
5737:
5708:
5649:
5612:
5579:
5526:
5494:
5467:
5408:
5385:
5356:
5327:
5300:
5220:
5194:
5161:
5157:. For example, when
5141:
5133:with respect to the
5093:
5057:
5037:
4997:
4884:
4839:
4803:
4725:
4689:
4665:
4627:
4593:
4562:
4558:, and similarly for
4531:
4500:
4430:
4385:
4365:
4345:
4325:
4297:
4214:
4188:
4168:
4097:
4074:
4042:
3984:
3953:
3922:
3902:
3878:
3850:
3826:
3802:
3756:
3736:
3732:of automorphisms of
3708:
3679:
3653:
3462:
3427:
3407:
3387:
3378:gauge transformation
3376:. Mathematically, a
3342:
3322:
2734:
2625:
2602:
2591:{\displaystyle A+ta}
2573:
2527:
2503:
2479:
2370:
2296:
2246:
2162:
2125:
2061:
2029:
1975:
1868:
1838:
1807:
1787:
1747:
1650:
1620:
1600:
1580:
1548:
1490:
1458:
1434:
1419:admits an invariant
1399:
1372:
1349:
1329:
1297:
1274:
1243:
1220:
1197:
1173:
1153:
1133:
1113:
1090:
1070:
1038:
998:
992:as small as possible
959:
932:
864:
837:
775:
749:
716:
642:
572:
457:
427:
421:gauge field strength
399:
346:
314:
275:
179:Yang–Mills equations
5980:Chern–Simons theory
5976:configuration space
5950:Chern–Simons theory
5944:Chern–Simons theory
5938:affine Gaudin model
5917:{\displaystyle 2+1}
5679:Hitchin's equations
5639:Bogomolny equations
5447:Donaldson's theorem
5441:Donaldson's theorem
5247:
5209:{\displaystyle k=1}
4911:
4824:
3256:
2142:
2078:
2047:Hodge star operator
1992:
1936:
1417:compact Lie algebra
1290:with values in the
1149:denote a principal
1106:, for some compact
1064:Riemannian manifold
629:Riemannian manifold
555:equations of motion
334:(or in physics the
269:Maxwell's equations
261:Yang–Mills theories
228:Donaldson's theorem
6010:
5964:
5914:
5896:Tzitzeica equation
5884:
5838:integrable systems
5823:
5790:
5745:
5723:
5664:
5643:magnetic monopoles
5637:, one obtains the
5627:
5594:
5544:
5512:
5473:
5423:
5397:{\displaystyle k,}
5394:
5371:
5342:
5313:
5282:
5229:
5206:
5180:
5147:
5123:
5079:
5043:
5019:
4980:
4893:
4867:
4825:
4806:
4785:
4707:
4671:
4651:
4610:
4579:
4548:
4517:
4464:
4416:
4371:
4351:
4331:
4303:
4280:
4200:{\displaystyle -1}
4197:
4174:
4147:
4080:
4048:
4001:
3970:
3939:
3908:
3884:
3860:
3836:
3812:
3788:
3742:
3718:
3691:
3659:
3636:
3445:
3413:
3393:
3348:
3328:
3305:
3303:
3242:
2704:
2608:
2588:
2559:
2509:
2489:
2449:
2347:de Rham cohomology
2327:
2275:
2194:
2143:
2128:
2095:
2064:
2035:
2015:
1978:
1957:
1922:
1851:
1820:
1793:
1769:
1730:
1633:
1606:
1586:
1566:
1534:
1476:
1440:
1405:
1385:
1355:
1335:
1315:
1280:
1256:
1226:
1203:
1179:
1159:
1139:
1119:
1096:
1076:
1044:
1011:
965:
945:
914:
850:
823:
761:
735:
702:
604:
536:
440:
405:
370:
320:
293:
5842:Ward's conjecture
5802:ADHM construction
5476:{\displaystyle X}
5150:{\displaystyle X}
5135:intersection form
5046:{\displaystyle X}
4674:{\displaystyle X}
4381:satisfies either
4374:{\displaystyle X}
4354:{\displaystyle G}
4334:{\displaystyle A}
4177:{\displaystyle 1}
4083:{\displaystyle X}
4051:{\displaystyle X}
3911:{\displaystyle G}
3887:{\displaystyle A}
3745:{\displaystyle P}
3662:{\displaystyle A}
3416:{\displaystyle P}
3396:{\displaystyle g}
3351:{\displaystyle a}
3331:{\displaystyle A}
3016:
2823:
2754:
2717:To determine the
2611:{\displaystyle A}
2512:{\displaystyle P}
2473:
2472:
2218:
2217:
2119:
2118:
1796:{\displaystyle X}
1609:{\displaystyle X}
1589:{\displaystyle X}
1443:{\displaystyle X}
1408:{\displaystyle G}
1358:{\displaystyle A}
1338:{\displaystyle P}
1283:{\displaystyle X}
1229:{\displaystyle A}
1206:{\displaystyle P}
1182:{\displaystyle X}
1162:{\displaystyle G}
1142:{\displaystyle P}
1122:{\displaystyle G}
1099:{\displaystyle X}
1079:{\displaystyle G}
1047:{\displaystyle X}
968:{\displaystyle A}
928:if the curvature
926:Chern–Weil theory
897:
423:is the curvature
408:{\displaystyle A}
382:principal bundles
323:{\displaystyle G}
169:, and especially
86:coefficient of a
32:Yang–Mills theory
16:(Redirected from
6328:
6295:
6292:
6286:
6283:
6277:
6276:
6258:
6252:
6249:
6243:
6240:
6234:
6231:
6225:
6222:
6216:
6213:
6207:
6204:
6198:
6195:
6186:
6183:
6177:
6174:
6165:
6162:
6156:
6153:
6147:
6143:
6137:
6134:
6119:
6116:
6110:
6107:
6090:
6083:
6052:Donaldson theory
6019:
6017:
6016:
6011:
5973:
5971:
5970:
5965:
5923:
5921:
5920:
5915:
5894:ASDYM gives the
5893:
5891:
5890:
5885:
5880:
5866:
5832:
5830:
5829:
5824:
5822:
5821:
5816:
5799:
5797:
5796:
5791:
5789:
5788:
5783:
5754:
5752:
5751:
5746:
5744:
5732:
5730:
5729:
5724:
5722:
5721:
5716:
5673:
5671:
5670:
5665:
5663:
5662:
5657:
5636:
5634:
5633:
5628:
5626:
5625:
5620:
5603:
5601:
5600:
5595:
5593:
5592:
5587:
5553:
5551:
5550:
5545:
5543:
5542:
5537:
5521:
5519:
5518:
5513:
5511:
5510:
5505:
5482:
5480:
5479:
5474:
5432:
5430:
5429:
5424:
5403:
5401:
5400:
5395:
5380:
5378:
5377:
5372:
5370:
5369:
5364:
5351:
5349:
5348:
5343:
5341:
5340:
5335:
5322:
5320:
5319:
5314:
5312:
5311:
5291:
5289:
5288:
5283:
5260:
5259:
5246:
5241:
5236:
5235:
5215:
5213:
5212:
5207:
5189:
5187:
5186:
5181:
5179:
5178:
5156:
5154:
5153:
5148:
5132:
5130:
5129:
5124:
5119:
5105:
5104:
5088:
5086:
5085:
5080:
5069:
5068:
5052:
5050:
5049:
5044:
5028:
5026:
5025:
5020:
5009:
5008:
4989:
4987:
4986:
4981:
4967:
4966:
4945:
4944:
4910:
4905:
4900:
4899:
4876:
4874:
4873:
4868:
4851:
4850:
4834:
4832:
4831:
4826:
4823:
4818:
4813:
4812:
4794:
4792:
4791:
4786:
4784:
4773:
4759:
4758:
4737:
4736:
4716:
4714:
4713:
4708:
4683:simply-connected
4680:
4678:
4677:
4672:
4660:
4658:
4657:
4652:
4619:
4617:
4616:
4611:
4609:
4608:
4603:
4602:
4588:
4586:
4585:
4580:
4578:
4577:
4572:
4571:
4557:
4555:
4554:
4549:
4547:
4546:
4541:
4540:
4526:
4524:
4523:
4518:
4516:
4515:
4510:
4509:
4473:
4471:
4470:
4465:
4463:
4462:
4461:
4442:
4441:
4425:
4423:
4422:
4417:
4415:
4414:
4413:
4397:
4396:
4380:
4378:
4377:
4372:
4360:
4358:
4357:
4352:
4340:
4338:
4337:
4332:
4312:
4310:
4309:
4304:
4289:
4287:
4286:
4281:
4270:
4269:
4248:
4247:
4226:
4225:
4206:
4204:
4203:
4198:
4183:
4181:
4180:
4175:
4156:
4154:
4153:
4148:
4137:
4136:
4115:
4114:
4089:
4087:
4086:
4081:
4057:
4055:
4054:
4049:
4024:Riemann surfaces
4010:
4008:
4007:
4002:
4000:
3999:
3994:
3993:
3979:
3977:
3976:
3971:
3969:
3968:
3963:
3962:
3948:
3946:
3945:
3940:
3938:
3937:
3932:
3931:
3917:
3915:
3914:
3909:
3893:
3891:
3890:
3885:
3869:
3867:
3866:
3861:
3859:
3858:
3845:
3843:
3842:
3837:
3835:
3834:
3821:
3819:
3818:
3813:
3811:
3810:
3797:
3795:
3794:
3789:
3787:
3786:
3780:
3775:
3774:
3765:
3764:
3751:
3749:
3748:
3743:
3727:
3725:
3724:
3719:
3717:
3716:
3700:
3698:
3697:
3692:
3668:
3666:
3665:
3660:
3645:
3643:
3642:
3637:
3617:
3616:
3611:
3595:
3594:
3585:
3584:
3572:
3571:
3559:
3558:
3553:
3537:
3536:
3527:
3526:
3514:
3513:
3498:
3497:
3454:
3452:
3451:
3446:
3422:
3420:
3419:
3414:
3402:
3400:
3399:
3394:
3364:) is satisfied.
3357:
3355:
3354:
3349:
3337:
3335:
3334:
3329:
3314:
3312:
3311:
3306:
3304:
3297:
3296:
3291:
3266:
3265:
3255:
3250:
3238:
3237:
3219:
3215:
3214:
3203:
3199:
3198:
3197:
3192:
3176:
3175:
3154:
3153:
3126:
3125:
3113:
3112:
3091:
3090:
3078:
3077:
3056:
3055:
3046:
3045:
3033:
3032:
3017:
3015:
3004:
2996:
2992:
2991:
2980:
2976:
2975:
2974:
2969:
2941:
2940:
2925:
2924:
2908:
2907:
2886:
2885:
2870:
2869:
2853:
2852:
2840:
2839:
2824:
2822:
2811:
2802:
2801:
2790:
2786:
2755:
2753:
2742:
2713:
2711:
2710:
2705:
2691:
2690:
2675:
2674:
2659:
2658:
2646:
2645:
2617:
2615:
2614:
2609:
2598:of a connection
2597:
2595:
2594:
2589:
2568:
2566:
2565:
2560:
2555:
2554:
2539:
2538:
2518:
2516:
2515:
2510:
2498:
2496:
2495:
2490:
2488:
2487:
2467:
2458:
2456:
2455:
2450:
2445:
2444:
2439:
2423:
2422:
2413:
2412:
2400:
2399:
2364:
2336:
2334:
2333:
2328:
2317:
2316:
2289:, which satisfy
2284:
2282:
2281:
2276:
2268:
2267:
2258:
2257:
2241:Bianchi identity
2212:
2203:
2201:
2200:
2195:
2187:
2186:
2174:
2173:
2156:
2152:
2150:
2149:
2144:
2141:
2136:
2113:
2104:
2102:
2101:
2096:
2088:
2087:
2077:
2072:
2055:
2044:
2042:
2041:
2036:
2024:
2022:
2021:
2016:
2011:
2010:
1991:
1986:
1966:
1964:
1963:
1958:
1956:
1955:
1954:
1953:
1935:
1930:
1909:
1908:
1907:
1906:
1883:
1882:
1860:
1858:
1857:
1852:
1850:
1849:
1832:adjoint operator
1829:
1827:
1826:
1821:
1819:
1818:
1802:
1800:
1799:
1794:
1778:
1776:
1775:
1770:
1768:
1767:
1739:
1737:
1736:
1731:
1729:
1728:
1694:
1693:
1681:
1680:
1679:
1678:
1642:
1640:
1639:
1634:
1632:
1631:
1615:
1613:
1612:
1607:
1595:
1593:
1592:
1587:
1575:
1573:
1572:
1567:
1543:
1541:
1540:
1535:
1530:
1529:
1520:
1519:
1485:
1483:
1482:
1477:
1452:cotangent bundle
1449:
1447:
1446:
1441:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1384:
1383:
1364:
1362:
1361:
1356:
1344:
1342:
1341:
1336:
1324:
1322:
1321:
1316:
1289:
1287:
1286:
1281:
1265:
1263:
1262:
1257:
1255:
1254:
1235:
1233:
1232:
1227:
1212:
1210:
1209:
1204:
1188:
1186:
1185:
1180:
1168:
1166:
1165:
1160:
1148:
1146:
1145:
1140:
1128:
1126:
1125:
1120:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1053:
1051:
1050:
1045:
1020:
1018:
1017:
1012:
1010:
1009:
974:
972:
971:
966:
954:
952:
951:
946:
944:
943:
923:
921:
920:
915:
898:
890:
876:
875:
859:
857:
856:
851:
849:
848:
832:
830:
829:
824:
816:
815:
803:
802:
790:
789:
770:
768:
767:
762:
744:
742:
741:
736:
731:
730:
711:
709:
708:
703:
680:
679:
667:
666:
654:
653:
613:
611:
610:
605:
597:
596:
584:
583:
545:
543:
542:
537:
532:
531:
526:
510:
509:
500:
499:
487:
486:
449:
447:
446:
441:
439:
438:
414:
412:
411:
406:
379:
377:
376:
371:
329:
327:
326:
321:
303:gauge theory by
302:
300:
299:
294:
257:gauge invariance
195:principal bundle
181:are a system of
134:compactification
109:(top left). The
73:
64:
53:
44:
21:
6336:
6335:
6331:
6330:
6329:
6327:
6326:
6325:
6301:
6300:
6299:
6298:
6293:
6289:
6284:
6280:
6273:
6260:
6259:
6255:
6250:
6246:
6241:
6237:
6232:
6228:
6223:
6219:
6214:
6210:
6205:
6201:
6196:
6189:
6184:
6180:
6175:
6168:
6163:
6159:
6154:
6150:
6144:
6140:
6135:
6122:
6117:
6113:
6108:
6104:
6099:
6094:
6093:
6084:
6080:
6075:
6038:
5984:
5983:
5956:
5955:
5952:
5946:
5900:
5899:
5854:
5853:
5811:
5806:
5805:
5778:
5773:
5772:
5735:
5734:
5711:
5706:
5705:
5700:on an interval.
5652:
5647:
5646:
5641:which describe
5615:
5610:
5609:
5582:
5577:
5576:
5569:
5529:
5524:
5523:
5497:
5492:
5491:
5465:
5464:
5457:Karen Uhlenbeck
5453:Clifford Taubes
5449:
5443:
5438:
5406:
5405:
5383:
5382:
5359:
5354:
5353:
5330:
5325:
5324:
5303:
5298:
5297:
5251:
5218:
5217:
5192:
5191:
5170:
5159:
5158:
5139:
5138:
5096:
5091:
5090:
5060:
5055:
5054:
5035:
5034:
5000:
4995:
4994:
4958:
4936:
4882:
4881:
4842:
4837:
4836:
4801:
4800:
4750:
4728:
4723:
4722:
4687:
4686:
4663:
4662:
4625:
4624:
4596:
4591:
4590:
4565:
4560:
4559:
4534:
4529:
4528:
4503:
4498:
4497:
4453:
4433:
4428:
4427:
4405:
4388:
4383:
4382:
4363:
4362:
4343:
4342:
4341:on a principal
4323:
4322:
4295:
4294:
4261:
4239:
4217:
4212:
4211:
4186:
4185:
4166:
4165:
4128:
4106:
4095:
4094:
4072:
4071:
4068:
4040:
4039:
4036:Kähler manifold
3987:
3982:
3981:
3956:
3951:
3950:
3925:
3920:
3919:
3900:
3899:
3876:
3875:
3848:
3847:
3824:
3823:
3800:
3799:
3754:
3753:
3734:
3733:
3706:
3705:
3677:
3676:
3651:
3650:
3600:
3586:
3576:
3563:
3542:
3528:
3515:
3505:
3489:
3460:
3459:
3425:
3424:
3405:
3404:
3385:
3384:
3374:gauge invariant
3370:
3340:
3339:
3320:
3319:
3318:The connection
3302:
3301:
3280:
3257:
3229:
3217:
3216:
3181:
3167:
3145:
3117:
3104:
3082:
3069:
3047:
3037:
3024:
3023:
3019:
3018:
3008:
2994:
2993:
2958:
2932:
2916:
2899:
2877:
2861:
2844:
2831:
2830:
2826:
2825:
2815:
2803:
2761:
2757:
2756:
2746:
2732:
2731:
2719:critical points
2682:
2666:
2650:
2628:
2623:
2622:
2600:
2599:
2571:
2570:
2530:
2525:
2524:
2501:
2500:
2477:
2476:
2465:
2428:
2414:
2404:
2391:
2368:
2367:
2355:
2308:
2294:
2293:
2259:
2249:
2244:
2243:
2210:
2178:
2165:
2160:
2159:
2123:
2122:
2111:
2079:
2059:
2058:
2027:
2026:
2002:
1973:
1972:
1945:
1940:
1898:
1893:
1874:
1866:
1865:
1841:
1836:
1835:
1810:
1805:
1804:
1785:
1784:
1759:
1745:
1744:
1720:
1685:
1670:
1665:
1648:
1647:
1623:
1618:
1617:
1598:
1597:
1578:
1577:
1546:
1545:
1521:
1511:
1488:
1487:
1456:
1455:
1432:
1431:
1397:
1396:
1375:
1370:
1369:
1347:
1346:
1327:
1326:
1295:
1294:
1272:
1271:
1246:
1241:
1240:
1218:
1217:
1195:
1194:
1171:
1170:
1151:
1150:
1131:
1130:
1111:
1110:
1088:
1087:
1068:
1067:
1036:
1035:
1032:
1023:critical points
1001:
996:
995:
977:flat connection
957:
956:
935:
930:
929:
867:
862:
861:
840:
835:
834:
807:
794:
778:
773:
772:
747:
746:
745:for the bundle
722:
714:
713:
671:
658:
645:
640:
639:
620:
588:
575:
570:
569:
515:
501:
491:
478:
455:
454:
430:
425:
424:
415:(in physics, a
397:
396:
344:
343:
332:structure group
312:
311:
273:
272:
241:
236:
224:Simon Donaldson
159:
158:
157:
156:
114:
103:
84:
76:
75:
74:
66:
65:
56:
55:
54:
46:
45:
34:
28:
23:
22:
15:
12:
11:
5:
6334:
6332:
6324:
6323:
6318:
6313:
6303:
6302:
6297:
6296:
6287:
6278:
6271:
6253:
6244:
6235:
6226:
6217:
6208:
6199:
6187:
6178:
6166:
6157:
6148:
6138:
6120:
6111:
6101:
6100:
6098:
6095:
6092:
6091:
6077:
6076:
6074:
6071:
6070:
6069:
6064:
6059:
6054:
6049:
6044:
6037:
6034:
6009:
6006:
6003:
6000:
5997:
5994:
5991:
5982:on a cylinder
5963:
5948:Main article:
5945:
5942:
5913:
5910:
5907:
5883:
5879:
5875:
5872:
5869:
5865:
5862:
5820:
5815:
5787:
5782:
5743:
5720:
5715:
5702:
5701:
5698:Nahm equations
5694:
5691:Hitchin system
5675:
5661:
5656:
5624:
5619:
5591:
5586:
5568:
5565:
5541:
5536:
5533:
5509:
5504:
5501:
5472:
5445:Main article:
5442:
5439:
5437:
5434:
5422:
5419:
5416:
5413:
5393:
5390:
5368:
5363:
5339:
5334:
5310:
5306:
5294:BPST instanton
5281:
5278:
5275:
5272:
5269:
5266:
5263:
5258:
5254:
5250:
5245:
5240:
5234:
5228:
5225:
5205:
5202:
5199:
5177:
5173:
5169:
5166:
5146:
5122:
5118:
5114:
5111:
5108:
5103:
5099:
5078:
5075:
5072:
5067:
5063:
5042:
5018:
5015:
5012:
5007:
5003:
4991:
4990:
4979:
4976:
4973:
4970:
4965:
4961:
4957:
4954:
4951:
4948:
4943:
4939:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4909:
4904:
4898:
4892:
4889:
4866:
4863:
4860:
4857:
4854:
4849:
4845:
4822:
4817:
4811:
4783:
4779:
4776:
4772:
4768:
4765:
4762:
4757:
4753:
4749:
4746:
4743:
4740:
4735:
4731:
4706:
4703:
4700:
4697:
4694:
4670:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4607:
4601:
4576:
4570:
4545:
4539:
4514:
4508:
4460:
4456:
4452:
4448:
4445:
4440:
4436:
4412:
4408:
4404:
4400:
4395:
4391:
4370:
4350:
4330:
4319:anti-self-dual
4302:
4291:
4290:
4279:
4276:
4273:
4268:
4264:
4260:
4257:
4254:
4251:
4246:
4242:
4238:
4235:
4232:
4229:
4224:
4220:
4196:
4193:
4173:
4159:
4158:
4146:
4143:
4140:
4135:
4131:
4127:
4124:
4121:
4118:
4113:
4109:
4105:
4102:
4079:
4067:
4064:
4047:
4030:. This is the
4016:Michael Atiyah
3998:
3992:
3967:
3961:
3936:
3930:
3907:
3883:
3857:
3833:
3809:
3785:
3779:
3773:
3768:
3763:
3741:
3715:
3690:
3687:
3684:
3658:
3647:
3646:
3635:
3632:
3629:
3626:
3623:
3620:
3615:
3610:
3607:
3604:
3599:
3593:
3589:
3583:
3579:
3575:
3570:
3566:
3562:
3557:
3552:
3549:
3546:
3541:
3535:
3531:
3525:
3522:
3518:
3512:
3508:
3504:
3501:
3496:
3492:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3444:
3441:
3438:
3435:
3432:
3412:
3392:
3369:
3366:
3347:
3327:
3316:
3315:
3300:
3295:
3290:
3287:
3284:
3279:
3275:
3272:
3269:
3264:
3260:
3254:
3249:
3245:
3241:
3236:
3232:
3228:
3225:
3222:
3220:
3218:
3213:
3210:
3207:
3202:
3196:
3191:
3188:
3185:
3180:
3174:
3170:
3166:
3163:
3160:
3157:
3152:
3148:
3144:
3141:
3138:
3135:
3132:
3129:
3124:
3120:
3116:
3111:
3107:
3103:
3100:
3097:
3094:
3089:
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3068:
3065:
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3054:
3050:
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3027:
3022:
3014:
3011:
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2999:
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2987:
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2979:
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2800:
2797:
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2715:
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2703:
2700:
2697:
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2669:
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2607:
2587:
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2432:
2427:
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2417:
2411:
2407:
2403:
2398:
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2390:
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2381:
2378:
2375:
2354:
2351:
2339:
2338:
2326:
2323:
2320:
2315:
2311:
2307:
2304:
2301:
2274:
2271:
2266:
2262:
2256:
2252:
2232:) is called a
2216:
2215:
2206:
2204:
2193:
2190:
2185:
2181:
2177:
2172:
2168:
2140:
2135:
2131:
2117:
2116:
2107:
2105:
2094:
2091:
2086:
2082:
2076:
2071:
2067:
2034:
2014:
2009:
2005:
2001:
1998:
1995:
1990:
1985:
1981:
1969:
1968:
1952:
1948:
1943:
1939:
1934:
1929:
1925:
1921:
1918:
1915:
1912:
1905:
1901:
1896:
1892:
1889:
1886:
1881:
1877:
1873:
1861:is defined by
1848:
1844:
1817:
1813:
1792:
1766:
1762:
1758:
1755:
1752:
1741:
1740:
1727:
1723:
1719:
1716:
1713:
1709:
1706:
1703:
1700:
1697:
1692:
1688:
1684:
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1673:
1668:
1664:
1661:
1658:
1655:
1630:
1626:
1605:
1585:
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1553:
1533:
1528:
1524:
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1514:
1510:
1507:
1504:
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1498:
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1469:
1466:
1463:
1439:
1404:
1382:
1378:
1354:
1334:
1314:
1311:
1308:
1305:
1302:
1292:adjoint bundle
1279:
1253:
1249:
1238:curvature form
1225:
1202:
1178:
1158:
1138:
1118:
1095:
1075:
1043:
1031:
1028:
1008:
1004:
964:
942:
938:
913:
910:
907:
904:
901:
896:
893:
888:
885:
882:
879:
874:
870:
847:
843:
822:
819:
814:
810:
806:
801:
797:
793:
788:
785:
781:
760:
757:
754:
734:
729:
725:
721:
701:
698:
695:
692:
689:
686:
683:
678:
674:
670:
665:
661:
657:
652:
648:
625:tangent bundle
619:
616:
615:
614:
603:
600:
595:
591:
587:
582:
578:
547:
546:
535:
530:
525:
522:
519:
514:
508:
504:
498:
494:
490:
485:
481:
477:
474:
471:
468:
465:
462:
437:
433:
404:
369:
366:
363:
360:
357:
354:
351:
319:
305:Wolfgang Pauli
292:
289:
286:
283:
280:
253:gauge symmetry
249:Chen-Ning Yang
240:
237:
235:
232:
112:
101:
88:BPST instanton
82:
78:
77:
68:
67:
59:
58:
57:
48:
47:
39:
38:
37:
36:
35:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6333:
6322:
6319:
6317:
6314:
6312:
6309:
6308:
6306:
6291:
6288:
6282:
6279:
6274:
6272:9780198570639
6268:
6264:
6257:
6254:
6248:
6245:
6239:
6236:
6230:
6227:
6221:
6218:
6212:
6209:
6203:
6200:
6194:
6192:
6188:
6182:
6179:
6173:
6171:
6167:
6161:
6158:
6152:
6149:
6142:
6139:
6133:
6131:
6129:
6127:
6125:
6121:
6115:
6112:
6106:
6103:
6096:
6088:
6082:
6079:
6072:
6068:
6065:
6063:
6060:
6058:
6055:
6053:
6050:
6048:
6045:
6043:
6040:
6039:
6035:
6033:
6031:
6027:
6026:Nigel Hitchin
6023:
6004:
6001:
5998:
5992:
5981:
5977:
5951:
5943:
5941:
5939:
5935:
5931:
5927:
5911:
5908:
5905:
5897:
5873:
5870:
5851:
5847:
5843:
5839:
5834:
5818:
5803:
5785:
5768:
5766:
5762:
5761:rational maps
5758:
5718:
5699:
5695:
5692:
5688:
5687:Higgs bundles
5684:
5680:
5676:
5659:
5644:
5640:
5622:
5607:
5606:
5605:
5589:
5574:
5566:
5564:
5562:
5559:surpassed by
5556:
5539:
5507:
5490:
5486:
5470:
5462:
5458:
5454:
5448:
5435:
5433:
5420:
5417:
5414:
5411:
5391:
5388:
5366:
5337:
5308:
5304:
5295:
5279:
5276:
5273:
5270:
5267:
5264:
5256:
5252:
5243:
5238:
5226:
5223:
5203:
5200:
5197:
5175:
5171:
5167:
5164:
5144:
5136:
5112:
5109:
5101:
5097:
5073:
5065:
5061:
5040:
5032:
5029:is the first
5013:
5005:
5001:
4971:
4963:
4959:
4955:
4949:
4941:
4937:
4933:
4930:
4924:
4921:
4918:
4915:
4912:
4907:
4902:
4890:
4887:
4880:
4879:
4878:
4864:
4861:
4855:
4847:
4843:
4820:
4815:
4798:
4777:
4766:
4763:
4755:
4751:
4747:
4741:
4733:
4729:
4720:
4701:
4695:
4692:
4684:
4668:
4645:
4639:
4636:
4633:
4630:
4621:
4605:
4574:
4543:
4512:
4495:
4491:
4487:
4483:
4479:
4478:
4458:
4454:
4450:
4446:
4443:
4438:
4434:
4410:
4406:
4402:
4398:
4393:
4389:
4368:
4348:
4328:
4320:
4316:
4300:
4274:
4266:
4258:
4252:
4244:
4236:
4230:
4222:
4210:
4209:
4208:
4194:
4191:
4171:
4164:
4141:
4133:
4119:
4111:
4103:
4100:
4093:
4092:
4091:
4077:
4065:
4063:
4061:
4045:
4037:
4033:
4029:
4025:
4021:
4017:
4012:
3996:
3965:
3934:
3905:
3897:
3881:
3873:
3777:
3766:
3739:
3731:
3702:
3688:
3685:
3682:
3674:
3673:
3656:
3630:
3624:
3621:
3618:
3613:
3597:
3591:
3581:
3577:
3568:
3564:
3560:
3555:
3539:
3533:
3523:
3520:
3516:
3510:
3506:
3502:
3494:
3490:
3486:
3480:
3477:
3474:
3468:
3465:
3458:
3457:
3456:
3439:
3433:
3430:
3410:
3390:
3383:
3379:
3375:
3367:
3365:
3363:
3362:
3345:
3325:
3298:
3293:
3277:
3270:
3267:
3262:
3258:
3252:
3247:
3243:
3234:
3230:
3226:
3223:
3221:
3211:
3208:
3205:
3200:
3194:
3178:
3172:
3164:
3161:
3158:
3150:
3146:
3142:
3136:
3133:
3130:
3127:
3122:
3118:
3109:
3105:
3101:
3098:
3092:
3087:
3083:
3079:
3074:
3070:
3063:
3060:
3057:
3052:
3042:
3038:
3029:
3025:
3020:
3012:
3009:
3005:
3000:
2998:
2988:
2985:
2982:
2977:
2971:
2955:
2948:
2945:
2942:
2937:
2933:
2929:
2926:
2921:
2917:
2912:
2909:
2904:
2900:
2896:
2893:
2890:
2887:
2882:
2878:
2874:
2871:
2866:
2862:
2857:
2854:
2849:
2845:
2836:
2832:
2827:
2819:
2816:
2812:
2807:
2805:
2798:
2795:
2792:
2787:
2780:
2777:
2774:
2771:
2765:
2762:
2758:
2750:
2747:
2743:
2730:
2729:
2728:
2726:
2725:
2720:
2701:
2698:
2695:
2692:
2687:
2683:
2679:
2676:
2671:
2667:
2663:
2660:
2655:
2651:
2647:
2642:
2639:
2636:
2633:
2629:
2621:
2620:
2619:
2605:
2585:
2582:
2579:
2576:
2546:
2543:
2535:
2522:
2506:
2469:
2462:
2460:
2446:
2441:
2425:
2419:
2409:
2405:
2396:
2392:
2388:
2382:
2376:
2373:
2366:
2365:
2362:
2361:, defined by
2360:
2352:
2350:
2348:
2344:
2324:
2321:
2318:
2313:
2309:
2305:
2302:
2299:
2292:
2291:
2290:
2288:
2272:
2269:
2264:
2260:
2254:
2250:
2242:
2237:
2235:
2231:
2230:
2225:
2224:
2214:
2207:
2205:
2191:
2188:
2183:
2179:
2175:
2170:
2166:
2158:
2157:
2154:
2138:
2133:
2129:
2115:
2108:
2106:
2092:
2089:
2084:
2080:
2074:
2069:
2065:
2057:
2056:
2053:
2050:
2048:
2032:
2012:
2007:
2003:
1999:
1996:
1993:
1988:
1983:
1979:
1950:
1946:
1937:
1932:
1927:
1923:
1919:
1916:
1910:
1903:
1899:
1890:
1887:
1884:
1879:
1875:
1864:
1863:
1862:
1846:
1842:
1833:
1815:
1811:
1803:. Using this
1790:
1782:
1764:
1760:
1756:
1753:
1750:
1725:
1721:
1717:
1714:
1711:
1704:
1701:
1698:
1690:
1686:
1682:
1675:
1671:
1662:
1659:
1656:
1646:
1645:
1644:
1628:
1624:
1603:
1583:
1560:
1554:
1551:
1531:
1526:
1522:
1516:
1508:
1502:
1496:
1493:
1470:
1464:
1461:
1453:
1437:
1428:
1426:
1422:
1421:inner product
1418:
1402:
1380:
1376:
1368:
1352:
1332:
1309:
1303:
1300:
1293:
1277:
1269:
1266:, which is a
1251:
1247:
1239:
1223:
1216:
1200:
1192:
1176:
1169:-bundle over
1156:
1136:
1116:
1109:
1093:
1086:-bundle over
1073:
1065:
1061:
1057:
1041:
1029:
1027:
1024:
1006:
1002:
993:
988:
986:
983:, which is a
982:
981:Chern classes
978:
962:
940:
936:
927:
908:
905:
902:
894:
891:
886:
883:
880:
877:
872:
868:
845:
841:
820:
812:
808:
804:
799:
795:
791:
786:
783:
779:
758:
752:
727:
723:
693:
687:
684:
681:
676:
672:
663:
655:
650:
646:
636:
634:
630:
626:
617:
601:
598:
593:
589:
585:
580:
576:
568:
567:
566:
564:
563:derived below
560:
556:
552:
533:
528:
512:
506:
496:
492:
483:
479:
475:
469:
463:
460:
453:
452:
451:
435:
431:
422:
418:
402:
394:
390:
385:
383:
364:
358:
352:
349:
341:
337:
333:
330:, called the
317:
310:
306:
287:
281:
270:
266:
265:James Maxwell
262:
258:
254:
250:
246:
238:
233:
231:
229:
225:
221:
217:
216:
211:
206:
204:
200:
196:
192:
191:vector bundle
188:
184:
180:
176:
172:
168:
164:
154:
150:
146:
142:
138:
135:
131:
127:
123:
119:
115:
108:
105:is the third
104:
97:
93:
89:
85:
72:
63:
52:
43:
33:
19:
6290:
6281:
6262:
6256:
6247:
6238:
6229:
6220:
6211:
6202:
6181:
6160:
6151:
6141:
6114:
6105:
6081:
5953:
5835:
5769:
5703:
5572:
5570:
5557:
5450:
5436:Applications
5031:Betti number
4992:
4622:
4493:
4489:
4485:
4481:
4475:
4318:
4314:
4292:
4160:
4069:
4013:
3703:
3670:
3648:
3382:automorphism
3377:
3371:
3359:
3317:
2722:
2716:
2521:affine space
2474:
2463:
2358:
2356:
2343:Hodge theory
2340:
2238:
2233:
2227:
2221:
2219:
2208:
2120:
2109:
2051:
1970:
1742:
1429:
1033:
991:
989:
976:
637:
621:
548:
420:
416:
392:
388:
386:
335:
331:
260:
256:
252:
245:Robert Mills
242:
220:moduli space
213:
209:
207:
202:
178:
175:gauge theory
160:
152:
144:
140:
136:
129:
125:
122:g=2, ρ=1,z=0
121:
117:
110:
107:Pauli matrix
99:
95:
91:
80:
5930:gauge group
5846:sine-Gordon
5767:to itself.
5757:Werner Nahm
4719:Chern class
4474:, then by (
4163:eigenvalues
3730:gauge group
3675:), so does
3669:satisfies (
2727:), compute
618:Mathematics
417:gauge field
336:gauge group
167:mathematics
6305:Categories
6097:References
4020:Raoul Bott
3752:. The set
3649:and so if
2353:Derivation
1423:under the
1191:connection
1030:Definition
234:Motivation
215:instantons
187:connection
94:-slice of
5993:×
5990:Σ
5962:Σ
5763:from the
5485:cobordism
5418:−
5271:−
5244:−
5227:
4934:−
4922:−
4908:−
4891:
4821:−
4778:≅
4748:∈
4696:
4640:
4606:±
4575:±
4544:−
4451:⋆
4447:−
4403:⋆
4315:self-dual
4301:⋆
4267:−
4263:Ω
4259:⊕
4241:Ω
4219:Ω
4192:−
4130:Ω
4126:→
4108:Ω
4101:⋆
3997:∗
3966:∗
3935:∗
3872:Hausdorff
3686:⋅
3625:
3588:‖
3574:‖
3565:∫
3530:‖
3521:−
3500:‖
3491:∫
3478:⋅
3469:
3434:
3274:⟩
3253:∗
3240:⟨
3231:∫
3169:‖
3162:∧
3156:‖
3140:⟩
3134:∧
3115:⟨
3096:⟩
3067:⟨
3049:‖
3035:‖
3026:∫
2952:⟩
2946:∧
2891:∧
2842:⟨
2833:∫
2766:
2696:∧
2532:Ω
2416:‖
2402:‖
2393:∫
2377:
2319:ω
2314:∗
2303:ω
2176:⋆
2139:∗
2075:∗
2033:⋆
2013:⋆
2000:⋆
1997:±
1989:∗
1942:⟩
1933:∗
1914:⟨
1895:⟩
1872:⟨
1708:⟩
1696:⟨
1687:∫
1667:⟩
1654:⟨
1555:
1527:∗
1513:Λ
1509:⊗
1497:
1465:
1304:
1189:. Then a
1108:Lie group
846:α
818:→
813:β
805:∩
800:α
787:β
784:α
756:→
728:α
688:
677:α
660:Ω
656:∈
651:α
586:⋆
503:‖
489:‖
480:∫
464:
359:
309:Lie group
282:
226:to prove
149:Uhlenbeck
6146:523–615.
6036:See also
5936:and the
5689:and the
5483:gives a
5461:definite
4877:, to be
4492:and the
3896:holonomy
1596:. Since
1268:two-form
1060:oriented
5683:Hitchin
2045:is the
1779:is the
1056:compact
239:Physics
201:of the
163:physics
132:on the
90:on the
6269:
6030:Witten
5840:, and
5053:, and
4993:where
4313:, the
3894:whose
3380:is an
2519:is an
2226:) or (
2025:where
1430:Since
1365:is an
389:fields
338:, see
218:. The
185:for a
177:, the
98:where
6073:Notes
5852:, of
1054:be a
975:is a
627:to a
189:on a
120:with
92:(x,x)
6267:ISBN
5848:and
5733:and
5455:and
5190:and
4661:and
4589:and
4527:and
4317:and
4184:and
4018:and
3980:and
3728:the
2721:of (
1034:Let
549:The
255:and
247:and
173:and
165:and
111:dx⊗σ
81:dx⊗σ
79:The
5978:of
5645:on
5224:dim
5137:on
5033:of
4888:dim
4681:is
4620:.
4484:or
4426:or
3870:is
3846:or
3701:.
1834:of
1783:of
1544:of
1325:of
1270:on
1193:on
267:on
212:or
193:or
161:In
139:of
126:z=0
6307::
6190:^
6169:^
6123:^
6032:.
5940:.
5563:.
5421:3.
4721:,
4693:SU
4637:SU
4062:.
4011:.
3622:YM
3466:YM
3431:ad
2763:YM
2374:YM
2236:.
2192:0.
2093:0.
1552:ad
1494:ad
1462:ad
1427:.
1301:ad
1062:,
1058:,
685:ad
602:0.
565::
461:YM
384:.
230:.
6275:.
6089:.
6008:]
6005:1
6002:,
5999:0
5996:[
5912:1
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5906:2
5882:)
5878:R
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5871:3
5868:(
5864:L
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5819:4
5814:R
5786:4
5781:R
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5714:R
5693:.
5674:.
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5585:R
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5415:k
5412:8
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5389:k
5367:4
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5280:5
5277:=
5274:3
5268:8
5265:=
5262:)
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5253:S
5249:(
5239:1
5233:M
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5201:=
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5110:X
5107:(
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5098:H
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5071:(
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4956:+
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4913:=
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4859:)
4856:P
4853:(
4848:2
4844:c
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4752:H
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4742:P
4739:(
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4634:=
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4507:A
4477:2
4459:A
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4444:=
4439:A
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4399:=
4394:A
4390:F
4369:X
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4278:)
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4272:(
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4245:+
4237:=
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4228:(
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4157:.
4145:)
4142:X
4139:(
4134:2
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4120:X
4117:(
4112:2
4104::
4078:X
4046:X
3991:M
3960:B
3929:A
3906:G
3882:A
3856:M
3832:B
3808:M
3784:G
3778:/
3772:A
3767:=
3762:B
3740:P
3714:G
3689:A
3683:g
3672:1
3657:A
3634:)
3631:A
3628:(
3619:=
3614:g
3609:l
3606:o
3603:v
3598:d
3592:2
3582:A
3578:F
3569:X
3561:=
3556:g
3551:l
3548:o
3545:v
3540:d
3534:2
3524:1
3517:g
3511:A
3507:F
3503:g
3495:X
3487:=
3484:)
3481:A
3475:g
3472:(
3443:)
3440:P
3437:(
3411:P
3391:g
3361:1
3346:a
3326:A
3299:.
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3271:a
3268:,
3263:A
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3248:A
3244:d
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3227:2
3224:=
3212:0
3209:=
3206:t
3201:)
3195:g
3190:l
3187:o
3184:v
3179:d
3173:2
3165:a
3159:a
3151:4
3147:t
3143:+
3137:a
3131:a
3128:,
3123:A
3119:F
3110:2
3106:t
3102:2
3099:+
3093:a
3088:A
3084:d
3080:,
3075:A
3071:F
3064:t
3061:2
3058:+
3053:2
3043:A
3039:F
3030:X
3021:(
3013:t
3010:d
3006:d
3001:=
2989:0
2986:=
2983:t
2978:)
2972:g
2967:l
2964:o
2961:v
2956:d
2949:a
2943:a
2938:2
2934:t
2930:+
2927:a
2922:A
2918:d
2913:t
2910:+
2905:A
2901:F
2897:,
2894:a
2888:a
2883:2
2879:t
2875:+
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2867:A
2863:d
2858:t
2855:+
2850:A
2846:F
2837:X
2828:(
2820:t
2817:d
2813:d
2808:=
2799:0
2796:=
2793:t
2788:)
2784:)
2781:a
2778:t
2775:+
2772:A
2769:(
2759:(
2751:t
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2744:d
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2702:.
2699:a
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2688:2
2684:t
2680:+
2677:a
2672:A
2668:d
2664:t
2661:+
2656:A
2652:F
2648:=
2643:a
2640:t
2637:+
2634:A
2630:F
2606:A
2586:a
2583:t
2580:+
2577:A
2557:)
2552:g
2547:;
2544:P
2541:(
2536:1
2507:P
2485:A
2468:)
2466:3
2464:(
2447:.
2442:g
2437:l
2434:o
2431:v
2426:d
2420:2
2410:A
2406:F
2397:X
2389:=
2386:)
2383:A
2380:(
2337:.
2325:0
2322:=
2310:d
2306:=
2300:d
2273:0
2270:=
2265:A
2261:F
2255:A
2251:d
2229:2
2223:1
2213:)
2211:2
2209:(
2189:=
2184:A
2180:F
2171:A
2167:d
2134:A
2130:d
2114:)
2112:1
2110:(
2090:=
2085:A
2081:F
2070:A
2066:d
2008:A
2004:d
1994:=
1984:A
1980:d
1967:.
1951:2
1947:L
1938:t
1928:A
1924:d
1920:,
1917:s
1911:=
1904:2
1900:L
1891:t
1888:,
1885:s
1880:A
1876:d
1847:A
1843:d
1816:2
1812:L
1791:X
1765:g
1761:l
1757:o
1754:v
1751:d
1726:g
1722:l
1718:o
1715:v
1712:d
1705:t
1702:,
1699:s
1691:X
1683:=
1676:2
1672:L
1663:t
1660:,
1657:s
1629:2
1625:L
1604:X
1584:X
1564:)
1561:P
1558:(
1532:X
1523:T
1517:2
1506:)
1503:P
1500:(
1474:)
1471:P
1468:(
1438:X
1403:G
1381:A
1377:d
1353:A
1333:P
1313:)
1310:P
1307:(
1278:X
1252:A
1248:F
1224:A
1201:P
1177:X
1157:G
1137:P
1117:G
1094:X
1074:G
1042:X
1007:2
1003:L
963:A
941:A
937:F
912:]
909:A
906:,
903:A
900:[
895:2
892:1
887:+
884:A
881:d
878:=
873:A
869:F
842:A
821:G
809:U
796:U
792::
780:g
759:X
753:P
733:}
724:U
720:{
700:)
697:)
694:P
691:(
682:,
673:U
669:(
664:1
647:A
599:=
594:A
590:F
581:A
577:d
534:.
529:g
524:l
521:o
518:v
513:d
507:2
497:A
493:F
484:X
476:=
473:)
470:A
467:(
436:A
432:F
403:A
368:)
365:1
362:(
356:U
353:=
350:G
318:G
291:)
288:1
285:(
279:U
155:.
153:S
145:R
141:R
137:S
130:z
118:A
113:3
102:3
100:σ
96:R
83:3
20:)
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