3302:
31:
40:
2722:
51:
3297:{\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}}}
60:
3633:
5547:
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
1014:
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.
4784:. 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
5543:
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.
611:
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
3450:
1954:
2446:
533:
1727:
699:
5759:
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
4977:
4782:
4277:
820:
1531:
760:. 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
5748:, 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
3785:
6134:
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),
5279:
624:, 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.
4144:
2701:
2727:
2012:
2556:
5881:
5440:
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
4648:
2324:
911:
367:
4822:
4704:
3442:
1563:
1473:
1312:
5120:
4607:
4576:
290:
6222:
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.
4461:
4413:
2191:
601:
4545:
4514:
3998:
3967:
3936:
5541:
5509:
2092:
5820:
5787:
5720:
5661:
5624:
5591:
5368:
5339:
732:
6007:
3857:
3833:
3809:
3715:
2486:
2338:
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.
4023:, 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
5742:
2272:
847:
4864:
5917:
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
2140:
5076:
5016:
3688:
1766:
5961:
5177:
4300:
2032:
758:
5420:
5310:
1848:
1817:
1630:
1382:
1253:
1008:
942:
437:
3628:{\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)}
2585:
5911:
5560:
Through the process of dimensional reduction, the Yang–Mills equations may be used to derive other important equations in differential geometry and gauge theory.
5203:
5391:
4194:
5470:
5144:
5040:
4668:
4368:
4348:
4328:
4171:
4077:
4045:
3905:
3881:
3739:
3656:
3410:
3390:
3345:
3325:
2605:
2506:
1790:
1603:
1583:
1437:
1402:
1352:
1332:
1277:
1223:
1200:
1176:
1156:
1136:
1116:
1093:
1073:
1041:
962:
402:
317:
1856:
376:
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
2358:
445:
1638:
240:
developed (essentially independent of the mathematical literature) the theory of principal bundles and connections in order to explain the concept of
630:
6174:
Donaldson, S. K. (1986). Connections, cohomology and the intersection forms of 4-manifolds. Journal of
Differential Geometry, 24(3), 275–341.
822:
are constant functions. Not every bundle is flat, so this is not possible in general. Instead one might ask that the local connection forms
4872:
5833:
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
4713:
4202:
6165:
Donaldson, S. K. (1983). An application of gauge theory to four-dimensional topology. Journal of
Differential Geometry, 18(2), 279–315.
6050:
763:
6240:
Donaldson, S. K. (1984). Nahm's equations and the classification of monopoles. Communications in
Mathematical Physics, 96(3), 387–408.
6195:
Taubes, C. H. (1982). Self-dual Yang–Mills connections on non-self-dual 4-manifolds. Journal of
Differential Geometry, 17(1), 139–170.
1478:
6283:
Axelrod, S., Della Pietra, S., & Witten, E. (1991). Geometric quantization of Chern Simons gauge theory. representations, 34, 39.
2041:
Assuming the above set up, the Yang–Mills equations are a system of (in general non-linear) partial differential equations given by
5922:
132:(bottom right). The BPST instanton is a solution to the anti-self duality equations, and therefore of the Yang–Mills equations, on
3744:
384:(change of local trivialisation of principal bundle), these physical fields must transform in precisely the way that a connection
6309:
6144:
Donaldson, S. K. (1983). A new proof of a theorem of
Narasimhan and Seshadri. Journal of Differential Geometry, 18(2), 269–277.
6153:
Friedman, R., & Morgan, J. W. (1998). Gauge theory and the topology of four-manifolds (Vol. 4). American
Mathematical Soc.
5208:
6098:
Yang, C.N. and Mills, R.L., 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical review, 96(1), p.191.
5753:
4085:
6274:
Hitchin, N. J. (1990). Flat connections and geometric quantization. Communications in mathematical physics, 131(2), 347–380.
6213:
Uhlenbeck, K. K. (1982). Removable singularities in Yang–Mills fields. Communications in
Mathematical Physics, 83(1), 11–29.
2613:
5370:
may be extended across the point at infinity using Uhlenbeck's removable singularity theorem. More generally, for positive
1203:
6045:
6055:
6204:
Uhlenbeck, K. K. (1982). Connections with L bounds on curvature. Communications in Mathematical Physics, 83(1), 31–42.
6259:
5964:
1963:
439:
of the connection, and the energy of the gauge field is given (up to a constant) by the Yang–Mills action functional
2515:
5844:
19:
This article discusses the Yang–Mills equations from a mathematical perspective. For the physics perspective, see
4785:
4020:
6231:
Hitchin, N. J. (1983). On the construction of monopoles. Communications in Mathematical Physics, 89(2), 145–190.
4612:
The moduli space of ASD connections, or instantons, was most intensively studied by Donaldson in the case where
4047:
is four. Here the Yang–Mills equations admit a simplification from a second-order PDE to a first-order PDE, the
6299:
5838:
5123:
122:
6107:
Pauli, W., 1941. Relativistic field theories of elementary particles. Reviews of Modern Physics, 13(3), p.203.
4615:
849:
are themselves constant. On a principal bundle the correct way to phrase this condition is that the curvature
296:
and others. The novelty of the work of Yang and Mills was to define gauge theories for an arbitrary choice of
6304:
6035:
5549:
2284:
1355:
1179:
852:
334:
171:
4791:
4677:
3415:
1536:
1446:
1285:
5081:
4581:
4550:
2707:
1011:
547:
263:
6125:
Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Oxford University Press.
4418:
5666:
By requiring the self-duality equations to be invariant under translation in two directions, one obtains
5597:
By requiring the anti-self-duality equations to be invariant under translations in a single direction of
4373:
2150:
560:
187:
4519:
4488:
3972:
3941:
3910:
6030:
5514:
5482:
2275:
2049:
979:
The best one can hope for is then to ask that instead of vanishing curvature, the bundle has curvature
163:
5796:
5763:
5696:
5637:
5600:
5567:
5344:
5315:
3718:
704:
539:
328:
5974:
5452:) the moduli space of ASD instantons on a smooth, compact, oriented, simply-connected four-manifold
4016:
3838:
3814:
3790:
3696:
2467:
369:
corresponding to electromagnetism, and the right framework to discuss such objects is the theory of
6186:
Donaldson, S. K. (1990). Polynomial invariants for smooth four-manifolds. Topology, 29(3), 257–315.
5477:
233:
175:
5725:
5685:
By requiring the anti-self-duality equations to be invariant in three directions, one obtains the
4027:. Moduli of Yang–Mills connections have been most studied when the dimension of the base manifold
2234:
5968:
5938:
5914:
5449:
4003:
Moduli spaces of Yang–Mills connections have been intensively studied in specific circumstances.
825:
4827:
6010:
5667:
5435:
3863:
or a smooth manifold. However, by restricting to irreducible connections, that is, connections
2113:
1769:
1413:
621:
257:
216:
5045:
4985:
3667:
1735:
976:
to the existence of flat connections: not every principal bundle can have a flat connection.
30:
5946:
5149:
914:
248:
as it applies to physical theories. The gauge theories Yang and Mills discovered, now called
159:
117:(bottom left). A visual representation of the field strength of a BPST instanton with center
39:
20:
4285:
2017:
737:
105:
coefficient (top right). These coefficients determine the restriction of the BPST instanton
5834:
5830:
5448:, Donaldson was able to show that in specific circumstances (when the intersection form is
5396:
5288:
1826:
1795:
1608:
1360:
1231:
1055:. The Yang–Mills equations can be phrased for a connection on a vector bundle or principal
986:
920:
415:
3693:
There is a moduli space of Yang–Mills connections modulo gauge transformations. Denote by
2561:
8:
5926:
5890:
5627:
5182:
2035:
1405:
1052:
617:
543:
253:
5373:
4176:
3327:
is a critical point of the Yang–Mills functional if and only if this vanishes for every
5884:
5631:
5455:
5129:
5025:
4653:
4353:
4333:
4313:
4156:
4062:
4030:
3890:
3866:
3724:
3641:
3395:
3375:
3330:
3310:
2590:
2491:
2335:
1775:
1588:
1568:
1422:
1387:
1337:
1317:
1262:
1208:
1185:
1161:
1141:
1121:
1101:
1078:
1058:
1026:
973:
947:
387:
302:
4024:
6255:
5826:
5790:
1949:{\displaystyle \langle d_{A}s,t\rangle _{L^{2}}=\langle s,d_{A}^{*}t\rangle _{L^{2}}}
1256:
5593:, and imposing that the solutions be invariant under a symmetry group. For example:
4079:
is four, a coincidence occurs: the Hodge star operator maps two-forms to two-forms,
3907:, one does obtain Hausdorff spaces. The space of irreducible connections is denoted
983:. The Yang–Mills action functional described above is precisely (the square of) the
6040:
5693:
There is a duality between solutions of the dimensionally reduced ASD equations on
4671:
2441:{\displaystyle \operatorname {YM} (A)=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}.}
2229:
1820:
1440:
528:{\displaystyle \operatorname {YM} (A)=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}.}
370:
183:
5564:
is the process of taking the Yang–Mills equations over a four-manifold, typically
4469:), the connection is a Yang–Mills connection. These connections are called either
1722:{\displaystyle \langle s,t\rangle _{L^{2}}=\int _{X}\langle s,t\rangle \,dvol_{g}}
140:'s removable singularity theorem to a topologically non-trivial ASD connection on
5445:
5441:
4012:
3860:
3362:
212:
137:
5686:
5679:
5282:
4004:
1280:
1226:
613:
293:
237:
76:
3787:
classifies all connections modulo gauge transformations, and the moduli space
694:{\displaystyle A_{\alpha }\in \Omega ^{1}(U_{\alpha },\operatorname {ad} (P))}
50:
6293:
6018:
6014:
5671:
5476:
between a copy of the manifold itself, and a disjoint union of copies of the
1409:
1048:
1044:
179:
5749:
5675:
5019:
3370:
2509:
2331:
208:
95:
1732:
where inside the integral the fiber-wise inner product is being used, and
331:
for more details). This group could be non-Abelian as opposed to the case
5918:
5745:
4707:
4150:
The Hodge star operator squares to the identity in this case, and so has
969:
155:
5943:
The moduli space of Yang–Mills equations over a compact Riemann surface
4485:. The spaces of self-dual and anti-self-dual connections are denoted by
4151:
4008:
2330:
In this sense the search for Yang–Mills connections can be compared to
5205:, the intersection form is trivial and the moduli space has dimension
1010:-norm of the curvature, and its Euler–Lagrange equations describe the
5473:
4972:{\displaystyle \dim {\mathcal {M}}_{k}^{-}=8k-3(1-b_{1}(X)+b_{+}(X))}
1096:
297:
203:
6075:
2274:, so Yang–Mills connections can be seen as a non-linear analogue of
2110:
Since the Hodge star is an isomorphism, by the explicit formula for
4777:{\displaystyle c_{2}(P)\in H^{4}(X,\mathbb {Z} )\cong \mathbb {Z} }
3884:
2464:
To derive the equations from the functional, recall that the space
5825:
Symmetry reductions of the ASD equations also lead to a number of
4272:{\displaystyle \Omega ^{2}(X)=\Omega _{+}(X)\oplus \Omega _{-}(X)}
1225:
on the total space of the principal bundle. This connection has a
4015:. There the moduli space obtains an alternative description as a
2346:
The Yang–Mills equations are the Euler–Lagrange equations of the
815:{\displaystyle g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G}
151:
59:
5555:
1526:{\displaystyle \operatorname {ad} (P)\otimes \Lambda ^{2}T^{*}X}
113:
to this slice. The corresponding field strength centered around
5921:
and symmetry reduction scheme. Other such master theories are
16:
Partial differential equations whose solutions are instantons
3780:{\displaystyle {\mathcal {B}}={\mathcal {A}}/{\mathcal {G}}}
232:
In their foundational paper on the topic of gauge theories,
4011:
studied the Yang–Mills equations for bundles over compact
3811:
of Yang–Mills connections is a subset. In general neither
968:), then the underlying principal bundle must have trivial
3356:
5274:{\displaystyle \dim {\mathcal {M}}_{1}^{-}(S^{4})=8-3=5}
1632:-inner product on the sections of this bundle. Namely,
194:. They have also found significant use in mathematics.
4139:{\displaystyle \star :\Omega ^{2}(X)\to \Omega ^{2}(X)}
550:
of this functional, which are the Yang–Mills equations
5078:
is the dimension of the positive-definite subspace of
2696:{\displaystyle F_{A+ta}=F_{A}+td_{A}a+t^{2}a\wedge a.}
6254:. Oxford: Oxford University Press. pp. 151–154.
5977:
5949:
5893:
5847:
5799:
5793:
can be thought of as a duality between instantons on
5766:
5728:
5699:
5640:
5603:
5570:
5517:
5485:
5458:
5399:
5376:
5347:
5318:
5291:
5211:
5185:
5152:
5132:
5084:
5048:
5028:
4988:
4875:
4830:
4794:
4716:
4680:
4656:
4618:
4584:
4553:
4522:
4491:
4421:
4376:
4356:
4336:
4316:
4288:
4205:
4179:
4159:
4088:
4065:
4033:
3975:
3944:
3913:
3893:
3869:
3841:
3817:
3793:
3747:
3727:
3699:
3670:
3644:
3453:
3418:
3398:
3378:
3333:
3313:
2725:
2616:
2593:
2564:
2518:
2494:
2470:
2361:
2287:
2237:
2153:
2142:
the Yang–Mills equations can equivalently be written
2116:
2052:
2020:
1966:
1859:
1829:
1798:
1778:
1738:
1641:
1611:
1591:
1571:
1539:
1481:
1449:
1425:
1390:
1384:, defined on the adjoint bundle. Additionally, since
1363:
1340:
1320:
1288:
1265:
1234:
1211:
1188:
1164:
1144:
1124:
1104:
1081:
1061:
1029:
989:
950:
923:
855:
828:
766:
740:
707:
633:
563:
448:
418:
390:
337:
305:
266:
5789:, instantons on dual four-dimensional tori, and the
2607:
in this affine space, the curvatures are related by
1443:, and combined with the invariant inner product on
6001:
5955:
5905:
5875:
5814:
5781:
5736:
5714:
5655:
5618:
5585:
5535:
5503:
5464:
5414:
5385:
5362:
5333:
5312:up to a 5 parameter family defining its centre in
5304:
5273:
5197:
5171:
5138:
5114:
5070:
5034:
5010:
4971:
4858:
4816:
4776:
4698:
4662:
4642:
4601:
4570:
4539:
4508:
4455:
4407:
4362:
4342:
4322:
4294:
4271:
4188:
4165:
4138:
4071:
4039:
3992:
3961:
3930:
3899:
3875:
3851:
3827:
3803:
3779:
3733:
3709:
3682:
3650:
3627:
3444:is invariant, the Yang–Mills functional satisfies
3436:
3404:
3384:
3339:
3319:
3296:
2695:
2599:
2579:
2550:
2500:
2480:
2440:
2318:
2266:
2185:
2134:
2086:
2026:
2006:
1948:
1842:
1811:
1784:
1760:
1721:
1624:
1597:
1577:
1557:
1525:
1467:
1431:
1396:
1376:
1346:
1326:
1306:
1271:
1247:
1217:
1194:
1170:
1150:
1130:
1110:
1087:
1067:
1035:
1002:
956:
936:
905:
841:
814:
752:
726:
693:
595:
527:
431:
396:
361:
311:
284:
5674:. These equations naturally lead to the study of
6291:
4048:
1439:is Riemannian, there is an inner product on the
546:for this physical theory should be given by the
2334:, which seeks a harmonic representative in the
2007:{\displaystyle d_{A}^{*}=\pm \star d_{A}\star }
1118:. Here the latter convention is presented. Let
4282:into the positive and negative eigenspaces of
4054:
2551:{\displaystyle \Omega ^{1}(P;{\mathfrak {g}})}
260:, which had been phrased in the language of a
5876:{\displaystyle \mathrm {SL} (3,\mathbb {R} )}
5822:and dual algebraic data over a single point.
5556:Dimensional reduction and other moduli spaces
2228:Every connection automatically satisfies the
3576:
3562:
3518:
3488:
3262:
3228:
3157:
3144:
3128:
3103:
3084:
3055:
3037:
3023:
2940:
2830:
2404:
2390:
1930:
1902:
1883:
1860:
1696:
1684:
1655:
1642:
721:
708:
491:
477:
4824:, the moduli space of ASD connections when
627:A connection is defined by its local forms
4196:. In particular, there is a decomposition
4017:moduli space of holomorphic vector bundles
6009:. In this case the moduli space admits a
5866:
5802:
5769:
5730:
5702:
5643:
5606:
5573:
5523:
5520:
5491:
5488:
5350:
5321:
5105:
4770:
4759:
3585:
3527:
3265:
3166:
2943:
2904:
2849:
2413:
1699:
500:
6249:
6243:
4788:, one may compute that the dimension of
4643:{\displaystyle G=\operatorname {SU} (2)}
4059:When the dimension of the base manifold
1475:there is an inner product on the bundle
408:) on a principal bundle transforms. The
6074:For a proof of this fact, see the post
6051:Deformed Hermitian Yang–Mills equations
5511:. We can count the number of copies of
5285:, which is the unique ASD instanton on
3938:, and so the moduli spaces are denoted
2319:{\displaystyle d\omega =d^{*}\omega =0}
906:{\displaystyle F_{A}=dA+{\frac {1}{2}}}
362:{\displaystyle G=\operatorname {U} (1)}
6292:
6182:
6180:
6161:
6159:
5932:
5429:
4817:{\displaystyle {\mathcal {M}}_{k}^{-}}
4699:{\displaystyle \operatorname {SU} (2)}
3437:{\displaystyle \operatorname {ad} (P)}
3357:Moduli space of Yang–Mills connections
1558:{\displaystyle \operatorname {ad} (P)}
1468:{\displaystyle \operatorname {ad} (P)}
1307:{\displaystyle \operatorname {ad} (P)}
620:, there is such a natural choice, the
197:Solutions of the equations are called
5115:{\displaystyle H_{2}(X,\mathbb {R} )}
4602:{\displaystyle {\mathcal {M}}^{\pm }}
4571:{\displaystyle {\mathcal {B}}^{\pm }}
285:{\displaystyle \operatorname {U} (1)}
6121:
6119:
6117:
6115:
6113:
5923:four-dimensional Chern–Simons theory
5281:. This agrees with existence of the
4706:-bundle is classified by its second
4456:{\displaystyle F_{A}=-{\star F_{A}}}
2352:
2144:
2043:
1204:Lie algebra-valued differential form
252:, generalised the classical work of
6177:
6156:
4408:{\displaystyle F_{A}={\star F_{A}}}
2540:
2186:{\displaystyle d_{A}\star F_{A}=0.}
596:{\displaystyle d_{A}\star F_{A}=0.}
136:. This solution can be extended by
13:
6252:Solitons, instantons, and twistors
5978:
5950:
5852:
5849:
5341:and its scale. Such instantons on
5221:
4885:
4798:
4588:
4557:
4540:{\displaystyle {\mathcal {A}}^{-}}
4526:
4509:{\displaystyle {\mathcal {A}}^{+}}
4495:
4251:
4229:
4207:
4118:
4096:
3993:{\displaystyle {\mathcal {M}}^{*}}
3979:
3962:{\displaystyle {\mathcal {B}}^{*}}
3948:
3931:{\displaystyle {\mathcal {A}}^{*}}
3917:
3844:
3820:
3796:
3772:
3760:
3750:
3702:
3597:
3594:
3591:
3539:
3536:
3533:
3347:, and this occurs precisely when (
3277:
3274:
3271:
3178:
3175:
3172:
2955:
2952:
2949:
2520:
2473:
2425:
2422:
2419:
1501:
648:
512:
509:
506:
344:
267:
14:
6321:
6110:
6076:https://mathoverflow.net/a/265399
5744:called the Nahm transform, after
5536:{\displaystyle \mathbb {CP} ^{2}}
5504:{\displaystyle \mathbb {CP} ^{2}}
4674:. In this setting, the principal
4483:anti-self-duality (ASD) equations
3412:, and since the inner product on
2087:{\displaystyle d_{A}^{*}F_{A}=0.}
5887:, and a particular reduction to
5815:{\displaystyle \mathbb {R} ^{4}}
5782:{\displaystyle \mathbb {R} ^{4}}
5715:{\displaystyle \mathbb {R} ^{3}}
5656:{\displaystyle \mathbb {R} ^{3}}
5619:{\displaystyle \mathbb {R} ^{4}}
5586:{\displaystyle \mathbb {R} ^{4}}
5363:{\displaystyle \mathbb {R} ^{4}}
5334:{\displaystyle \mathbb {R} ^{4}}
58:
49:
38:
29:
6277:
6268:
6234:
6225:
6216:
6207:
6198:
6189:
6068:
5424:
5393:the moduli space has dimension
1334:. Associated to the connection
727:{\displaystyle \{U_{\alpha }\}}
186:. They arise in physics as the
6310:Partial differential equations
6168:
6147:
6138:
6128:
6101:
6092:
6046:Hermitian Yang–Mills equations
6013:, discovered independently by
6002:{\displaystyle \Sigma \times }
5996:
5984:
5870:
5856:
5250:
5237:
5109:
5095:
5065:
5059:
5005:
4999:
4966:
4963:
4957:
4941:
4935:
4916:
4847:
4841:
4763:
4749:
4733:
4727:
4693:
4687:
4637:
4631:
4266:
4260:
4244:
4238:
4222:
4216:
4133:
4127:
4114:
4111:
4105:
3852:{\displaystyle {\mathcal {M}}}
3828:{\displaystyle {\mathcal {B}}}
3804:{\displaystyle {\mathcal {M}}}
3710:{\displaystyle {\mathcal {G}}}
3622:
3616:
3472:
3460:
3431:
3425:
2772:
2757:
2545:
2529:
2481:{\displaystyle {\mathcal {A}}}
2374:
2368:
1552:
1546:
1494:
1488:
1462:
1456:
1301:
1295:
900:
888:
806:
744:
701:for a trivialising open cover
688:
685:
679:
657:
606:
551:
461:
455:
356:
350:
279:
273:
172:partial differential equations
1:
6085:
6036:Connection (principal bundle)
4350:-bundle over a four-manifold
3361:The Yang–Mills equations are
2512:modelled on the vector space
2341:
1356:exterior covariant derivative
1018:
222:
5737:{\displaystyle \mathbb {R} }
2558:. Given a small deformation
2267:{\displaystyle d_{A}F_{A}=0}
1960:Explicitly this is given by
192:Yang–Mills action functional
7:
6024:
4786:Atiyah–Singer index theorem
4479:self-duality (SD) equations
4465:
4310:two-forms. If a connection
4055:Anti-self-duality equations
4049:anti-self-duality equations
4021:Narasimhan–Seshadri theorem
3660:
3349:
2712:
2454:
2276:harmonic differential forms
2217:
2211:
2199:
2100:
1819:-inner product, the formal
1404:is compact, its associated
842:{\displaystyle A_{\alpha }}
380:, and derives that under a
10:
6326:
6056:Yang–Mills–Higgs equations
6031:Connection (vector bundle)
5936:
5839:Korteweg–de Vries equation
5433:
4859:{\displaystyle c_{2}(P)=k}
4475:anti-self-dual connections
944:vanishes (that is to say,
542:dictates that the correct
382:local gauge transformation
227:
211:of instantons was used by
18:
6250:Dunajski, Maciej (2010).
6017:and Axelrod–Della Pietra–
5550:Seiberg–Witten invariants
3887:group is given by all of
2209:A connection satisfying (
2135:{\displaystyle d_{A}^{*}}
1605:is oriented, there is an
540:principle of least action
329:Gauge group (mathematics)
6061:
5478:complex projective plane
5071:{\displaystyle b_{+}(X)}
5011:{\displaystyle b_{1}(X)}
4477:, and the equations the
3683:{\displaystyle g\cdot A}
3392:of the principal bundle
1761:{\displaystyle dvol_{g}}
548:Euler–Lagrange equations
188:Euler–Lagrange equations
5956:{\displaystyle \Sigma }
5915:integrable chiral model
5754:complex projective line
5172:{\displaystyle X=S^{4}}
974:topological obstruction
6011:geometric quantization
6003:
5957:
5907:
5877:
5816:
5783:
5738:
5716:
5670:first investigated by
5657:
5620:
5587:
5537:
5505:
5466:
5416:
5387:
5364:
5335:
5306:
5275:
5199:
5173:
5140:
5116:
5072:
5036:
5012:
4973:
4860:
4818:
4778:
4700:
4664:
4644:
4603:
4572:
4541:
4510:
4457:
4409:
4364:
4344:
4324:
4296:
4295:{\displaystyle \star }
4273:
4190:
4167:
4140:
4073:
4041:
3994:
3963:
3932:
3901:
3877:
3853:
3829:
3805:
3781:
3735:
3711:
3684:
3652:
3629:
3438:
3406:
3386:
3341:
3321:
3298:
2697:
2601:
2581:
2552:
2502:
2488:of all connections on
2482:
2442:
2320:
2268:
2187:
2136:
2088:
2028:
2027:{\displaystyle \star }
2008:
1950:
1844:
1813:
1786:
1770:Riemannian volume form
1762:
1723:
1626:
1599:
1579:
1559:
1527:
1469:
1433:
1414:adjoint representation
1398:
1378:
1348:
1328:
1308:
1273:
1249:
1219:
1202:may be specified by a
1196:
1172:
1152:
1132:
1112:
1089:
1069:
1037:
1004:
958:
938:
913:vanishes. However, by
907:
843:
816:
754:
753:{\displaystyle P\to X}
728:
695:
622:Levi-Civita connection
597:
529:
433:
398:
363:
313:
286:
199:Yang–Mills connections
6300:Differential geometry
6004:
5963:can be viewed as the
5958:
5913:dimensions gives the
5908:
5878:
5817:
5784:
5739:
5717:
5658:
5621:
5588:
5562:Dimensional reduction
5538:
5506:
5467:
5417:
5415:{\displaystyle 8k-3.}
5388:
5365:
5336:
5307:
5305:{\displaystyle S^{4}}
5276:
5200:
5174:
5141:
5117:
5073:
5037:
5013:
4974:
4861:
4819:
4779:
4701:
4665:
4645:
4604:
4573:
4542:
4511:
4471:self-dual connections
4458:
4410:
4365:
4345:
4325:
4297:
4274:
4191:
4168:
4141:
4074:
4042:
3995:
3964:
3933:
3902:
3878:
3854:
3830:
3806:
3782:
3736:
3712:
3685:
3653:
3630:
3439:
3407:
3387:
3342:
3322:
3299:
2698:
2602:
2582:
2553:
2503:
2483:
2443:
2348:Yang–Mills functional
2321:
2269:
2223:Yang–Mills connection
2188:
2137:
2089:
2038:acting on two-forms.
2029:
2009:
1951:
1845:
1843:{\displaystyle d_{A}}
1814:
1812:{\displaystyle L^{2}}
1787:
1763:
1724:
1627:
1625:{\displaystyle L^{2}}
1600:
1580:
1565:-valued two-forms on
1560:
1528:
1470:
1434:
1399:
1379:
1377:{\displaystyle d_{A}}
1349:
1329:
1309:
1274:
1250:
1248:{\displaystyle F_{A}}
1220:
1197:
1173:
1153:
1133:
1113:
1090:
1070:
1038:
1005:
1003:{\displaystyle L^{2}}
959:
939:
937:{\displaystyle F_{A}}
908:
844:
817:
755:
729:
696:
598:
530:
434:
432:{\displaystyle F_{A}}
399:
364:
314:
287:
160:differential geometry
6305:Mathematical physics
5975:
5947:
5891:
5845:
5797:
5764:
5726:
5697:
5638:
5601:
5568:
5515:
5483:
5456:
5397:
5374:
5345:
5316:
5289:
5209:
5183:
5150:
5146:. For example, when
5130:
5122:with respect to the
5082:
5046:
5026:
4986:
4873:
4828:
4792:
4714:
4678:
4654:
4616:
4582:
4551:
4547:, and similarly for
4520:
4489:
4419:
4374:
4354:
4334:
4314:
4286:
4203:
4177:
4157:
4086:
4063:
4031:
3973:
3942:
3911:
3891:
3867:
3839:
3815:
3791:
3745:
3725:
3721:of automorphisms of
3697:
3668:
3642:
3451:
3416:
3396:
3376:
3367:gauge transformation
3365:. Mathematically, a
3331:
3311:
2723:
2614:
2591:
2580:{\displaystyle A+ta}
2562:
2516:
2492:
2468:
2359:
2285:
2235:
2151:
2114:
2050:
2018:
1964:
1857:
1827:
1796:
1776:
1736:
1639:
1609:
1589:
1569:
1537:
1479:
1447:
1423:
1408:admits an invariant
1388:
1361:
1338:
1318:
1286:
1263:
1232:
1209:
1186:
1162:
1142:
1122:
1102:
1079:
1059:
1027:
987:
981:as small as possible
948:
921:
853:
826:
764:
738:
705:
631:
561:
446:
416:
410:gauge field strength
388:
335:
303:
264:
168:Yang–Mills equations
5969:Chern–Simons theory
5965:configuration space
5939:Chern–Simons theory
5933:Chern–Simons theory
5927:affine Gaudin model
5906:{\displaystyle 2+1}
5668:Hitchin's equations
5628:Bogomolny equations
5436:Donaldson's theorem
5430:Donaldson's theorem
5236:
5198:{\displaystyle k=1}
4900:
4813:
3245:
2131:
2067:
2036:Hodge star operator
1981:
1925:
1406:compact Lie algebra
1279:with values in the
1138:denote a principal
1095:, for some compact
1053:Riemannian manifold
618:Riemannian manifold
544:equations of motion
323:(or in physics the
258:Maxwell's equations
250:Yang–Mills theories
217:Donaldson's theorem
5999:
5953:
5903:
5885:Tzitzeica equation
5873:
5827:integrable systems
5812:
5779:
5734:
5712:
5653:
5632:magnetic monopoles
5626:, one obtains the
5616:
5583:
5533:
5501:
5462:
5412:
5386:{\displaystyle k,}
5383:
5360:
5331:
5302:
5271:
5218:
5195:
5169:
5136:
5112:
5068:
5032:
5008:
4969:
4882:
4856:
4814:
4795:
4774:
4696:
4660:
4640:
4599:
4568:
4537:
4506:
4453:
4405:
4360:
4340:
4320:
4292:
4269:
4189:{\displaystyle -1}
4186:
4163:
4136:
4069:
4037:
3990:
3959:
3928:
3897:
3873:
3849:
3825:
3801:
3777:
3731:
3707:
3680:
3648:
3625:
3434:
3402:
3382:
3337:
3317:
3294:
3292:
3231:
2693:
2597:
2577:
2548:
2498:
2478:
2438:
2336:de Rham cohomology
2316:
2264:
2183:
2132:
2117:
2084:
2053:
2024:
2004:
1967:
1946:
1911:
1840:
1809:
1782:
1758:
1719:
1622:
1595:
1575:
1555:
1523:
1465:
1429:
1394:
1374:
1344:
1324:
1304:
1269:
1245:
1215:
1192:
1168:
1148:
1128:
1108:
1085:
1065:
1033:
1000:
954:
934:
903:
839:
812:
750:
724:
691:
593:
525:
429:
394:
359:
309:
282:
5831:Ward's conjecture
5791:ADHM construction
5465:{\displaystyle X}
5139:{\displaystyle X}
5124:intersection form
5035:{\displaystyle X}
4663:{\displaystyle X}
4370:satisfies either
4363:{\displaystyle X}
4343:{\displaystyle G}
4323:{\displaystyle A}
4166:{\displaystyle 1}
4072:{\displaystyle X}
4040:{\displaystyle X}
3900:{\displaystyle G}
3876:{\displaystyle A}
3734:{\displaystyle P}
3651:{\displaystyle A}
3405:{\displaystyle P}
3385:{\displaystyle g}
3340:{\displaystyle a}
3320:{\displaystyle A}
3005:
2812:
2743:
2706:To determine the
2600:{\displaystyle A}
2501:{\displaystyle P}
2462:
2461:
2207:
2206:
2108:
2107:
1785:{\displaystyle X}
1598:{\displaystyle X}
1578:{\displaystyle X}
1432:{\displaystyle X}
1397:{\displaystyle G}
1347:{\displaystyle A}
1327:{\displaystyle P}
1272:{\displaystyle X}
1218:{\displaystyle A}
1195:{\displaystyle P}
1171:{\displaystyle X}
1151:{\displaystyle G}
1131:{\displaystyle P}
1111:{\displaystyle G}
1088:{\displaystyle X}
1068:{\displaystyle G}
1036:{\displaystyle X}
957:{\displaystyle A}
917:if the curvature
915:Chern–Weil theory
886:
412:is the curvature
397:{\displaystyle A}
371:principal bundles
312:{\displaystyle G}
158:, and especially
75:coefficient of a
21:Yang–Mills theory
6317:
6284:
6281:
6275:
6272:
6266:
6265:
6247:
6241:
6238:
6232:
6229:
6223:
6220:
6214:
6211:
6205:
6202:
6196:
6193:
6187:
6184:
6175:
6172:
6166:
6163:
6154:
6151:
6145:
6142:
6136:
6132:
6126:
6123:
6108:
6105:
6099:
6096:
6079:
6072:
6041:Donaldson theory
6008:
6006:
6005:
6000:
5962:
5960:
5959:
5954:
5912:
5910:
5909:
5904:
5883:ASDYM gives the
5882:
5880:
5879:
5874:
5869:
5855:
5821:
5819:
5818:
5813:
5811:
5810:
5805:
5788:
5786:
5785:
5780:
5778:
5777:
5772:
5743:
5741:
5740:
5735:
5733:
5721:
5719:
5718:
5713:
5711:
5710:
5705:
5662:
5660:
5659:
5654:
5652:
5651:
5646:
5625:
5623:
5622:
5617:
5615:
5614:
5609:
5592:
5590:
5589:
5584:
5582:
5581:
5576:
5542:
5540:
5539:
5534:
5532:
5531:
5526:
5510:
5508:
5507:
5502:
5500:
5499:
5494:
5471:
5469:
5468:
5463:
5421:
5419:
5418:
5413:
5392:
5390:
5389:
5384:
5369:
5367:
5366:
5361:
5359:
5358:
5353:
5340:
5338:
5337:
5332:
5330:
5329:
5324:
5311:
5309:
5308:
5303:
5301:
5300:
5280:
5278:
5277:
5272:
5249:
5248:
5235:
5230:
5225:
5224:
5204:
5202:
5201:
5196:
5178:
5176:
5175:
5170:
5168:
5167:
5145:
5143:
5142:
5137:
5121:
5119:
5118:
5113:
5108:
5094:
5093:
5077:
5075:
5074:
5069:
5058:
5057:
5041:
5039:
5038:
5033:
5017:
5015:
5014:
5009:
4998:
4997:
4978:
4976:
4975:
4970:
4956:
4955:
4934:
4933:
4899:
4894:
4889:
4888:
4865:
4863:
4862:
4857:
4840:
4839:
4823:
4821:
4820:
4815:
4812:
4807:
4802:
4801:
4783:
4781:
4780:
4775:
4773:
4762:
4748:
4747:
4726:
4725:
4705:
4703:
4702:
4697:
4672:simply-connected
4669:
4667:
4666:
4661:
4649:
4647:
4646:
4641:
4608:
4606:
4605:
4600:
4598:
4597:
4592:
4591:
4577:
4575:
4574:
4569:
4567:
4566:
4561:
4560:
4546:
4544:
4543:
4538:
4536:
4535:
4530:
4529:
4515:
4513:
4512:
4507:
4505:
4504:
4499:
4498:
4462:
4460:
4459:
4454:
4452:
4451:
4450:
4431:
4430:
4414:
4412:
4411:
4406:
4404:
4403:
4402:
4386:
4385:
4369:
4367:
4366:
4361:
4349:
4347:
4346:
4341:
4329:
4327:
4326:
4321:
4301:
4299:
4298:
4293:
4278:
4276:
4275:
4270:
4259:
4258:
4237:
4236:
4215:
4214:
4195:
4193:
4192:
4187:
4172:
4170:
4169:
4164:
4145:
4143:
4142:
4137:
4126:
4125:
4104:
4103:
4078:
4076:
4075:
4070:
4046:
4044:
4043:
4038:
4013:Riemann surfaces
3999:
3997:
3996:
3991:
3989:
3988:
3983:
3982:
3968:
3966:
3965:
3960:
3958:
3957:
3952:
3951:
3937:
3935:
3934:
3929:
3927:
3926:
3921:
3920:
3906:
3904:
3903:
3898:
3882:
3880:
3879:
3874:
3858:
3856:
3855:
3850:
3848:
3847:
3834:
3832:
3831:
3826:
3824:
3823:
3810:
3808:
3807:
3802:
3800:
3799:
3786:
3784:
3783:
3778:
3776:
3775:
3769:
3764:
3763:
3754:
3753:
3740:
3738:
3737:
3732:
3716:
3714:
3713:
3708:
3706:
3705:
3689:
3687:
3686:
3681:
3657:
3655:
3654:
3649:
3634:
3632:
3631:
3626:
3606:
3605:
3600:
3584:
3583:
3574:
3573:
3561:
3560:
3548:
3547:
3542:
3526:
3525:
3516:
3515:
3503:
3502:
3487:
3486:
3443:
3441:
3440:
3435:
3411:
3409:
3408:
3403:
3391:
3389:
3388:
3383:
3353:) is satisfied.
3346:
3344:
3343:
3338:
3326:
3324:
3323:
3318:
3303:
3301:
3300:
3295:
3293:
3286:
3285:
3280:
3255:
3254:
3244:
3239:
3227:
3226:
3208:
3204:
3203:
3192:
3188:
3187:
3186:
3181:
3165:
3164:
3143:
3142:
3115:
3114:
3102:
3101:
3080:
3079:
3067:
3066:
3045:
3044:
3035:
3034:
3022:
3021:
3006:
3004:
2993:
2985:
2981:
2980:
2969:
2965:
2964:
2963:
2958:
2930:
2929:
2914:
2913:
2897:
2896:
2875:
2874:
2859:
2858:
2842:
2841:
2829:
2828:
2813:
2811:
2800:
2791:
2790:
2779:
2775:
2744:
2742:
2731:
2702:
2700:
2699:
2694:
2680:
2679:
2664:
2663:
2648:
2647:
2635:
2634:
2606:
2604:
2603:
2598:
2587:of a connection
2586:
2584:
2583:
2578:
2557:
2555:
2554:
2549:
2544:
2543:
2528:
2527:
2507:
2505:
2504:
2499:
2487:
2485:
2484:
2479:
2477:
2476:
2456:
2447:
2445:
2444:
2439:
2434:
2433:
2428:
2412:
2411:
2402:
2401:
2389:
2388:
2353:
2325:
2323:
2322:
2317:
2306:
2305:
2278:, which satisfy
2273:
2271:
2270:
2265:
2257:
2256:
2247:
2246:
2230:Bianchi identity
2201:
2192:
2190:
2189:
2184:
2176:
2175:
2163:
2162:
2145:
2141:
2139:
2138:
2133:
2130:
2125:
2102:
2093:
2091:
2090:
2085:
2077:
2076:
2066:
2061:
2044:
2033:
2031:
2030:
2025:
2013:
2011:
2010:
2005:
2000:
1999:
1980:
1975:
1955:
1953:
1952:
1947:
1945:
1944:
1943:
1942:
1924:
1919:
1898:
1897:
1896:
1895:
1872:
1871:
1849:
1847:
1846:
1841:
1839:
1838:
1821:adjoint operator
1818:
1816:
1815:
1810:
1808:
1807:
1791:
1789:
1788:
1783:
1767:
1765:
1764:
1759:
1757:
1756:
1728:
1726:
1725:
1720:
1718:
1717:
1683:
1682:
1670:
1669:
1668:
1667:
1631:
1629:
1628:
1623:
1621:
1620:
1604:
1602:
1601:
1596:
1584:
1582:
1581:
1576:
1564:
1562:
1561:
1556:
1532:
1530:
1529:
1524:
1519:
1518:
1509:
1508:
1474:
1472:
1471:
1466:
1441:cotangent bundle
1438:
1436:
1435:
1430:
1403:
1401:
1400:
1395:
1383:
1381:
1380:
1375:
1373:
1372:
1353:
1351:
1350:
1345:
1333:
1331:
1330:
1325:
1313:
1311:
1310:
1305:
1278:
1276:
1275:
1270:
1254:
1252:
1251:
1246:
1244:
1243:
1224:
1222:
1221:
1216:
1201:
1199:
1198:
1193:
1177:
1175:
1174:
1169:
1157:
1155:
1154:
1149:
1137:
1135:
1134:
1129:
1117:
1115:
1114:
1109:
1094:
1092:
1091:
1086:
1074:
1072:
1071:
1066:
1042:
1040:
1039:
1034:
1009:
1007:
1006:
1001:
999:
998:
963:
961:
960:
955:
943:
941:
940:
935:
933:
932:
912:
910:
909:
904:
887:
879:
865:
864:
848:
846:
845:
840:
838:
837:
821:
819:
818:
813:
805:
804:
792:
791:
779:
778:
759:
757:
756:
751:
733:
731:
730:
725:
720:
719:
700:
698:
697:
692:
669:
668:
656:
655:
643:
642:
602:
600:
599:
594:
586:
585:
573:
572:
534:
532:
531:
526:
521:
520:
515:
499:
498:
489:
488:
476:
475:
438:
436:
435:
430:
428:
427:
403:
401:
400:
395:
368:
366:
365:
360:
318:
316:
315:
310:
292:gauge theory by
291:
289:
288:
283:
246:gauge invariance
184:principal bundle
170:are a system of
123:compactification
98:(top left). The
62:
53:
42:
33:
6325:
6324:
6320:
6319:
6318:
6316:
6315:
6314:
6290:
6289:
6288:
6287:
6282:
6278:
6273:
6269:
6262:
6248:
6244:
6239:
6235:
6230:
6226:
6221:
6217:
6212:
6208:
6203:
6199:
6194:
6190:
6185:
6178:
6173:
6169:
6164:
6157:
6152:
6148:
6143:
6139:
6133:
6129:
6124:
6111:
6106:
6102:
6097:
6093:
6088:
6083:
6082:
6073:
6069:
6064:
6027:
5976:
5973:
5972:
5948:
5945:
5944:
5941:
5935:
5892:
5889:
5888:
5865:
5848:
5846:
5843:
5842:
5806:
5801:
5800:
5798:
5795:
5794:
5773:
5768:
5767:
5765:
5762:
5761:
5729:
5727:
5724:
5723:
5706:
5701:
5700:
5698:
5695:
5694:
5689:on an interval.
5647:
5642:
5641:
5639:
5636:
5635:
5630:which describe
5610:
5605:
5604:
5602:
5599:
5598:
5577:
5572:
5571:
5569:
5566:
5565:
5558:
5527:
5519:
5518:
5516:
5513:
5512:
5495:
5487:
5486:
5484:
5481:
5480:
5457:
5454:
5453:
5446:Karen Uhlenbeck
5442:Clifford Taubes
5438:
5432:
5427:
5398:
5395:
5394:
5375:
5372:
5371:
5354:
5349:
5348:
5346:
5343:
5342:
5325:
5320:
5319:
5317:
5314:
5313:
5296:
5292:
5290:
5287:
5286:
5244:
5240:
5231:
5226:
5220:
5219:
5210:
5207:
5206:
5184:
5181:
5180:
5163:
5159:
5151:
5148:
5147:
5131:
5128:
5127:
5104:
5089:
5085:
5083:
5080:
5079:
5053:
5049:
5047:
5044:
5043:
5027:
5024:
5023:
4993:
4989:
4987:
4984:
4983:
4951:
4947:
4929:
4925:
4895:
4890:
4884:
4883:
4874:
4871:
4870:
4835:
4831:
4829:
4826:
4825:
4808:
4803:
4797:
4796:
4793:
4790:
4789:
4769:
4758:
4743:
4739:
4721:
4717:
4715:
4712:
4711:
4679:
4676:
4675:
4655:
4652:
4651:
4617:
4614:
4613:
4593:
4587:
4586:
4585:
4583:
4580:
4579:
4562:
4556:
4555:
4554:
4552:
4549:
4548:
4531:
4525:
4524:
4523:
4521:
4518:
4517:
4500:
4494:
4493:
4492:
4490:
4487:
4486:
4446:
4442:
4438:
4426:
4422:
4420:
4417:
4416:
4398:
4394:
4390:
4381:
4377:
4375:
4372:
4371:
4355:
4352:
4351:
4335:
4332:
4331:
4330:on a principal
4315:
4312:
4311:
4287:
4284:
4283:
4254:
4250:
4232:
4228:
4210:
4206:
4204:
4201:
4200:
4178:
4175:
4174:
4158:
4155:
4154:
4121:
4117:
4099:
4095:
4087:
4084:
4083:
4064:
4061:
4060:
4057:
4032:
4029:
4028:
4025:Kähler manifold
3984:
3978:
3977:
3976:
3974:
3971:
3970:
3953:
3947:
3946:
3945:
3943:
3940:
3939:
3922:
3916:
3915:
3914:
3912:
3909:
3908:
3892:
3889:
3888:
3868:
3865:
3864:
3843:
3842:
3840:
3837:
3836:
3819:
3818:
3816:
3813:
3812:
3795:
3794:
3792:
3789:
3788:
3771:
3770:
3765:
3759:
3758:
3749:
3748:
3746:
3743:
3742:
3726:
3723:
3722:
3701:
3700:
3698:
3695:
3694:
3669:
3666:
3665:
3643:
3640:
3639:
3601:
3590:
3589:
3579:
3575:
3569:
3565:
3556:
3552:
3543:
3532:
3531:
3521:
3517:
3508:
3504:
3498:
3494:
3482:
3478:
3452:
3449:
3448:
3417:
3414:
3413:
3397:
3394:
3393:
3377:
3374:
3373:
3363:gauge invariant
3359:
3332:
3329:
3328:
3312:
3309:
3308:
3307:The connection
3291:
3290:
3281:
3270:
3269:
3250:
3246:
3240:
3235:
3222:
3218:
3206:
3205:
3193:
3182:
3171:
3170:
3160:
3156:
3138:
3134:
3110:
3106:
3097:
3093:
3075:
3071:
3062:
3058:
3040:
3036:
3030:
3026:
3017:
3013:
3012:
3008:
3007:
2997:
2992:
2983:
2982:
2970:
2959:
2948:
2947:
2925:
2921:
2909:
2905:
2892:
2888:
2870:
2866:
2854:
2850:
2837:
2833:
2824:
2820:
2819:
2815:
2814:
2804:
2799:
2792:
2780:
2750:
2746:
2745:
2735:
2730:
2726:
2724:
2721:
2720:
2708:critical points
2675:
2671:
2659:
2655:
2643:
2639:
2621:
2617:
2615:
2612:
2611:
2592:
2589:
2588:
2563:
2560:
2559:
2539:
2538:
2523:
2519:
2517:
2514:
2513:
2493:
2490:
2489:
2472:
2471:
2469:
2466:
2465:
2429:
2418:
2417:
2407:
2403:
2397:
2393:
2384:
2380:
2360:
2357:
2356:
2344:
2301:
2297:
2286:
2283:
2282:
2252:
2248:
2242:
2238:
2236:
2233:
2232:
2171:
2167:
2158:
2154:
2152:
2149:
2148:
2126:
2121:
2115:
2112:
2111:
2072:
2068:
2062:
2057:
2051:
2048:
2047:
2019:
2016:
2015:
1995:
1991:
1976:
1971:
1965:
1962:
1961:
1938:
1934:
1933:
1929:
1920:
1915:
1891:
1887:
1886:
1882:
1867:
1863:
1858:
1855:
1854:
1834:
1830:
1828:
1825:
1824:
1803:
1799:
1797:
1794:
1793:
1777:
1774:
1773:
1752:
1748:
1737:
1734:
1733:
1713:
1709:
1678:
1674:
1663:
1659:
1658:
1654:
1640:
1637:
1636:
1616:
1612:
1610:
1607:
1606:
1590:
1587:
1586:
1570:
1567:
1566:
1538:
1535:
1534:
1514:
1510:
1504:
1500:
1480:
1477:
1476:
1448:
1445:
1444:
1424:
1421:
1420:
1389:
1386:
1385:
1368:
1364:
1362:
1359:
1358:
1339:
1336:
1335:
1319:
1316:
1315:
1287:
1284:
1283:
1264:
1261:
1260:
1239:
1235:
1233:
1230:
1229:
1210:
1207:
1206:
1187:
1184:
1183:
1163:
1160:
1159:
1143:
1140:
1139:
1123:
1120:
1119:
1103:
1100:
1099:
1080:
1077:
1076:
1060:
1057:
1056:
1028:
1025:
1024:
1021:
1012:critical points
994:
990:
988:
985:
984:
966:flat connection
949:
946:
945:
928:
924:
922:
919:
918:
878:
860:
856:
854:
851:
850:
833:
829:
827:
824:
823:
800:
796:
787:
783:
771:
767:
765:
762:
761:
739:
736:
735:
734:for the bundle
715:
711:
706:
703:
702:
664:
660:
651:
647:
638:
634:
632:
629:
628:
609:
581:
577:
568:
564:
562:
559:
558:
516:
505:
504:
494:
490:
484:
480:
471:
467:
447:
444:
443:
423:
419:
417:
414:
413:
404:(in physics, a
389:
386:
385:
336:
333:
332:
321:structure group
304:
301:
300:
265:
262:
261:
230:
225:
213:Simon Donaldson
148:
147:
146:
145:
103:
92:
73:
65:
64:
63:
55:
54:
45:
44:
43:
35:
34:
23:
17:
12:
11:
5:
6323:
6313:
6312:
6307:
6302:
6286:
6285:
6276:
6267:
6260:
6242:
6233:
6224:
6215:
6206:
6197:
6188:
6176:
6167:
6155:
6146:
6137:
6127:
6109:
6100:
6090:
6089:
6087:
6084:
6081:
6080:
6066:
6065:
6063:
6060:
6059:
6058:
6053:
6048:
6043:
6038:
6033:
6026:
6023:
5998:
5995:
5992:
5989:
5986:
5983:
5980:
5971:on a cylinder
5952:
5937:Main article:
5934:
5931:
5902:
5899:
5896:
5872:
5868:
5864:
5861:
5858:
5854:
5851:
5809:
5804:
5776:
5771:
5732:
5709:
5704:
5691:
5690:
5687:Nahm equations
5683:
5680:Hitchin system
5664:
5650:
5645:
5613:
5608:
5580:
5575:
5557:
5554:
5530:
5525:
5522:
5498:
5493:
5490:
5461:
5434:Main article:
5431:
5428:
5426:
5423:
5411:
5408:
5405:
5402:
5382:
5379:
5357:
5352:
5328:
5323:
5299:
5295:
5283:BPST instanton
5270:
5267:
5264:
5261:
5258:
5255:
5252:
5247:
5243:
5239:
5234:
5229:
5223:
5217:
5214:
5194:
5191:
5188:
5166:
5162:
5158:
5155:
5135:
5111:
5107:
5103:
5100:
5097:
5092:
5088:
5067:
5064:
5061:
5056:
5052:
5031:
5007:
5004:
5001:
4996:
4992:
4980:
4979:
4968:
4965:
4962:
4959:
4954:
4950:
4946:
4943:
4940:
4937:
4932:
4928:
4924:
4921:
4918:
4915:
4912:
4909:
4906:
4903:
4898:
4893:
4887:
4881:
4878:
4855:
4852:
4849:
4846:
4843:
4838:
4834:
4811:
4806:
4800:
4772:
4768:
4765:
4761:
4757:
4754:
4751:
4746:
4742:
4738:
4735:
4732:
4729:
4724:
4720:
4695:
4692:
4689:
4686:
4683:
4659:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4596:
4590:
4565:
4559:
4534:
4528:
4503:
4497:
4449:
4445:
4441:
4437:
4434:
4429:
4425:
4401:
4397:
4393:
4389:
4384:
4380:
4359:
4339:
4319:
4308:anti-self-dual
4291:
4280:
4279:
4268:
4265:
4262:
4257:
4253:
4249:
4246:
4243:
4240:
4235:
4231:
4227:
4224:
4221:
4218:
4213:
4209:
4185:
4182:
4162:
4148:
4147:
4135:
4132:
4129:
4124:
4120:
4116:
4113:
4110:
4107:
4102:
4098:
4094:
4091:
4068:
4056:
4053:
4036:
4019:. This is the
4005:Michael Atiyah
3987:
3981:
3956:
3950:
3925:
3919:
3896:
3872:
3846:
3822:
3798:
3774:
3768:
3762:
3757:
3752:
3730:
3704:
3679:
3676:
3673:
3647:
3636:
3635:
3624:
3621:
3618:
3615:
3612:
3609:
3604:
3599:
3596:
3593:
3588:
3582:
3578:
3572:
3568:
3564:
3559:
3555:
3551:
3546:
3541:
3538:
3535:
3530:
3524:
3520:
3514:
3511:
3507:
3501:
3497:
3493:
3490:
3485:
3481:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3433:
3430:
3427:
3424:
3421:
3401:
3381:
3358:
3355:
3336:
3316:
3305:
3304:
3289:
3284:
3279:
3276:
3273:
3268:
3264:
3261:
3258:
3253:
3249:
3243:
3238:
3234:
3230:
3225:
3221:
3217:
3214:
3211:
3209:
3207:
3202:
3199:
3196:
3191:
3185:
3180:
3177:
3174:
3169:
3163:
3159:
3155:
3152:
3149:
3146:
3141:
3137:
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3130:
3127:
3124:
3121:
3118:
3113:
3109:
3105:
3100:
3096:
3092:
3089:
3086:
3083:
3078:
3074:
3070:
3065:
3061:
3057:
3054:
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3048:
3043:
3039:
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3029:
3025:
3020:
3016:
3011:
3003:
3000:
2996:
2991:
2988:
2986:
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2979:
2976:
2973:
2968:
2962:
2957:
2954:
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2946:
2942:
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2928:
2924:
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2917:
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2903:
2900:
2895:
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2878:
2873:
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2865:
2862:
2857:
2853:
2848:
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2840:
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2823:
2818:
2810:
2807:
2803:
2798:
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2789:
2786:
2783:
2778:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2749:
2741:
2738:
2734:
2729:
2728:
2704:
2703:
2692:
2689:
2686:
2683:
2678:
2674:
2670:
2667:
2662:
2658:
2654:
2651:
2646:
2642:
2638:
2633:
2630:
2627:
2624:
2620:
2596:
2576:
2573:
2570:
2567:
2547:
2542:
2537:
2534:
2531:
2526:
2522:
2497:
2475:
2460:
2459:
2450:
2448:
2437:
2432:
2427:
2424:
2421:
2416:
2410:
2406:
2400:
2396:
2392:
2387:
2383:
2379:
2376:
2373:
2370:
2367:
2364:
2343:
2340:
2328:
2327:
2315:
2312:
2309:
2304:
2300:
2296:
2293:
2290:
2263:
2260:
2255:
2251:
2245:
2241:
2221:) is called a
2205:
2204:
2195:
2193:
2182:
2179:
2174:
2170:
2166:
2161:
2157:
2129:
2124:
2120:
2106:
2105:
2096:
2094:
2083:
2080:
2075:
2071:
2065:
2060:
2056:
2023:
2003:
1998:
1994:
1990:
1987:
1984:
1979:
1974:
1970:
1958:
1957:
1941:
1937:
1932:
1928:
1923:
1918:
1914:
1910:
1907:
1904:
1901:
1894:
1890:
1885:
1881:
1878:
1875:
1870:
1866:
1862:
1850:is defined by
1837:
1833:
1806:
1802:
1781:
1755:
1751:
1747:
1744:
1741:
1730:
1729:
1716:
1712:
1708:
1705:
1702:
1698:
1695:
1692:
1689:
1686:
1681:
1677:
1673:
1666:
1662:
1657:
1653:
1650:
1647:
1644:
1619:
1615:
1594:
1574:
1554:
1551:
1548:
1545:
1542:
1522:
1517:
1513:
1507:
1503:
1499:
1496:
1493:
1490:
1487:
1484:
1464:
1461:
1458:
1455:
1452:
1428:
1393:
1371:
1367:
1343:
1323:
1303:
1300:
1297:
1294:
1291:
1281:adjoint bundle
1268:
1242:
1238:
1227:curvature form
1214:
1191:
1167:
1147:
1127:
1107:
1084:
1064:
1032:
1020:
1017:
997:
993:
953:
931:
927:
902:
899:
896:
893:
890:
885:
882:
877:
874:
871:
868:
863:
859:
836:
832:
811:
808:
803:
799:
795:
790:
786:
782:
777:
774:
770:
749:
746:
743:
723:
718:
714:
710:
690:
687:
684:
681:
678:
675:
672:
667:
663:
659:
654:
650:
646:
641:
637:
614:tangent bundle
608:
605:
604:
603:
592:
589:
584:
580:
576:
571:
567:
536:
535:
524:
519:
514:
511:
508:
503:
497:
493:
487:
483:
479:
474:
470:
466:
463:
460:
457:
454:
451:
426:
422:
393:
358:
355:
352:
349:
346:
343:
340:
308:
294:Wolfgang Pauli
281:
278:
275:
272:
269:
242:gauge symmetry
238:Chen-Ning Yang
229:
226:
224:
221:
101:
90:
77:BPST instanton
71:
67:
66:
57:
56:
48:
47:
46:
37:
36:
28:
27:
26:
25:
24:
15:
9:
6:
4:
3:
2:
6322:
6311:
6308:
6306:
6303:
6301:
6298:
6297:
6295:
6280:
6271:
6263:
6261:9780198570639
6257:
6253:
6246:
6237:
6228:
6219:
6210:
6201:
6192:
6183:
6181:
6171:
6162:
6160:
6150:
6141:
6131:
6122:
6120:
6118:
6116:
6114:
6104:
6095:
6091:
6077:
6071:
6067:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6028:
6022:
6020:
6016:
6015:Nigel Hitchin
6012:
5993:
5990:
5987:
5981:
5970:
5966:
5940:
5930:
5928:
5924:
5920:
5916:
5900:
5897:
5894:
5886:
5862:
5859:
5840:
5836:
5832:
5828:
5823:
5807:
5792:
5774:
5757:
5755:
5751:
5750:rational maps
5747:
5707:
5688:
5684:
5681:
5677:
5676:Higgs bundles
5673:
5669:
5665:
5648:
5633:
5629:
5611:
5596:
5595:
5594:
5578:
5563:
5553:
5551:
5548:surpassed by
5545:
5528:
5496:
5479:
5475:
5459:
5451:
5447:
5443:
5437:
5422:
5409:
5406:
5403:
5400:
5380:
5377:
5355:
5326:
5297:
5293:
5284:
5268:
5265:
5262:
5259:
5256:
5253:
5245:
5241:
5232:
5227:
5215:
5212:
5192:
5189:
5186:
5164:
5160:
5156:
5153:
5133:
5125:
5101:
5098:
5090:
5086:
5062:
5054:
5050:
5029:
5021:
5018:is the first
5002:
4994:
4990:
4960:
4952:
4948:
4944:
4938:
4930:
4926:
4922:
4919:
4913:
4910:
4907:
4904:
4901:
4896:
4891:
4879:
4876:
4869:
4868:
4867:
4853:
4850:
4844:
4836:
4832:
4809:
4804:
4787:
4766:
4755:
4752:
4744:
4740:
4736:
4730:
4722:
4718:
4709:
4690:
4684:
4681:
4673:
4657:
4634:
4628:
4625:
4622:
4619:
4610:
4594:
4563:
4532:
4501:
4484:
4480:
4476:
4472:
4468:
4467:
4447:
4443:
4439:
4435:
4432:
4427:
4423:
4399:
4395:
4391:
4387:
4382:
4378:
4357:
4337:
4317:
4309:
4305:
4289:
4263:
4255:
4247:
4241:
4233:
4225:
4219:
4211:
4199:
4198:
4197:
4183:
4180:
4160:
4153:
4130:
4122:
4108:
4100:
4092:
4089:
4082:
4081:
4080:
4066:
4052:
4050:
4034:
4026:
4022:
4018:
4014:
4010:
4006:
4001:
3985:
3954:
3923:
3894:
3886:
3870:
3862:
3766:
3755:
3728:
3720:
3691:
3677:
3674:
3671:
3663:
3662:
3645:
3619:
3613:
3610:
3607:
3602:
3586:
3580:
3570:
3566:
3557:
3553:
3549:
3544:
3528:
3522:
3512:
3509:
3505:
3499:
3495:
3491:
3483:
3479:
3475:
3469:
3466:
3463:
3457:
3454:
3447:
3446:
3445:
3428:
3422:
3419:
3399:
3379:
3372:
3368:
3364:
3354:
3352:
3351:
3334:
3314:
3287:
3282:
3266:
3259:
3256:
3251:
3247:
3241:
3236:
3232:
3223:
3219:
3215:
3212:
3210:
3200:
3197:
3194:
3189:
3183:
3167:
3161:
3153:
3150:
3147:
3139:
3135:
3131:
3125:
3122:
3119:
3116:
3111:
3107:
3098:
3094:
3090:
3087:
3081:
3076:
3072:
3068:
3063:
3059:
3052:
3049:
3046:
3041:
3031:
3027:
3018:
3014:
3009:
3001:
2998:
2994:
2989:
2987:
2977:
2974:
2971:
2966:
2960:
2944:
2937:
2934:
2931:
2926:
2922:
2918:
2915:
2910:
2906:
2901:
2898:
2893:
2889:
2885:
2882:
2879:
2876:
2871:
2867:
2863:
2860:
2855:
2851:
2846:
2843:
2838:
2834:
2825:
2821:
2816:
2808:
2805:
2801:
2796:
2794:
2787:
2784:
2781:
2776:
2769:
2766:
2763:
2760:
2754:
2751:
2747:
2739:
2736:
2732:
2719:
2718:
2717:
2715:
2714:
2709:
2690:
2687:
2684:
2681:
2676:
2672:
2668:
2665:
2660:
2656:
2652:
2649:
2644:
2640:
2636:
2631:
2628:
2625:
2622:
2618:
2610:
2609:
2608:
2594:
2574:
2571:
2568:
2565:
2535:
2532:
2524:
2511:
2495:
2458:
2451:
2449:
2435:
2430:
2414:
2408:
2398:
2394:
2385:
2381:
2377:
2371:
2365:
2362:
2355:
2354:
2351:
2350:, defined by
2349:
2339:
2337:
2333:
2313:
2310:
2307:
2302:
2298:
2294:
2291:
2288:
2281:
2280:
2279:
2277:
2261:
2258:
2253:
2249:
2243:
2239:
2231:
2226:
2224:
2220:
2219:
2214:
2213:
2203:
2196:
2194:
2180:
2177:
2172:
2168:
2164:
2159:
2155:
2147:
2146:
2143:
2127:
2122:
2118:
2104:
2097:
2095:
2081:
2078:
2073:
2069:
2063:
2058:
2054:
2046:
2045:
2042:
2039:
2037:
2021:
2001:
1996:
1992:
1988:
1985:
1982:
1977:
1972:
1968:
1939:
1935:
1926:
1921:
1916:
1912:
1908:
1905:
1899:
1892:
1888:
1879:
1876:
1873:
1868:
1864:
1853:
1852:
1851:
1835:
1831:
1822:
1804:
1800:
1792:. Using this
1779:
1771:
1753:
1749:
1745:
1742:
1739:
1714:
1710:
1706:
1703:
1700:
1693:
1690:
1687:
1679:
1675:
1671:
1664:
1660:
1651:
1648:
1645:
1635:
1634:
1633:
1617:
1613:
1592:
1572:
1549:
1543:
1540:
1520:
1515:
1511:
1505:
1497:
1491:
1485:
1482:
1459:
1453:
1450:
1442:
1426:
1417:
1415:
1411:
1410:inner product
1407:
1391:
1369:
1365:
1357:
1341:
1321:
1298:
1292:
1289:
1282:
1266:
1258:
1255:, which is a
1240:
1236:
1228:
1212:
1205:
1189:
1181:
1165:
1158:-bundle over
1145:
1125:
1105:
1098:
1082:
1075:-bundle over
1062:
1054:
1050:
1046:
1030:
1016:
1013:
995:
991:
982:
977:
975:
972:, which is a
971:
970:Chern classes
967:
951:
929:
925:
916:
897:
894:
891:
883:
880:
875:
872:
869:
866:
861:
857:
834:
830:
809:
801:
797:
793:
788:
784:
780:
775:
772:
768:
747:
741:
716:
712:
682:
676:
673:
670:
665:
661:
652:
644:
639:
635:
625:
623:
619:
615:
590:
587:
582:
578:
574:
569:
565:
557:
556:
555:
553:
552:derived below
549:
545:
541:
522:
517:
501:
495:
485:
481:
472:
468:
464:
458:
452:
449:
442:
441:
440:
424:
420:
411:
407:
391:
383:
379:
374:
372:
353:
347:
341:
338:
330:
326:
322:
319:, called the
306:
299:
295:
276:
270:
259:
255:
254:James Maxwell
251:
247:
243:
239:
235:
220:
218:
214:
210:
206:
205:
200:
195:
193:
189:
185:
181:
180:vector bundle
177:
173:
169:
165:
161:
157:
153:
143:
139:
135:
131:
127:
124:
120:
116:
112:
108:
104:
97:
94:is the third
93:
86:
82:
78:
74:
61:
52:
41:
32:
22:
6279:
6270:
6251:
6245:
6236:
6227:
6218:
6209:
6200:
6191:
6170:
6149:
6140:
6130:
6103:
6094:
6070:
5942:
5824:
5758:
5692:
5561:
5559:
5546:
5439:
5425:Applications
5020:Betti number
4981:
4611:
4482:
4478:
4474:
4470:
4464:
4307:
4303:
4281:
4149:
4058:
4002:
3692:
3659:
3637:
3371:automorphism
3366:
3360:
3348:
3306:
2711:
2705:
2510:affine space
2463:
2452:
2347:
2345:
2332:Hodge theory
2329:
2227:
2222:
2216:
2210:
2208:
2197:
2109:
2098:
2040:
1959:
1731:
1418:
1022:
980:
978:
965:
626:
610:
537:
409:
405:
381:
377:
375:
324:
320:
249:
245:
241:
234:Robert Mills
231:
209:moduli space
202:
198:
196:
191:
167:
164:gauge theory
149:
141:
133:
129:
125:
118:
114:
111:g=2, ρ=1,z=0
110:
106:
99:
96:Pauli matrix
88:
84:
80:
69:
5919:gauge group
5835:sine-Gordon
5756:to itself.
5746:Werner Nahm
4708:Chern class
4463:, then by (
4152:eigenvalues
3719:gauge group
3664:), so does
3658:satisfies (
2716:), compute
607:Mathematics
406:gauge field
325:gauge group
156:mathematics
6294:Categories
6086:References
4009:Raoul Bott
3741:. The set
3638:and so if
2342:Derivation
1412:under the
1180:connection
1019:Definition
223:Motivation
204:instantons
176:connection
83:-slice of
5982:×
5979:Σ
5951:Σ
5752:from the
5474:cobordism
5407:−
5260:−
5233:−
5216:
4923:−
4911:−
4897:−
4880:
4810:−
4767:≅
4737:∈
4685:
4629:
4595:±
4564:±
4533:−
4440:⋆
4436:−
4392:⋆
4304:self-dual
4290:⋆
4256:−
4252:Ω
4248:⊕
4230:Ω
4208:Ω
4181:−
4119:Ω
4115:→
4097:Ω
4090:⋆
3986:∗
3955:∗
3924:∗
3861:Hausdorff
3675:⋅
3614:
3577:‖
3563:‖
3554:∫
3519:‖
3510:−
3489:‖
3480:∫
3467:⋅
3458:
3423:
3263:⟩
3242:∗
3229:⟨
3220:∫
3158:‖
3151:∧
3145:‖
3129:⟩
3123:∧
3104:⟨
3085:⟩
3056:⟨
3038:‖
3024:‖
3015:∫
2941:⟩
2935:∧
2880:∧
2831:⟨
2822:∫
2755:
2685:∧
2521:Ω
2405:‖
2391:‖
2382:∫
2366:
2308:ω
2303:∗
2292:ω
2165:⋆
2128:∗
2064:∗
2022:⋆
2002:⋆
1989:⋆
1986:±
1978:∗
1931:⟩
1922:∗
1903:⟨
1884:⟩
1861:⟨
1697:⟩
1685:⟨
1676:∫
1656:⟩
1643:⟨
1544:
1516:∗
1502:Λ
1498:⊗
1486:
1454:
1293:
1178:. Then a
1097:Lie group
835:α
807:→
802:β
794:∩
789:α
776:β
773:α
745:→
717:α
677:
666:α
649:Ω
645:∈
640:α
575:⋆
492:‖
478:‖
469:∫
453:
348:
298:Lie group
271:
215:to prove
138:Uhlenbeck
6135:523–615.
6025:See also
5925:and the
5678:and the
5472:gives a
5450:definite
4866:, to be
4481:and the
3885:holonomy
1585:. Since
1257:two-form
1049:oriented
5672:Hitchin
2034:is the
1768:is the
1045:compact
228:Physics
190:of the
152:physics
121:on the
79:on the
6258:
6019:Witten
5829:, and
5042:, and
4982:where
4302:, the
3883:whose
3369:is an
2508:is an
2215:) or (
2014:where
1419:Since
1354:is an
378:fields
327:, see
207:. The
174:for a
166:, the
87:where
6062:Notes
5841:, of
1043:be a
964:is a
616:to a
178:on a
109:with
81:(x,x)
6256:ISBN
5837:and
5722:and
5444:and
5179:and
4650:and
4578:and
4516:and
4306:and
4173:and
4007:and
3969:and
3717:the
2710:of (
1023:Let
538:The
244:and
236:and
162:and
154:and
100:dx⊗σ
70:dx⊗σ
68:The
5967:of
5634:on
5213:dim
5126:on
5022:of
4877:dim
4670:is
4609:.
4473:or
4415:or
3859:is
3835:or
3690:.
1823:of
1772:of
1533:of
1314:of
1259:on
1182:on
256:on
201:or
182:or
150:In
128:of
115:z=0
6296::
6179:^
6158:^
6112:^
6021:.
5929:.
5552:.
5410:3.
4710:,
4682:SU
4626:SU
4051:.
4000:.
3611:YM
3455:YM
3420:ad
2752:YM
2363:YM
2225:.
2181:0.
2082:0.
1541:ad
1483:ad
1451:ad
1416:.
1290:ad
1051:,
1047:,
674:ad
591:0.
554::
450:YM
373:.
219:.
6264:.
6078:.
5997:]
5994:1
5991:,
5988:0
5985:[
5901:1
5898:+
5895:2
5871:)
5867:R
5863:,
5860:3
5857:(
5853:L
5850:S
5808:4
5803:R
5775:4
5770:R
5731:R
5708:3
5703:R
5682:.
5663:.
5649:3
5644:R
5612:4
5607:R
5579:4
5574:R
5529:2
5524:P
5521:C
5497:2
5492:P
5489:C
5460:X
5404:k
5401:8
5381:,
5378:k
5356:4
5351:R
5327:4
5322:R
5298:4
5294:S
5269:5
5266:=
5263:3
5257:8
5254:=
5251:)
5246:4
5242:S
5238:(
5228:1
5222:M
5193:1
5190:=
5187:k
5165:4
5161:S
5157:=
5154:X
5134:X
5110:)
5106:R
5102:,
5099:X
5096:(
5091:2
5087:H
5066:)
5063:X
5060:(
5055:+
5051:b
5030:X
5006:)
5003:X
5000:(
4995:1
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4961:X
4958:(
4953:+
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4939:X
4936:(
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4917:(
4914:3
4908:k
4905:8
4902:=
4892:k
4886:M
4854:k
4851:=
4848:)
4845:P
4842:(
4837:2
4833:c
4805:k
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4756:,
4753:X
4750:(
4745:4
4741:H
4734:)
4731:P
4728:(
4723:2
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4694:)
4691:2
4688:(
4658:X
4638:)
4635:2
4632:(
4623:=
4620:G
4589:M
4558:B
4527:A
4502:+
4496:A
4466:2
4448:A
4444:F
4433:=
4428:A
4424:F
4400:A
4396:F
4388:=
4383:A
4379:F
4358:X
4338:G
4318:A
4267:)
4264:X
4261:(
4245:)
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4239:(
4234:+
4226:=
4223:)
4220:X
4217:(
4212:2
4184:1
4161:1
4146:.
4134:)
4131:X
4128:(
4123:2
4112:)
4109:X
4106:(
4101:2
4093::
4067:X
4035:X
3980:M
3949:B
3918:A
3895:G
3871:A
3845:M
3821:B
3797:M
3773:G
3767:/
3761:A
3756:=
3751:B
3729:P
3703:G
3678:A
3672:g
3661:1
3646:A
3623:)
3620:A
3617:(
3608:=
3603:g
3598:l
3595:o
3592:v
3587:d
3581:2
3571:A
3567:F
3558:X
3550:=
3545:g
3540:l
3537:o
3534:v
3529:d
3523:2
3513:1
3506:g
3500:A
3496:F
3492:g
3484:X
3476:=
3473:)
3470:A
3464:g
3461:(
3432:)
3429:P
3426:(
3400:P
3380:g
3350:1
3335:a
3315:A
3288:.
3283:g
3278:l
3275:o
3272:v
3267:d
3260:a
3257:,
3252:A
3248:F
3237:A
3233:d
3224:X
3216:2
3213:=
3201:0
3198:=
3195:t
3190:)
3184:g
3179:l
3176:o
3173:v
3168:d
3162:2
3154:a
3148:a
3140:4
3136:t
3132:+
3126:a
3120:a
3117:,
3112:A
3108:F
3099:2
3095:t
3091:2
3088:+
3082:a
3077:A
3073:d
3069:,
3064:A
3060:F
3053:t
3050:2
3047:+
3042:2
3032:A
3028:F
3019:X
3010:(
3002:t
2999:d
2995:d
2990:=
2978:0
2975:=
2972:t
2967:)
2961:g
2956:l
2953:o
2950:v
2945:d
2938:a
2932:a
2927:2
2923:t
2919:+
2916:a
2911:A
2907:d
2902:t
2899:+
2894:A
2890:F
2886:,
2883:a
2877:a
2872:2
2868:t
2864:+
2861:a
2856:A
2852:d
2847:t
2844:+
2839:A
2835:F
2826:X
2817:(
2809:t
2806:d
2802:d
2797:=
2788:0
2785:=
2782:t
2777:)
2773:)
2770:a
2767:t
2764:+
2761:A
2758:(
2748:(
2740:t
2737:d
2733:d
2713:3
2691:.
2688:a
2682:a
2677:2
2673:t
2669:+
2666:a
2661:A
2657:d
2653:t
2650:+
2645:A
2641:F
2637:=
2632:a
2629:t
2626:+
2623:A
2619:F
2595:A
2575:a
2572:t
2569:+
2566:A
2546:)
2541:g
2536:;
2533:P
2530:(
2525:1
2496:P
2474:A
2457:)
2455:3
2453:(
2436:.
2431:g
2426:l
2423:o
2420:v
2415:d
2409:2
2399:A
2395:F
2386:X
2378:=
2375:)
2372:A
2369:(
2326:.
2314:0
2311:=
2299:d
2295:=
2289:d
2262:0
2259:=
2254:A
2250:F
2244:A
2240:d
2218:2
2212:1
2202:)
2200:2
2198:(
2178:=
2173:A
2169:F
2160:A
2156:d
2123:A
2119:d
2103:)
2101:1
2099:(
2079:=
2074:A
2070:F
2059:A
2055:d
1997:A
1993:d
1983:=
1973:A
1969:d
1956:.
1940:2
1936:L
1927:t
1917:A
1913:d
1909:,
1906:s
1900:=
1893:2
1889:L
1880:t
1877:,
1874:s
1869:A
1865:d
1836:A
1832:d
1805:2
1801:L
1780:X
1754:g
1750:l
1746:o
1743:v
1740:d
1715:g
1711:l
1707:o
1704:v
1701:d
1694:t
1691:,
1688:s
1680:X
1672:=
1665:2
1661:L
1652:t
1649:,
1646:s
1618:2
1614:L
1593:X
1573:X
1553:)
1550:P
1547:(
1521:X
1512:T
1506:2
1495:)
1492:P
1489:(
1463:)
1460:P
1457:(
1427:X
1392:G
1370:A
1366:d
1342:A
1322:P
1302:)
1299:P
1296:(
1267:X
1241:A
1237:F
1213:A
1190:P
1166:X
1146:G
1126:P
1106:G
1083:X
1063:G
1031:X
996:2
992:L
952:A
930:A
926:F
901:]
898:A
895:,
892:A
889:[
884:2
881:1
876:+
873:A
870:d
867:=
862:A
858:F
831:A
810:G
798:U
785:U
781::
769:g
748:X
742:P
722:}
713:U
709:{
689:)
686:)
683:P
680:(
671:,
662:U
658:(
653:1
636:A
588:=
583:A
579:F
570:A
566:d
523:.
518:g
513:l
510:o
507:v
502:d
496:2
486:A
482:F
473:X
465:=
462:)
459:A
456:(
425:A
421:F
392:A
357:)
354:1
351:(
345:U
342:=
339:G
307:G
280:)
277:1
274:(
268:U
144:.
142:S
134:R
130:R
126:S
119:z
107:A
102:3
91:3
89:σ
85:R
72:3
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