22:
944:
3768:
does not apply. However, about any point, a generalized complex manifold is, up to diffeomorphism and B-field, a product of a symplectic manifold with a generalized complex manifold which is of complex type at the point, much like
Weinstein's theorem for the local structure of
4125:
Up to a shift of the B-field, which corresponds to multiplying the pure spinor by the exponential of a closed, real 2-form, symplectic manifolds are the only type 0 generalized complex manifolds. Manifolds which are symplectic up to a shift of the B-field are sometimes called
2217:
698:
3892:, 0)-forms are pure spinors, as they are annihilated by antiholomorphic tangent vectors and by holomorphic one-forms. Thus this line bundle can be used as a canonical bundle to define a generalized complex structure. Restricting the annihilator from
539:
1331:
455:
3314:, or more generally if its exterior derivative is equal to the action of a gamma matrix on itself, then the almost complex structure is integrable and so such pure spinors correspond to generalized complex structures.
2641:. In other words, the type of a maximal isotropic subbundle is the codimension of its projection onto the tangent bundle. In the complex case one uses the complex dimension and the type is sometimes referred to as the
1168:
In generalized complex geometry one is not interested in vector fields, but rather in the formal sums of vector fields and one-forms. A kind of Lie bracket for such formal sums was introduced in 1990 and is called the
4356:
3595:
3289:
3231:
3173:
2705:
4194:
3992:
3941:
3504:
3067:
2918:
807:
3830:
3000:
4086:
The above pure spinor is globally defined, and so the canonical bundle is trivial. This means that symplectic manifolds are not only generalized complex manifolds but in fact are generalized Calabi-Yau manifolds.
3645:
815:
2068:
2841:, which means that each point has an open neighborhood in which the type does not increase. In practice this means that subsets of greater type than the ambient type occur on submanifolds with positive
3119:
2817:
2760:
2060:
1632:
1459:
1060:
316:
3537:
3449:
3317:
If one further imposes that the canonical bundle is holomorphically trivial, meaning that it is global sections which are closed forms, then it defines a generalized Calabi-Yau structure and
592:
4491:
Notice that a generalized Calabi metric structure, which was introduced by Marco
Gualtieri, is a stronger condition than a generalized Calabi–Yau structure, which was introduced by
2635:
2586:
2493:
2294:
2257:
1779:
1595:
4291:
3400:
1558:
1493:
3675:
Every regular point in a generalized complex manifold has an open neighborhood which, after a diffeomorphism and shift of the B-field, has the same generalized complex structure as the
1918:
1137:-eigenbundle of a unique generalized almost complex structure, so that the properties (i), (ii) can be considered as an alternative definition of generalized almost complex structure.
1100:
999:
4297:
2452:
600:
3773:. The remaining question of the local structure is: what does a generalized complex structure look like near a point of complex type? In fact, it will be induced by a holomorphic
3750:
4482:
4074:
2328:
1365:
2409:
4403:
3709:
1967:
1836:
1740:
1708:
474:
1865:
3855:
2538:
2353:
2515:
2373:
1521:
1135:
732:
1179:
4023:
3018:. Vectors have an action on differential forms given by the interior product. One-forms have an action on forms given by the wedge product. Thus sections of the bundle
2023:
1997:
1661:
4108:
1938:
1807:
4406:
756:
2945:
3295:
of the pure spinor and its complex conjugate contains a top-dimensional component. Such pure spinors determine generalized almost complex structures.
367:
115:
4645:
119:
5097:
4307:
3546:
3240:
3182:
3124:
2656:
4699:
4148:
3946:
3895:
3458:
3021:
2872:
761:
51:
4495:. In particular a generalized Calabi–Yau metric structure implies the existence of two commuting generalized almost complex structures.
3793:
2963:
3943:
to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore, this generalized complex structure on
939:{\displaystyle L=\{X+\xi \in (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} \ :\ {\mathcal {J}}(X+\xi )={\sqrt {-1}}(X+\xi )\}}
4005:. In fact complex manifolds, and the manifolds obtained by multiplying the pure spinor bundle defining a complex manifold by a complex,
3603:
2212:{\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X))={\frac {1}{2}}(\varepsilon (Y,X)+\varepsilon (X,Y))=0}
168:, which relate 10-dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.
1388:
165:
3087:
4304:
generalized complex structures such that minus the product of the corresponding tensors is a positive definite metric on
3311:
3084:
if it is annihilated by half of a set of a set of generators of the
Clifford algebra. Spinors are sections of our bundle
2936:. The complex line bundle corresponding to a particular generalized almost complex structure is often referred to as the
2932:
As in the case of ordinary complex geometry, there is a correspondence between generalized almost complex structures and
2768:
2728:
2028:
1600:
1427:
284:
3298:
Given a generalized almost complex structure, one can also determine a pure spinor up to multiplication by an arbitrary
4692:
4083:
defines a symplectic structure on the tangent space. Thus symplectic manifolds are also generalized complex manifolds.
3509:
3424:
547:
73:
3233:
Such subbundles are always isotropic, so to define an almost complex structure one must only impose that the sum of
1022:
44:
5887:
5064:
4510:
2603:
2554:
2461:
2262:
2225:
1747:
1563:
5294:
5029:
4226:
4110:
is related to a pure spinor which is just a number by an imaginary shift of the B-field, which is a shift of the
3360:
1529:
1464:
1870:
5882:
5656:
3885:, 0)-forms are related locally by multiplication by a complex function and so they form a complex line bundle.
3597:
However the intersection of their projections need not be trivial. In general this intersection is of the form
693:{\displaystyle \langle {\mathcal {J}}(X+\xi ),{\mathcal {J}}(Y+\eta )\rangle =\langle X+\xi ,Y+\eta \rangle .}
4742:
4685:
965:
5681:
5006:
4737:
4431:
Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to
2414:
279:
3714:
735:
534:{\displaystyle {\mathcal {J}}:\mathbf {T} \oplus \mathbf {T} ^{*}\to \mathbf {T} \oplus \mathbf {T} ^{*}}
5261:
4434:
4043:
2299:
1339:
1065:
5182:
4887:
4834:
3654:
3014:
between differential forms and spinors. In particular even and odd forms map to the two chiralities of
2833:
The type of a generalized almost complex structure is in general not constant, it can jump by any even
2382:
4361:
4118:
pure spinor. A scalar is annihilated by the entire tangent space, and so these structures are of type
3685:
1943:
1812:
1713:
1684:
5476:
3539:
In the definition of a generalized almost complex structure we have imposed that the intersection of
2862:
1841:
150:
34:
5521:
1326:{\displaystyle =+{\mathcal {L}}_{X}\eta -{\mathcal {L}}_{Y}\xi -{\frac {1}{2}}d(i(X)\eta -i(Y)\xi )}
5701:
5621:
5436:
5370:
4732:
4114:. Therefore, these generalized complex structures are of the same type as those corresponding to a
3995:
3835:
3680:
3543:
and its conjugate contains only the origin, otherwise they would be unable to span the entirety of
2597:
2520:
2333:
1154:
1146:
704:
465:
126:
103:
38:
30:
5581:
4138:
Some of the almost structures in generalized complex geometry may be rephrased in the language of
2498:
2358:
1506:
1113:
710:
274:
of the cotangent bundle. Generalized geometry unites these two fields by treating sections of the
271:
5841:
5651:
5365:
5203:
5177:
5049:
4918:
4807:
4749:
2710:
2376:
5208:
5606:
5347:
5153:
5044:
5016:
4839:
4414:
4008:
4001:
As only half of a basis of vector fields are holomorphic, these complex structures are of type
3765:
3753:
125:
These structures first arose in
Hitchin's program of characterizing geometrical structures via
55:
318:
of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.
5461:
5401:
5342:
5309:
5304:
5102:
5092:
5059:
4923:
4800:
4795:
4790:
4775:
4765:
4556:
4410:
4301:
2718:
2002:
1012:
99:
91:
5756:
1976:
1640:
5846:
5631:
4844:
4829:
4785:
4425:
4115:
4093:
3858:
3670:
3070:
1162:
154:
107:
4589:
Graña, Mariana (2006). "Flux compactifications in string theory: a comprehensive review".
1923:
1792:
8:
5761:
5646:
5299:
5198:
4824:
2933:
1380:
267:
4208:
structure. A generalized almost complex structure is a reduction of this structure to a
3302:. These choices of pure spinors are defined to be the sections of the canonical bundle.
5806:
5726:
5626:
5586:
5516:
5466:
5431:
5266:
5143:
5039:
4780:
4654:
4616:
4598:
4565:
4536:
741:
130:
5193:
5771:
5676:
5511:
5421:
5391:
5187:
5082:
5034:
4928:
4628:
4620:
3676:
1150:
959:
252:
134:
4405:
Generalized KĂ€hler manifolds, and their twisted counterparts, are equivalent to the
1403:
is a generalized almost complex structure such that the space of smooth sections of
220:
5781:
5716:
5686:
5566:
5506:
5471:
5416:
5406:
5319:
5271:
5229:
5134:
5127:
5120:
5113:
5106:
5024:
4814:
4722:
4664:
4608:
4575:
4519:
3774:
3770:
3299:
3074:
2941:
326:
263:
236:
5386:
4612:
3338:
Locally all pure spinors can be written in the same form, depending on an integer
5861:
5816:
5766:
5751:
5741:
5636:
5601:
5426:
4996:
4770:
3862:
2838:
2714:
1170:
5706:
4580:
4551:
1523:
is a 2-form. This correspondence extends straightforwardly to the complex case.
450:{\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X)).}
5836:
5831:
5791:
5731:
5561:
5551:
5546:
5541:
5456:
5451:
5446:
5411:
5396:
5324:
5001:
4864:
4669:
4640:
4220:
structure. Therefore, the space of generalized complex structures is the coset
3007:
2645:. While the type of a subbundle can in principle be any integer between 0 and 2
1368:
1158:
212:
192:
142:
5721:
5641:
4111:
3175:
Therefore, a given pure spinor is annihilated by a half-dimensional subbundle
5876:
5826:
5811:
5786:
5776:
5746:
5691:
5666:
5596:
5591:
5556:
5531:
5491:
5281:
4933:
4849:
4727:
4708:
4505:
4492:
4422:
3406:
3292:
2827:
2458:. Gualtieri has proven that all maximal isotropic subbundles are of the form
468:
of the generalized tangent bundle which preserves the natural inner product:
322:
204:
162:
158:
111:
5611:
3015:
157:. Today generalized complex structures also play a leading role in physical
5856:
5696:
5576:
5526:
5496:
5481:
5289:
5256:
5148:
5074:
5054:
4991:
4854:
4632:
4523:
4418:
1664:
228:
138:
4351:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}
3590:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}
3284:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}
3226:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}
3168:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}
3121:
and generators of the
Clifford algebra are the fibers of our other bundle
2960:
The canonical bundle is a one complex dimensional subbundle of the bundle
2700:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}
2653:
because the sum of the subbundle and its complex conjugate must be all of
5851:
5821:
5801:
5661:
5616:
5571:
5536:
5486:
5251:
5220:
4981:
4938:
4636:
4189:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }
4139:
3987:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }
3936:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }
3499:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }
3351:
3062:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }
3011:
2949:
2913:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }
2842:
802:{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }
146:
87:
4541:
2649:, generalized almost complex structures cannot have a type greater than
5796:
5736:
5671:
5334:
5314:
5213:
5172:
4986:
4659:
4603:
4358:
Generalized KĂ€hler structures are reductions of the structure group to
3825:{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }
2995:{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }
1679:
1635:
707:, a generalized almost complex structure is uniquely determined by its
3657:
in which the dimension of the fibers of Î is constant is said to be a
5711:
5501:
5441:
5087:
4961:
4882:
4877:
4872:
3354:Ί which generates the canonical bundle may always be put in the form
1421:
3832:
has a complex conjugation operation given by complex conjugation in
5357:
5246:
5241:
4971:
4966:
4903:
4819:
3873:)-forms, which are homogeneous polynomials in these one-forms with
3651:
3310:
If a pure spinor that determines a particular complex structure is
2722:
182:
4570:
4976:
4956:
4913:
4908:
4641:"Topological M-theory as unification of form theories of gravity"
3640:{\displaystyle E\cap {\overline {E}}=\Delta \otimes \mathbb {C} }
2834:
266:
one considers structures on the tangent bundles of manifolds. In
4677:
4142:. The word "almost" is removed if the structure is integrable.
3756:
of the two by two off-diagonal matrices with entries 1 and â1.
1420:
There is a one-to-one correspondence between maximal isotropic
4948:
2869:
with its complex conjugate. A maximal isotropic subspace of
1165:
subbundle is another section of the holomorphic subbundle.
4627:
2920:
is a generalized almost complex structure if and only if
2955:
2826:
is an arbitrary closed 2-form called the B-field in the
2588:
is the real dimension of the subbundle that annihilates
2944:
in the ordinary case. It is sometimes also called the
4437:
4364:
4310:
4229:
4151:
4096:
4046:
4011:
3949:
3898:
3838:
3796:
3717:
3688:
3606:
3549:
3512:
3461:
3427:
3363:
3305:
3243:
3185:
3127:
3090:
3024:
2966:
2875:
2771:
2731:
2659:
2606:
2557:
2523:
2501:
2464:
2417:
2385:
2361:
2336:
2302:
2265:
2228:
2071:
2031:
2005:
1979:
1946:
1926:
1873:
1844:
1815:
1795:
1750:
1716:
1687:
1643:
1603:
1566:
1532:
1509:
1467:
1430:
1342:
1182:
1116:
1068:
1025:
968:
818:
764:
744:
713:
603:
550:
477:
370:
287:
3114:{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} ,}
110:. Generalized complex structures were introduced by
2812:{\displaystyle X+\xi \longrightarrow X+\xi +i_{X}B}
2755:{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}
2055:{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}
1627:{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}
1454:{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}
311:{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}
4486:
4476:
4397:
4350:
4285:
4188:
4102:
4068:
4017:
3986:
3935:
3849:
3824:
3744:
3703:
3639:
3589:
3531:
3498:
3451:to be the projection of the holomorphic subbundle
3443:
3394:
3283:
3225:
3167:
3113:
3061:
2994:
2912:
2811:
2754:
2699:
2629:
2580:
2532:
2509:
2487:
2446:
2403:
2367:
2347:
2322:
2288:
2251:
2211:
2054:
2017:
1991:
1961:
1932:
1912:
1859:
1830:
1801:
1773:
1734:
1702:
1655:
1634:as follows. The elements of the subbundle are the
1626:
1589:
1560:one can construct a maximally isotropic subbundle
1552:
1515:
1487:
1453:
1359:
1325:
1129:
1094:
1054:
993:
938:
801:
750:
726:
692:
586:
533:
449:
310:
3532:{\displaystyle \mathbf {T} \otimes \mathbb {C} .}
176:
5874:
4646:Advances in Theoretical and Mathematical Physics
3752:with the standard symplectic form, which is the
3444:{\displaystyle \mathbf {T} \otimes \mathbb {C} }
3350:-form Ω. In a local neighborhood of any point a
43:but its sources remain unclear because it lacks
1410:
758:of the complexified generalized tangent bundle
587:{\displaystyle {\mathcal {J}}^{2}=-{\rm {Id}},}
1055:{\displaystyle \langle \ell ,\ell '\rangle =0}
114:in 2002 and further developed by his students
4693:
3881:antiholomorphic factors. In particular, all (
2709:The type of a maximal isotropic subbundle is
4508:(2003). "Generalized Calabi-Yau manifolds".
3069:act on differential forms. This action is a
2630:{\displaystyle L(\mathbf {E} ,\varepsilon )}
2581:{\displaystyle L(\mathbf {E} ,\varepsilon )}
2488:{\displaystyle L(\mathbf {E} ,\varepsilon )}
2289:{\displaystyle L(\mathbf {E} ,\varepsilon )}
2252:{\displaystyle L(\mathbf {E} ,\varepsilon )}
2096:
2072:
1774:{\displaystyle L(\mathbf {E} ,\varepsilon )}
1590:{\displaystyle L(\mathbf {E} ,\varepsilon )}
1043:
1026:
933:
825:
684:
660:
654:
604:
395:
371:
4286:{\displaystyle {\frac {O(2n,2n)}{U(n,n)}}.}
4133:
3650:for some subbundle Î. A point which has an
3395:{\displaystyle \Phi =e^{B+i\omega }\Omega }
1553:{\displaystyle (\mathbf {E} ,\varepsilon )}
1488:{\displaystyle (\mathbf {E} ,\varepsilon )}
4700:
4686:
3346:, a nondegenerate symplectic form Ï and a
1913:{\displaystyle \xi (Y)=\varepsilon (X,Y),}
4668:
4658:
4602:
4579:
4569:
4549:
4540:
4530:
4341:
4182:
3980:
3929:
3840:
3818:
3720:
3691:
3633:
3580:
3522:
3492:
3437:
3274:
3216:
3158:
3055:
2988:
2906:
2690:
870:
795:
133:, a connection which formed the basis of
74:Learn how and when to remove this message
3790:The space of complex differential forms
353:are one-forms then the inner product of
4504:
4032:
3759:
3328:
994:{\displaystyle L\cap {\overline {L}}=0}
5875:
4025:-closed (2,0)-form, are the only type
2454:Thus the total (complex) dimension in
321:The fibers are endowed with a natural
4681:
4588:
4487:Calabi versus Calabi–Yau metric
3333:
2956:Generalized almost complex structures
2447:{\displaystyle n-\dim(\mathbf {E} ).}
1407:is closed under the Courant bracket.
5098:Bogomol'nyiâPrasadâSommerfield bound
4037:The pure spinor bundle generated by
3785:
3745:{\displaystyle \mathbb {R} ^{2n-2k}}
3664:
3237:and its complex conjugate is all of
2330:(complex) dimensions of choices for
953:satisfies the following properties:
462:generalized almost complex structure
15:
4298:generalized almost KĂ€hler structure
4196:with the above inner product is an
3764:Near non-regular points, the above
3421:of the complexified tangent bundle
2927:
13:
4477:{\displaystyle SU(n)\times SU(n).}
4069:{\displaystyle \phi =e^{i\omega }}
4012:
3711:and the standard symplectic space
3626:
3389:
3364:
3306:Integrability and other structures
2323:{\displaystyle \dim(\mathbf {E} )}
1360:{\displaystyle {\mathcal {L}}_{X}}
1346:
1254:
1234:
1140:
1095:{\displaystyle \ell ,\ell '\in L.}
885:
634:
609:
576:
573:
554:
480:
14:
5899:
4707:
3002:of complex differential forms on
2551:of a maximal isotropic subbundle
2404:{\displaystyle \mathbf {E} ^{*},}
1415:
1145:In ordinary complex geometry, an
102:that includes as special cases a
4511:Quarterly Journal of Mathematics
4421:in the context of 2-dimensional
4398:{\displaystyle U(n)\times U(n).}
4324:
4315:
4165:
4156:
3963:
3954:
3912:
3903:
3810:
3799:
3704:{\displaystyle \mathbb {C} ^{k}}
3563:
3554:
3514:
3475:
3466:
3429:
3412:
3257:
3248:
3199:
3190:
3141:
3132:
3104:
3093:
3038:
3029:
2980:
2969:
2889:
2880:
2861:is the complex dimension of the
2857:of a maximal isotropic subspace
2742:
2733:
2673:
2664:
2614:
2596:minus the real dimension of the
2565:
2503:
2472:
2434:
2411:which is of (complex) dimension
2388:
2338:
2313:
2273:
2236:
2042:
2033:
1962:{\displaystyle \mathbf {E} ^{*}}
1949:
1831:{\displaystyle \mathbf {E} ^{*}}
1818:
1758:
1735:{\displaystyle \varepsilon (X).}
1703:{\displaystyle \mathbf {E} ^{*}}
1690:
1614:
1605:
1574:
1537:
1472:
1441:
1432:
853:
844:
778:
769:
703:Like in the case of an ordinary
521:
512:
498:
489:
298:
289:
20:
5295:Eleven-dimensional supergravity
4029:generalized complex manifolds.
3405:where Ω is decomposable as the
3323:generalized Calabi-Yau manifold
1860:{\displaystyle \varepsilon (X)}
1394:
4552:"Generalized complex geometry"
4468:
4462:
4450:
4444:
4389:
4383:
4374:
4368:
4334:
4311:
4274:
4262:
4254:
4236:
4175:
4152:
3973:
3950:
3922:
3899:
3573:
3550:
3485:
3462:
3267:
3244:
3209:
3186:
3151:
3128:
3048:
3025:
2899:
2876:
2781:
2683:
2660:
2624:
2610:
2575:
2561:
2482:
2468:
2438:
2430:
2317:
2309:
2283:
2269:
2246:
2232:
2200:
2197:
2185:
2176:
2164:
2158:
2142:
2139:
2133:
2124:
2118:
2112:
1904:
1892:
1883:
1877:
1854:
1848:
1768:
1754:
1726:
1720:
1584:
1570:
1547:
1533:
1482:
1468:
1320:
1314:
1308:
1296:
1290:
1284:
1225:
1213:
1207:
1183:
930:
918:
902:
890:
863:
840:
788:
765:
651:
639:
626:
614:
508:
441:
438:
432:
423:
417:
411:
177:The generalized tangent bundle
1:
4743:Second superstring revolution
4613:10.1016/j.physrep.2005.10.008
4498:
4079:for a nondegenerate two-form
3850:{\displaystyle \mathbb {C} .}
2848:
2717:and also under shifts of the
2533:{\displaystyle \varepsilon .}
2348:{\displaystyle \mathbf {E} ,}
2296:is maximal because there are
1781:is isotropic, notice that if
1401:generalized complex structure
270:one is instead interested in
171:
96:generalized complex structure
5237:Generalized complex manifold
4738:First superstring revolution
4533:Generalized complex geometry
3618:
2510:{\displaystyle \mathbf {E} }
2368:{\displaystyle \varepsilon }
1516:{\displaystyle \varepsilon }
1411:Maximal isotropic subbundles
1130:{\displaystyle {\sqrt {-1}}}
1110:satisfying (i), (ii) is the
980:
727:{\displaystyle {\sqrt {-1}}}
247:, is the vector bundle over
211:whose fibers consist of all
7:
4581:10.4007/annals.2011.174.1.3
3780:
2259:is isotropic. Furthermore,
151:topological string theories
10:
5904:
4835:Non-critical string theory
4670:10.4310/ATMP.2005.v9.n4.a5
3857:This allows one to define
3668:
3291:This is true whenever the
1106:Vice versa, any subbundle
958:the intersection with its
276:generalized tangent bundle
5379:
5356:
5333:
5280:
5165:
5073:
5015:
4947:
4896:
4863:
4758:
4715:
4550:Gualtieri, Marco (2011).
4531:Gualtieri, Marco (2004).
4018:{\displaystyle \partial }
3080:A spinor is said to be a
1710:is equal to the one-form
5371:Introduction to M-theory
5065:WessâZuminoâWitten model
5007:HananyâWitten transition
4733:History of string theory
4134:Relation to G-structures
3877:holomorphic factors and
2940:, as it generalizes the
2637:onto the tangent bundle
1147:almost complex structure
705:almost complex structure
466:almost complex structure
199:, which will be denoted
29:This article includes a
5888:Structures on manifolds
5050:Vertex operator algebra
4750:String theory landscape
3998:on the tangent bundle.
2542:
2375:is unrestricted on the
2018:{\displaystyle Y+\eta }
1371:along the vector field
1161:of two sections of the
153:are special cases of a
58:more precise citations.
5348:AdS/CFT correspondence
5103:Exceptional Lie groups
5045:Superconformal algebra
5017:Conformal field theory
4888:MontonenâOlive duality
4840:Non-linear sigma model
4478:
4426:quantum field theories
4399:
4352:
4287:
4190:
4104:
4070:
4019:
3988:
3937:
3851:
3826:
3766:classification theorem
3746:
3705:
3641:
3591:
3533:
3500:
3445:
3396:
3285:
3227:
3169:
3115:
3063:
2996:
2948:, as its sections are
2914:
2837:. However it is upper
2813:
2756:
2701:
2631:
2592:. Equivalently it is 2
2582:
2534:
2511:
2489:
2448:
2405:
2369:
2349:
2324:
2290:
2253:
2213:
2056:
2019:
1993:
1992:{\displaystyle X+\xi }
1963:
1934:
1914:
1861:
1832:
1803:
1775:
1736:
1704:
1657:
1656:{\displaystyle X+\xi }
1628:
1591:
1554:
1517:
1489:
1455:
1361:
1327:
1131:
1096:
1056:
995:
940:
803:
752:
728:
694:
588:
535:
451:
345:are vector fields and
312:
166:flux compactifications
149:'s 2004 proposal that
5883:Differential geometry
5343:Holographic principle
5310:Type IIB supergravity
5305:Type IIA supergravity
5157:-form electrodynamics
4776:Bosonic string theory
4557:Annals of Mathematics
4479:
4411:Sylvester James Gates
4407:bihermitian manifolds
4400:
4353:
4288:
4191:
4105:
4103:{\displaystyle \phi }
4071:
4020:
3989:
3938:
3852:
3827:
3747:
3706:
3642:
3592:
3534:
3501:
3446:
3417:Define the subbundle
3397:
3342:, the B-field 2-form
3286:
3228:
3170:
3116:
3073:of the action of the
3064:
2997:
2915:
2814:
2757:
2702:
2632:
2583:
2535:
2512:
2490:
2449:
2406:
2370:
2350:
2325:
2291:
2254:
2214:
2057:
2020:
1994:
1964:
1935:
1915:
1862:
1833:
1804:
1776:
1737:
1705:
1658:
1629:
1592:
1555:
1518:
1490:
1456:
1362:
1328:
1132:
1097:
1057:
996:
962:is the zero section:
941:
804:
753:
729:
695:
589:
536:
452:
313:
100:differential manifold
92:differential geometry
5262:HoĆavaâWitten theory
5209:HyperkÀhler manifold
4897:Particles and fields
4845:Tachyon condensation
4830:Matrix string theory
4524:10.1093/qmath/hag025
4435:
4362:
4308:
4227:
4149:
4094:
4044:
4033:Symplectic manifolds
4009:
3994:defines an ordinary
3947:
3896:
3836:
3794:
3760:Local holomorphicity
3715:
3686:
3681:complex vector space
3604:
3547:
3510:
3459:
3425:
3361:
3329:Local classification
3241:
3183:
3125:
3088:
3022:
2964:
2934:complex line bundles
2873:
2769:
2729:
2657:
2604:
2555:
2521:
2499:
2462:
2415:
2383:
2359:
2334:
2300:
2263:
2226:
2069:
2029:
2003:
1977:
1944:
1933:{\displaystyle \xi }
1924:
1871:
1842:
1813:
1802:{\displaystyle \xi }
1793:
1748:
1714:
1685:
1641:
1601:
1564:
1530:
1507:
1465:
1428:
1340:
1180:
1173:which is defined by
1114:
1066:
1023:
966:
816:
762:
742:
711:
601:
548:
475:
368:
285:
155:topological M-theory
108:symplectic structure
5300:Type I supergravity
5204:CalabiâYau manifold
5199:Ricci-flat manifold
5178:KaluzaâKlein theory
4919:RamondâRamond field
4825:String field theory
4635:; Neitzke, Andrew;
1381:exterior derivative
1157:if and only if the
1011:, i.e. its complex
738:, i.e. a subbundle
268:symplectic geometry
251:whose sections are
98:is a property of a
5267:K-theory (physics)
5144:ADE classification
4781:Superstring theory
4629:Dijkgraaf, Robbert
4474:
4395:
4348:
4283:
4186:
4100:
4066:
4015:
3984:
3933:
3847:
3822:
3742:
3701:
3637:
3587:
3529:
3496:
3441:
3392:
3281:
3223:
3165:
3111:
3059:
3006:. Recall that the
2992:
2946:pure spinor bundle
2910:
2809:
2752:
2697:
2627:
2578:
2530:
2507:
2485:
2444:
2401:
2365:
2345:
2320:
2286:
2249:
2209:
2052:
2015:
1989:
1959:
1930:
1910:
1857:
1828:
1799:
1771:
1732:
1700:
1678:restricted to the
1653:
1624:
1587:
1550:
1513:
1499:is a subbundle of
1485:
1451:
1357:
1323:
1127:
1092:
1052:
991:
936:
799:
748:
724:
690:
584:
531:
447:
308:
131:differential forms
31:list of references
5870:
5869:
5652:van Nieuwenhuizen
5188:Why 10 dimensions
5093:ChernâSimons form
5060:KacâMoody algebra
5040:Conformal algebra
5035:Conformal anomaly
4929:Magnetic monopole
4924:KalbâRamond field
4766:NambuâGoto action
4278:
3996:complex structure
3786:Complex manifolds
3775:Poisson structure
3771:Poisson manifolds
3677:Cartesian product
3671:Darboux's theorem
3665:Darboux's theorem
3621:
2156:
2110:
1674:and the one-form
1279:
1155:complex structure
1125:
1009:maximal isotropic
983:
960:complex conjugate
916:
882:
876:
751:{\displaystyle L}
722:
409:
135:Robbert Dijkgraaf
104:complex structure
84:
83:
76:
5895:
5380:String theorists
5320:Lie superalgebra
5272:Twisted K-theory
5230:Spin(7)-manifold
5183:Compactification
5025:Virasoro algebra
4808:Heterotic string
4702:
4695:
4688:
4679:
4678:
4674:
4672:
4662:
4624:
4606:
4585:
4583:
4573:
4546:
4544:
4527:
4483:
4481:
4480:
4475:
4404:
4402:
4401:
4396:
4357:
4355:
4354:
4349:
4344:
4333:
4332:
4327:
4318:
4292:
4290:
4289:
4284:
4279:
4277:
4257:
4231:
4219:
4207:
4195:
4193:
4192:
4187:
4185:
4174:
4173:
4168:
4159:
4109:
4107:
4106:
4101:
4090:The pure spinor
4075:
4073:
4072:
4067:
4065:
4064:
4024:
4022:
4021:
4016:
3993:
3991:
3990:
3985:
3983:
3972:
3971:
3966:
3957:
3942:
3940:
3939:
3934:
3932:
3921:
3920:
3915:
3906:
3856:
3854:
3853:
3848:
3843:
3831:
3829:
3828:
3823:
3821:
3813:
3808:
3807:
3802:
3751:
3749:
3748:
3743:
3741:
3740:
3723:
3710:
3708:
3707:
3702:
3700:
3699:
3694:
3646:
3644:
3643:
3638:
3636:
3622:
3614:
3596:
3594:
3593:
3588:
3583:
3572:
3571:
3566:
3557:
3538:
3536:
3535:
3530:
3525:
3517:
3505:
3503:
3502:
3497:
3495:
3484:
3483:
3478:
3469:
3450:
3448:
3447:
3442:
3440:
3432:
3401:
3399:
3398:
3393:
3388:
3387:
3334:Canonical bundle
3321:is said to be a
3300:complex function
3290:
3288:
3287:
3282:
3277:
3266:
3265:
3260:
3251:
3232:
3230:
3229:
3224:
3219:
3208:
3207:
3202:
3193:
3174:
3172:
3171:
3166:
3161:
3150:
3149:
3144:
3135:
3120:
3118:
3117:
3112:
3107:
3102:
3101:
3096:
3075:Clifford algebra
3068:
3066:
3065:
3060:
3058:
3047:
3046:
3041:
3032:
3001:
2999:
2998:
2993:
2991:
2983:
2978:
2977:
2972:
2942:canonical bundle
2938:canonical bundle
2928:Canonical bundle
2919:
2917:
2916:
2911:
2909:
2898:
2897:
2892:
2883:
2818:
2816:
2815:
2810:
2805:
2804:
2761:
2759:
2758:
2753:
2751:
2750:
2745:
2736:
2706:
2704:
2703:
2698:
2693:
2682:
2681:
2676:
2667:
2636:
2634:
2633:
2628:
2617:
2587:
2585:
2584:
2579:
2568:
2539:
2537:
2536:
2531:
2516:
2514:
2513:
2508:
2506:
2494:
2492:
2491:
2486:
2475:
2453:
2451:
2450:
2445:
2437:
2410:
2408:
2407:
2402:
2397:
2396:
2391:
2374:
2372:
2371:
2366:
2354:
2352:
2351:
2346:
2341:
2329:
2327:
2326:
2321:
2316:
2295:
2293:
2292:
2287:
2276:
2258:
2256:
2255:
2250:
2239:
2218:
2216:
2215:
2210:
2157:
2149:
2111:
2103:
2061:
2059:
2058:
2053:
2051:
2050:
2045:
2036:
2025:are sections of
2024:
2022:
2021:
2016:
1998:
1996:
1995:
1990:
1968:
1966:
1965:
1960:
1958:
1957:
1952:
1939:
1937:
1936:
1931:
1919:
1917:
1916:
1911:
1866:
1864:
1863:
1858:
1837:
1835:
1834:
1829:
1827:
1826:
1821:
1808:
1806:
1805:
1800:
1785:is a section of
1780:
1778:
1777:
1772:
1761:
1741:
1739:
1738:
1733:
1709:
1707:
1706:
1701:
1699:
1698:
1693:
1670:is a section of
1662:
1660:
1659:
1654:
1633:
1631:
1630:
1625:
1623:
1622:
1617:
1608:
1596:
1594:
1593:
1588:
1577:
1559:
1557:
1556:
1551:
1540:
1522:
1520:
1519:
1514:
1494:
1492:
1491:
1486:
1475:
1460:
1458:
1457:
1452:
1450:
1449:
1444:
1435:
1389:interior product
1366:
1364:
1363:
1358:
1356:
1355:
1350:
1349:
1332:
1330:
1329:
1324:
1280:
1272:
1264:
1263:
1258:
1257:
1244:
1243:
1238:
1237:
1136:
1134:
1133:
1128:
1126:
1118:
1101:
1099:
1098:
1093:
1082:
1061:
1059:
1058:
1053:
1042:
1000:
998:
997:
992:
984:
976:
945:
943:
942:
937:
917:
909:
889:
888:
880:
874:
873:
862:
861:
856:
847:
808:
806:
805:
800:
798:
787:
786:
781:
772:
757:
755:
754:
749:
733:
731:
730:
725:
723:
715:
699:
697:
696:
691:
638:
637:
613:
612:
593:
591:
590:
585:
580:
579:
564:
563:
558:
557:
540:
538:
537:
532:
530:
529:
524:
515:
507:
506:
501:
492:
484:
483:
456:
454:
453:
448:
410:
402:
317:
315:
314:
309:
307:
306:
301:
292:
264:complex geometry
237:cotangent bundle
86:In the field of
79:
72:
68:
65:
59:
54:this article by
45:inline citations
24:
23:
16:
5903:
5902:
5898:
5897:
5896:
5894:
5893:
5892:
5873:
5872:
5871:
5866:
5375:
5352:
5329:
5276:
5224:
5194:KĂ€hler manifold
5161:
5138:
5131:
5124:
5117:
5110:
5069:
5030:Mirror symmetry
5011:
4997:Brane cosmology
4943:
4892:
4859:
4815:N=2 superstring
4801:Type IIB string
4796:Type IIA string
4771:Polyakov action
4754:
4711:
4706:
4542:math.DG/0401221
4501:
4489:
4436:
4433:
4432:
4363:
4360:
4359:
4340:
4328:
4323:
4322:
4314:
4309:
4306:
4305:
4258:
4232:
4230:
4228:
4225:
4224:
4209:
4197:
4181:
4169:
4164:
4163:
4155:
4150:
4147:
4146:
4136:
4095:
4092:
4091:
4057:
4053:
4045:
4042:
4041:
4035:
4010:
4007:
4006:
3979:
3967:
3962:
3961:
3953:
3948:
3945:
3944:
3928:
3916:
3911:
3910:
3902:
3897:
3894:
3893:
3865:one-forms and (
3863:antiholomorphic
3839:
3837:
3834:
3833:
3817:
3809:
3803:
3798:
3797:
3795:
3792:
3791:
3788:
3783:
3762:
3724:
3719:
3718:
3716:
3713:
3712:
3695:
3690:
3689:
3687:
3684:
3683:
3673:
3667:
3632:
3613:
3605:
3602:
3601:
3579:
3567:
3562:
3561:
3553:
3548:
3545:
3544:
3521:
3513:
3511:
3508:
3507:
3491:
3479:
3474:
3473:
3465:
3460:
3457:
3456:
3436:
3428:
3426:
3423:
3422:
3415:
3374:
3370:
3362:
3359:
3358:
3336:
3331:
3308:
3273:
3261:
3256:
3255:
3247:
3242:
3239:
3238:
3215:
3203:
3198:
3197:
3189:
3184:
3181:
3180:
3157:
3145:
3140:
3139:
3131:
3126:
3123:
3122:
3103:
3097:
3092:
3091:
3089:
3086:
3085:
3054:
3042:
3037:
3036:
3028:
3023:
3020:
3019:
2987:
2979:
2973:
2968:
2967:
2965:
2962:
2961:
2958:
2930:
2905:
2893:
2888:
2887:
2879:
2874:
2871:
2870:
2853:The real index
2851:
2839:semi-continuous
2800:
2796:
2770:
2767:
2766:
2746:
2741:
2740:
2732:
2730:
2727:
2726:
2715:diffeomorphisms
2689:
2677:
2672:
2671:
2663:
2658:
2655:
2654:
2613:
2605:
2602:
2601:
2564:
2556:
2553:
2552:
2545:
2522:
2519:
2518:
2502:
2500:
2497:
2496:
2471:
2463:
2460:
2459:
2433:
2416:
2413:
2412:
2392:
2387:
2386:
2384:
2381:
2380:
2360:
2357:
2356:
2337:
2335:
2332:
2331:
2312:
2301:
2298:
2297:
2272:
2264:
2261:
2260:
2235:
2227:
2224:
2223:
2148:
2102:
2070:
2067:
2066:
2046:
2041:
2040:
2032:
2030:
2027:
2026:
2004:
2001:
2000:
1978:
1975:
1974:
1973:. Thesefore if
1953:
1948:
1947:
1945:
1942:
1941:
1925:
1922:
1921:
1920:as the part of
1872:
1869:
1868:
1843:
1840:
1839:
1822:
1817:
1816:
1814:
1811:
1810:
1794:
1791:
1790:
1757:
1749:
1746:
1745:
1715:
1712:
1711:
1694:
1689:
1688:
1686:
1683:
1682:
1642:
1639:
1638:
1618:
1613:
1612:
1604:
1602:
1599:
1598:
1573:
1565:
1562:
1561:
1536:
1531:
1528:
1527:
1508:
1505:
1504:
1471:
1466:
1463:
1462:
1445:
1440:
1439:
1431:
1429:
1426:
1425:
1418:
1413:
1397:
1351:
1345:
1344:
1343:
1341:
1338:
1337:
1271:
1259:
1253:
1252:
1251:
1239:
1233:
1232:
1231:
1181:
1178:
1177:
1171:Courant bracket
1143:
1141:Courant bracket
1117:
1115:
1112:
1111:
1104:
1075:
1067:
1064:
1063:
1035:
1024:
1021:
1020:
975:
967:
964:
963:
949:Such subbundle
908:
884:
883:
869:
857:
852:
851:
843:
817:
814:
813:
794:
782:
777:
776:
768:
763:
760:
759:
743:
740:
739:
714:
712:
709:
708:
633:
632:
608:
607:
602:
599:
598:
572:
571:
559:
553:
552:
551:
549:
546:
545:
525:
520:
519:
511:
502:
497:
496:
488:
479:
478:
476:
473:
472:
401:
369:
366:
365:
302:
297:
296:
288:
286:
283:
282:
278:, which is the
272:exterior powers
213:tangent vectors
179:
174:
116:Marco Gualtieri
80:
69:
63:
60:
49:
35:related reading
25:
21:
12:
11:
5:
5901:
5891:
5890:
5885:
5868:
5867:
5865:
5864:
5859:
5854:
5849:
5844:
5839:
5834:
5829:
5824:
5819:
5814:
5809:
5804:
5799:
5794:
5789:
5784:
5779:
5774:
5769:
5764:
5759:
5754:
5749:
5744:
5739:
5734:
5729:
5724:
5719:
5714:
5709:
5704:
5702:Randjbar-Daemi
5699:
5694:
5689:
5684:
5679:
5674:
5669:
5664:
5659:
5654:
5649:
5644:
5639:
5634:
5629:
5624:
5619:
5614:
5609:
5604:
5599:
5594:
5589:
5584:
5579:
5574:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5534:
5529:
5524:
5519:
5514:
5509:
5504:
5499:
5494:
5489:
5484:
5479:
5474:
5469:
5464:
5459:
5454:
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5389:
5383:
5381:
5377:
5376:
5374:
5373:
5368:
5362:
5360:
5354:
5353:
5351:
5350:
5345:
5339:
5337:
5331:
5330:
5328:
5327:
5325:Lie supergroup
5322:
5317:
5312:
5307:
5302:
5297:
5292:
5286:
5284:
5278:
5277:
5275:
5274:
5269:
5264:
5259:
5254:
5249:
5244:
5239:
5234:
5233:
5232:
5227:
5222:
5218:
5217:
5216:
5206:
5196:
5191:
5185:
5180:
5175:
5169:
5167:
5163:
5162:
5160:
5159:
5151:
5146:
5141:
5136:
5129:
5122:
5115:
5108:
5100:
5095:
5090:
5085:
5079:
5077:
5071:
5070:
5068:
5067:
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5027:
5021:
5019:
5013:
5012:
5010:
5009:
5004:
5002:Quiver diagram
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4953:
4951:
4945:
4944:
4942:
4941:
4936:
4931:
4926:
4921:
4916:
4911:
4906:
4900:
4898:
4894:
4893:
4891:
4890:
4885:
4880:
4875:
4869:
4867:
4865:String duality
4861:
4860:
4858:
4857:
4852:
4847:
4842:
4837:
4832:
4827:
4822:
4817:
4812:
4811:
4810:
4805:
4804:
4803:
4798:
4791:Type II string
4788:
4778:
4773:
4768:
4762:
4760:
4756:
4755:
4753:
4752:
4747:
4746:
4745:
4740:
4730:
4728:Cosmic strings
4725:
4719:
4717:
4713:
4712:
4705:
4704:
4697:
4690:
4682:
4676:
4675:
4660:hep-th/0411073
4653:(4): 603â665.
4625:
4604:hep-th/0509003
4586:
4547:
4535:(PhD Thesis).
4528:
4518:(3): 281â308.
4506:Hitchin, Nigel
4500:
4497:
4488:
4485:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4423:supersymmetric
4409:discovered by
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4347:
4343:
4339:
4336:
4331:
4326:
4321:
4317:
4313:
4294:
4293:
4282:
4276:
4273:
4270:
4267:
4264:
4261:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4184:
4180:
4177:
4172:
4167:
4162:
4158:
4154:
4135:
4132:
4099:
4077:
4076:
4063:
4060:
4056:
4052:
4049:
4034:
4031:
4014:
3982:
3978:
3975:
3970:
3965:
3960:
3956:
3952:
3931:
3927:
3924:
3919:
3914:
3909:
3905:
3901:
3846:
3842:
3820:
3816:
3812:
3806:
3801:
3787:
3784:
3782:
3779:
3761:
3758:
3739:
3736:
3733:
3730:
3727:
3722:
3698:
3693:
3669:Main article:
3666:
3663:
3648:
3647:
3635:
3631:
3628:
3625:
3620:
3617:
3612:
3609:
3586:
3582:
3578:
3575:
3570:
3565:
3560:
3556:
3552:
3528:
3524:
3520:
3516:
3494:
3490:
3487:
3482:
3477:
3472:
3468:
3464:
3439:
3435:
3431:
3414:
3411:
3409:of one-forms.
3403:
3402:
3391:
3386:
3383:
3380:
3377:
3373:
3369:
3366:
3335:
3332:
3330:
3327:
3307:
3304:
3280:
3276:
3272:
3269:
3264:
3259:
3254:
3250:
3246:
3222:
3218:
3214:
3211:
3206:
3201:
3196:
3192:
3188:
3164:
3160:
3156:
3153:
3148:
3143:
3138:
3134:
3130:
3110:
3106:
3100:
3095:
3071:representation
3057:
3053:
3050:
3045:
3040:
3035:
3031:
3027:
3008:gamma matrices
2990:
2986:
2982:
2976:
2971:
2957:
2954:
2929:
2926:
2908:
2904:
2901:
2896:
2891:
2886:
2882:
2878:
2850:
2847:
2820:
2819:
2808:
2803:
2799:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2749:
2744:
2739:
2735:
2696:
2692:
2688:
2685:
2680:
2675:
2670:
2666:
2662:
2626:
2623:
2620:
2616:
2612:
2609:
2577:
2574:
2571:
2567:
2563:
2560:
2544:
2541:
2529:
2526:
2505:
2484:
2481:
2478:
2474:
2470:
2467:
2443:
2440:
2436:
2432:
2429:
2426:
2423:
2420:
2400:
2395:
2390:
2364:
2344:
2340:
2319:
2315:
2311:
2308:
2305:
2285:
2282:
2279:
2275:
2271:
2268:
2248:
2245:
2242:
2238:
2234:
2231:
2220:
2219:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2155:
2152:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2109:
2106:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2049:
2044:
2039:
2035:
2014:
2011:
2008:
1988:
1985:
1982:
1956:
1951:
1940:orthogonal to
1929:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1856:
1853:
1850:
1847:
1825:
1820:
1809:restricted to
1798:
1770:
1767:
1764:
1760:
1756:
1753:
1731:
1728:
1725:
1722:
1719:
1697:
1692:
1652:
1649:
1646:
1621:
1616:
1611:
1607:
1586:
1583:
1580:
1576:
1572:
1569:
1549:
1546:
1543:
1539:
1535:
1512:
1484:
1481:
1478:
1474:
1470:
1448:
1443:
1438:
1434:
1417:
1416:Classification
1414:
1412:
1409:
1396:
1393:
1369:Lie derivative
1354:
1348:
1334:
1333:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1278:
1275:
1270:
1267:
1262:
1256:
1250:
1247:
1242:
1236:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1142:
1139:
1124:
1121:
1103:
1102:
1091:
1088:
1085:
1081:
1078:
1074:
1071:
1051:
1048:
1045:
1041:
1038:
1034:
1031:
1028:
1002:
990:
987:
982:
979:
974:
971:
955:
947:
946:
935:
932:
929:
926:
923:
920:
915:
912:
907:
904:
901:
898:
895:
892:
887:
879:
872:
868:
865:
860:
855:
850:
846:
842:
839:
836:
833:
830:
827:
824:
821:
797:
793:
790:
785:
780:
775:
771:
767:
747:
721:
718:
701:
700:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
636:
631:
628:
625:
622:
619:
616:
611:
606:
583:
578:
575:
570:
567:
562:
556:
542:
541:
528:
523:
518:
514:
510:
505:
500:
495:
491:
487:
482:
458:
457:
446:
443:
440:
437:
434:
431:
428:
425:
422:
419:
416:
413:
408:
405:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
361:is defined as
305:
300:
295:
291:
193:tangent bundle
178:
175:
173:
170:
163:supersymmetric
143:Andrew Neitzke
120:Gil Cavalcanti
82:
81:
39:external links
28:
26:
19:
9:
6:
4:
3:
2:
5900:
5889:
5886:
5884:
5881:
5880:
5878:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5847:Zamolodchikov
5845:
5843:
5842:Zamolodchikov
5840:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5740:
5738:
5735:
5733:
5730:
5728:
5725:
5723:
5720:
5718:
5715:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5683:
5680:
5678:
5675:
5673:
5670:
5668:
5665:
5663:
5660:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5638:
5635:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5615:
5613:
5610:
5608:
5605:
5603:
5600:
5598:
5595:
5593:
5590:
5588:
5585:
5583:
5580:
5578:
5575:
5573:
5570:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5518:
5515:
5513:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5384:
5382:
5378:
5372:
5369:
5367:
5366:Matrix theory
5364:
5363:
5361:
5359:
5355:
5349:
5346:
5344:
5341:
5340:
5338:
5336:
5332:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5291:
5288:
5287:
5285:
5283:
5282:Supersymmetry
5279:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5231:
5228:
5226:
5219:
5215:
5212:
5211:
5210:
5207:
5205:
5202:
5201:
5200:
5197:
5195:
5192:
5189:
5186:
5184:
5181:
5179:
5176:
5174:
5171:
5170:
5168:
5164:
5158:
5156:
5152:
5150:
5147:
5145:
5142:
5139:
5132:
5125:
5118:
5111:
5104:
5101:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5080:
5078:
5076:
5072:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5022:
5020:
5018:
5014:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4954:
4952:
4950:
4946:
4940:
4937:
4935:
4934:Dual graviton
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4901:
4899:
4895:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4870:
4868:
4866:
4862:
4856:
4853:
4851:
4850:RNS formalism
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4818:
4816:
4813:
4809:
4806:
4802:
4799:
4797:
4794:
4793:
4792:
4789:
4787:
4786:Type I string
4784:
4783:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4763:
4761:
4757:
4751:
4748:
4744:
4741:
4739:
4736:
4735:
4734:
4731:
4729:
4726:
4724:
4721:
4720:
4718:
4714:
4710:
4709:String theory
4703:
4698:
4696:
4691:
4689:
4684:
4683:
4680:
4671:
4666:
4661:
4656:
4652:
4648:
4647:
4642:
4638:
4634:
4633:Gukov, Sergei
4630:
4626:
4622:
4618:
4614:
4610:
4605:
4600:
4597:(3): 91â158.
4596:
4592:
4587:
4582:
4577:
4572:
4567:
4564:(1): 75â123.
4563:
4559:
4558:
4553:
4548:
4543:
4538:
4534:
4529:
4525:
4521:
4517:
4513:
4512:
4507:
4503:
4502:
4496:
4494:
4493:Nigel Hitchin
4484:
4471:
4465:
4459:
4456:
4453:
4447:
4441:
4438:
4429:
4427:
4424:
4420:
4416:
4412:
4408:
4392:
4386:
4380:
4377:
4371:
4365:
4345:
4337:
4329:
4319:
4303:
4300:is a pair of
4299:
4280:
4271:
4268:
4265:
4259:
4251:
4248:
4245:
4242:
4239:
4233:
4223:
4222:
4221:
4217:
4213:
4205:
4201:
4178:
4170:
4160:
4143:
4141:
4131:
4129:
4123:
4121:
4117:
4113:
4097:
4088:
4084:
4082:
4061:
4058:
4054:
4050:
4047:
4040:
4039:
4038:
4030:
4028:
4004:
3999:
3997:
3976:
3968:
3958:
3925:
3917:
3907:
3891:
3886:
3884:
3880:
3876:
3872:
3868:
3864:
3860:
3844:
3814:
3804:
3778:
3776:
3772:
3767:
3757:
3755:
3737:
3734:
3731:
3728:
3725:
3696:
3682:
3678:
3672:
3662:
3660:
3659:regular point
3656:
3653:
3629:
3623:
3615:
3610:
3607:
3600:
3599:
3598:
3584:
3576:
3568:
3558:
3542:
3526:
3518:
3488:
3480:
3470:
3454:
3433:
3420:
3413:Regular point
3410:
3408:
3407:wedge product
3384:
3381:
3378:
3375:
3371:
3367:
3357:
3356:
3355:
3353:
3349:
3345:
3341:
3326:
3324:
3320:
3315:
3313:
3303:
3301:
3296:
3294:
3293:wedge product
3278:
3270:
3262:
3252:
3236:
3220:
3212:
3204:
3194:
3178:
3162:
3154:
3146:
3136:
3108:
3098:
3083:
3078:
3076:
3072:
3051:
3043:
3033:
3017:
3013:
3009:
3005:
2984:
2974:
2953:
2951:
2947:
2943:
2939:
2935:
2925:
2923:
2902:
2894:
2884:
2868:
2864:
2860:
2856:
2846:
2844:
2840:
2836:
2831:
2829:
2828:string theory
2825:
2806:
2801:
2797:
2793:
2790:
2787:
2784:
2778:
2775:
2772:
2765:
2764:
2763:
2747:
2737:
2724:
2720:
2716:
2712:
2707:
2694:
2686:
2678:
2668:
2652:
2648:
2644:
2640:
2621:
2618:
2607:
2599:
2595:
2591:
2572:
2569:
2558:
2550:
2540:
2527:
2524:
2479:
2476:
2465:
2457:
2441:
2427:
2424:
2421:
2418:
2398:
2393:
2378:
2362:
2342:
2306:
2303:
2280:
2277:
2266:
2243:
2240:
2229:
2206:
2203:
2194:
2191:
2188:
2182:
2179:
2173:
2170:
2167:
2161:
2153:
2150:
2145:
2136:
2130:
2127:
2121:
2115:
2107:
2104:
2099:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2065:
2064:
2063:
2047:
2037:
2012:
2009:
2006:
1986:
1983:
1980:
1972:
1954:
1927:
1907:
1901:
1898:
1895:
1889:
1886:
1880:
1874:
1851:
1845:
1823:
1796:
1788:
1784:
1765:
1762:
1751:
1742:
1729:
1723:
1717:
1695:
1681:
1677:
1673:
1669:
1666:
1650:
1647:
1644:
1637:
1619:
1609:
1581:
1578:
1567:
1544:
1541:
1526:Given a pair
1524:
1510:
1502:
1498:
1479:
1476:
1446:
1436:
1423:
1408:
1406:
1402:
1392:
1390:
1386:
1382:
1378:
1374:
1370:
1352:
1317:
1311:
1305:
1302:
1299:
1293:
1287:
1281:
1276:
1273:
1268:
1265:
1260:
1248:
1245:
1240:
1228:
1222:
1219:
1216:
1210:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1176:
1175:
1174:
1172:
1166:
1164:
1160:
1156:
1152:
1148:
1138:
1122:
1119:
1109:
1089:
1086:
1083:
1079:
1076:
1072:
1069:
1049:
1046:
1039:
1036:
1032:
1029:
1018:
1014:
1010:
1006:
1003:
988:
985:
977:
972:
969:
961:
957:
956:
954:
952:
927:
924:
921:
913:
910:
905:
899:
896:
893:
877:
866:
858:
848:
837:
834:
831:
828:
822:
819:
812:
811:
810:
791:
783:
773:
745:
737:
719:
716:
706:
687:
681:
678:
675:
672:
669:
666:
663:
657:
648:
645:
642:
629:
623:
620:
617:
597:
596:
595:
581:
568:
565:
560:
526:
516:
503:
493:
485:
471:
470:
469:
467:
463:
444:
435:
429:
426:
420:
414:
406:
403:
398:
392:
389:
386:
383:
380:
377:
374:
364:
363:
362:
360:
356:
352:
348:
344:
340:
336:
332:
328:
324:
323:inner product
319:
303:
293:
281:
277:
273:
269:
265:
260:
258:
254:
250:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
205:vector bundle
202:
198:
194:
190:
187:
185:
169:
167:
164:
160:
159:string theory
156:
152:
148:
144:
140:
136:
132:
128:
123:
121:
117:
113:
112:Nigel Hitchin
109:
105:
101:
97:
93:
89:
78:
75:
67:
57:
53:
47:
46:
40:
36:
32:
27:
18:
17:
5392:Arkani-Hamed
5290:Supergravity
5257:Moduli space
5236:
5154:
5149:Dirac string
5075:Gauge theory
5055:Loop algebra
4992:Black string
4855:GS formalism
4650:
4644:
4637:Vafa, Cumrun
4594:
4590:
4561:
4555:
4532:
4515:
4509:
4490:
4430:
4419:Martin RoÄek
4295:
4215:
4211:
4203:
4199:
4144:
4140:G-structures
4137:
4128:B-symplectic
4127:
4124:
4119:
4089:
4085:
4080:
4078:
4036:
4026:
4002:
4000:
3889:
3887:
3882:
3878:
3874:
3870:
3866:
3789:
3763:
3674:
3658:
3655:neighborhood
3649:
3540:
3452:
3418:
3416:
3404:
3347:
3343:
3339:
3337:
3322:
3318:
3316:
3309:
3297:
3234:
3176:
3081:
3079:
3077:on spinors.
3016:Weyl spinors
3003:
2959:
2950:pure spinors
2937:
2931:
2921:
2866:
2863:intersection
2858:
2854:
2852:
2832:
2830:literature.
2823:
2821:
2762:of the form
2721:, which are
2708:
2650:
2646:
2643:complex type
2642:
2638:
2593:
2589:
2548:
2546:
2455:
2221:
1970:
1969:annihilates
1786:
1782:
1744:To see that
1743:
1675:
1671:
1667:
1665:vector field
1525:
1500:
1496:
1419:
1404:
1400:
1398:
1384:
1376:
1372:
1335:
1167:
1144:
1107:
1105:
1016:
1008:
1004:
950:
948:
702:
543:
461:
459:
358:
354:
350:
346:
342:
338:
334:
330:
320:
275:
261:
256:
248:
244:
240:
232:
229:vector field
224:
216:
208:
200:
196:
188:
183:
181:Consider an
180:
139:Sergei Gukov
124:
95:
85:
70:
61:
50:Please help
42:
5752:Silverstein
5252:Orientifold
4987:Black holes
4982:Black brane
4939:Dual photon
4145:The bundle
4112:KĂ€hler form
3859:holomorphic
3352:pure spinor
3082:pure spinor
3012:isomorphism
2843:codimension
1636:formal sums
1163:holomorphic
1159:Lie bracket
736:eigenbundle
464:is just an
147:Cumrun Vafa
127:functionals
88:mathematics
56:introducing
5877:Categories
5772:Strominger
5767:Steinhardt
5762:Staudacher
5677:Polchinski
5627:Nanopoulos
5587:Mandelstam
5567:Kontsevich
5407:Berenstein
5335:Holography
5315:Superspace
5214:K3 surface
5173:Worldsheet
5088:Instantons
4716:Background
4499:References
4415:Chris Hull
3754:direct sum
3010:define an
2849:Real index
2723:isometries
2598:projection
2377:complement
1680:dual space
1663:where the
1461:and pairs
1395:Definition
1151:integrable
544:such that
280:direct sum
243:, denoted
172:Definition
5807:Veneziano
5687:Rajaraman
5582:Maldacena
5472:Gopakumar
5422:Dijkgraaf
5417:Curtright
5083:Anomalies
4962:NS5-brane
4883:U-duality
4878:S-duality
4873:T-duality
4621:119508517
4591:Phys. Rep
4571:0911.0993
4454:×
4428:in 1984.
4378:×
4338:⊗
4330:∗
4320:⊕
4302:commuting
4179:⊗
4171:∗
4161:⊕
4098:ϕ
4062:ω
4048:ϕ
4013:∂
3977:⊗
3969:∗
3959:⊕
3926:⊗
3918:∗
3908:⊕
3815:⊗
3805:∗
3800:Λ
3732:−
3630:⊗
3627:Δ
3619:¯
3611:∩
3577:⊗
3569:∗
3559:⊕
3519:⊗
3489:⊗
3481:∗
3471:⊕
3434:⊗
3390:Ω
3385:ω
3365:Φ
3271:⊗
3263:∗
3253:⊕
3213:⊗
3205:∗
3195:⊕
3155:⊗
3147:∗
3137:⊕
3099:∗
3094:Λ
3052:⊗
3044:∗
3034:⊕
2985:⊗
2975:∗
2970:Λ
2903:⊗
2895:∗
2885:⊕
2791:ξ
2782:⟶
2779:ξ
2748:∗
2738:⊕
2711:invariant
2687:⊗
2679:∗
2669:⊕
2622:ε
2573:ε
2525:ε
2495:for some
2480:ε
2428:
2422:−
2394:∗
2363:ε
2307:
2281:ε
2244:ε
2183:ε
2162:ε
2131:η
2116:ξ
2097:⟩
2094:η
2082:ξ
2073:⟨
2048:∗
2038:⊕
2013:η
1987:ξ
1955:∗
1928:ξ
1890:ε
1875:ξ
1846:ε
1824:∗
1797:ξ
1766:ε
1718:ε
1696:∗
1651:ξ
1620:∗
1610:⊕
1582:ε
1545:ε
1511:ε
1480:ε
1447:∗
1437:⊕
1422:subbundle
1318:ξ
1303:−
1300:η
1269:−
1266:ξ
1249:−
1246:η
1205:η
1193:ξ
1120:−
1084:∈
1077:ℓ
1070:ℓ
1044:⟩
1037:ℓ
1030:ℓ
1027:⟨
981:¯
973:∩
928:ξ
911:−
900:ξ
867:⊗
859:∗
849:⊕
838:∈
835:ξ
809:given by
792:⊗
784:∗
774:⊕
717:−
685:⟩
682:η
670:ξ
661:⟨
655:⟩
649:η
624:ξ
605:⟨
569:−
527:∗
517:⊕
509:→
504:∗
494:⊕
430:η
415:ξ
396:⟩
393:η
381:ξ
372:⟨
327:signature
304:∗
294:⊕
253:one-forms
203:, is the
186:-manifold
90:known as
64:June 2020
5862:Zwiebach
5817:Verlinde
5812:Verlinde
5787:Townsend
5782:Susskind
5717:Sagnotti
5682:Polyakov
5637:Nekrasov
5602:Minwalla
5597:Martinec
5562:Knizhnik
5557:Klebanov
5552:Kapustin
5517:'t Hooft
5452:Fischler
5387:AganagiÄ
5358:M-theory
5247:Conifold
5242:Orbifold
5225:manifold
5166:Geometry
4972:M5-brane
4967:M2-brane
4904:Graviton
4820:F-theory
4639:(2005).
3781:Examples
1080:′
1062:for all
1040:′
5792:Trivedi
5777:Sundrum
5742:Shenker
5732:Seiberg
5727:Schwarz
5697:Randall
5657:Novikov
5647:Nielsen
5632:NÄstase
5542:Kallosh
5527:Gibbons
5467:Gliozzi
5457:Friedan
5447:Ferrara
5432:Douglas
5427:Distler
4977:S-brane
4957:D-brane
4914:Tachyon
4909:Dilaton
4723:Strings
4560:. (2).
3679:of the
2835:integer
2719:B-field
2222:and so
1387:is the
1379:is the
1367:is the
1015:equals
333:,
221:section
52:improve
5857:Zumino
5852:Zaslow
5837:Yoneya
5827:Witten
5747:Siegel
5722:Scherk
5692:Ramond
5667:Ooguri
5592:Marolf
5547:Kaluza
5532:Kachru
5522:HoĆava
5512:Harvey
5507:Hanson
5492:Gubser
5482:Greene
5412:Bousso
5397:Atiyah
4949:Branes
4759:Theory
4619:
4116:scalar
3312:closed
2822:where
2713:under
1495:where
1336:where
881:
875:
337:). If
235:. The
191:. The
106:and a
5797:Turok
5707:RoÄek
5672:Ovrut
5662:Olive
5642:Neveu
5622:Myers
5617:Mukhi
5607:Moore
5577:Linde
5572:Klein
5497:Gukov
5487:Gross
5477:Green
5462:Gates
5442:Dvali
5402:Banks
4655:arXiv
4617:S2CID
4599:arXiv
4566:arXiv
4537:arXiv
2924:= 0.
2062:then
1867:then
1153:to a
594:and
325:with
227:is a
207:over
161:, as
37:, or
5822:Wess
5802:Vafa
5712:Rohm
5612:Motl
5537:Kaku
5502:Guth
5437:Duff
4417:and
3861:and
3652:open
2549:type
2547:The
2543:Type
2517:and
2355:and
1999:and
1789:and
1503:and
1383:and
1019:and
1013:rank
357:and
349:and
341:and
219:. A
145:and
118:and
94:, a
5832:Yau
5757:SÆĄn
5737:Sen
4665:doi
4609:doi
4595:423
4576:doi
4562:174
4520:doi
4202:, 2
4198:O(2
3506:to
3455:of
3179:of
2865:of
2725:of
2600:of
2425:dim
2379:of
2304:dim
1838:is
1597:of
1424:of
1149:is
1007:is
359:Y+η
355:X+Ο
262:In
255:on
239:of
231:on
223:of
215:to
195:of
129:of
5879::
5133:,
5126:,
5119:,
5112:,
4663:.
4649:.
4643:.
4631:;
4615:.
4607:.
4593:.
4574:.
4554:.
4516:54
4514:.
4413:,
4296:A
4214:,
4210:U(
4130:.
4122:.
3869:,
3777:.
3661:.
3325:.
2952:.
2845:.
1399:A
1391:.
1375:,
460:A
259:.
141:,
137:,
122:.
41:,
33:,
5223:2
5221:G
5190:?
5155:p
5140:)
5137:8
5135:E
5130:7
5128:E
5123:6
5121:E
5116:4
5114:F
5109:2
5107:G
5105:(
4701:e
4694:t
4687:v
4673:.
4667::
4657::
4651:9
4623:.
4611::
4601::
4584:.
4578::
4568::
4545:.
4539::
4526:.
4522::
4472:.
4469:)
4466:n
4463:(
4460:U
4457:S
4451:)
4448:n
4445:(
4442:U
4439:S
4393:.
4390:)
4387:n
4384:(
4381:U
4375:)
4372:n
4369:(
4366:U
4346:.
4342:C
4335:)
4325:T
4316:T
4312:(
4281:.
4275:)
4272:n
4269:,
4266:n
4263:(
4260:U
4255:)
4252:n
4249:2
4246:,
4243:n
4240:2
4237:(
4234:O
4218:)
4216:n
4212:n
4206:)
4204:n
4200:n
4183:C
4176:)
4166:T
4157:T
4153:(
4120:0
4081:Ï
4059:i
4055:e
4051:=
4027:N
4003:N
3981:C
3974:)
3964:T
3955:T
3951:(
3930:C
3923:)
3913:T
3904:T
3900:(
3890:n
3888:(
3883:n
3879:n
3875:m
3871:n
3867:m
3845:.
3841:C
3819:C
3811:T
3738:k
3735:2
3729:n
3726:2
3721:R
3697:k
3692:C
3634:C
3624:=
3616:E
3608:E
3585:.
3581:C
3574:)
3564:T
3555:T
3551:(
3541:L
3527:.
3523:C
3515:T
3493:C
3486:)
3476:T
3467:T
3463:(
3453:L
3438:C
3430:T
3419:E
3382:i
3379:+
3376:B
3372:e
3368:=
3348:k
3344:B
3340:k
3319:M
3279:.
3275:C
3268:)
3258:T
3249:T
3245:(
3235:E
3221:.
3217:C
3210:)
3200:T
3191:T
3187:(
3177:E
3163:.
3159:C
3152:)
3142:T
3133:T
3129:(
3109:,
3105:T
3056:C
3049:)
3039:T
3030:T
3026:(
3004:M
2989:C
2981:T
2922:r
2907:C
2900:)
2890:T
2881:T
2877:(
2867:L
2859:L
2855:r
2824:B
2807:B
2802:X
2798:i
2794:+
2788:+
2785:X
2776:+
2773:X
2743:T
2734:T
2695:.
2691:C
2684:)
2674:T
2665:T
2661:(
2651:N
2647:N
2639:T
2625:)
2619:,
2615:E
2611:(
2608:L
2594:N
2590:E
2576:)
2570:,
2566:E
2562:(
2559:L
2528:.
2504:E
2483:)
2477:,
2473:E
2469:(
2466:L
2456:n
2442:.
2439:)
2435:E
2431:(
2419:n
2399:,
2389:E
2343:,
2339:E
2318:)
2314:E
2310:(
2284:)
2278:,
2274:E
2270:(
2267:L
2247:)
2241:,
2237:E
2233:(
2230:L
2207:0
2204:=
2201:)
2198:)
2195:Y
2192:,
2189:X
2186:(
2180:+
2177:)
2174:X
2171:,
2168:Y
2165:(
2159:(
2154:2
2151:1
2146:=
2143:)
2140:)
2137:X
2134:(
2128:+
2125:)
2122:Y
2119:(
2113:(
2108:2
2105:1
2100:=
2091:+
2088:Y
2085:,
2079:+
2076:X
2043:T
2034:T
2010:+
2007:Y
1984:+
1981:X
1971:Y
1950:E
1908:,
1905:)
1902:Y
1899:,
1896:X
1893:(
1887:=
1884:)
1881:Y
1878:(
1855:)
1852:X
1849:(
1819:E
1787:E
1783:Y
1769:)
1763:,
1759:E
1755:(
1752:L
1730:.
1727:)
1724:X
1721:(
1691:E
1676:Ο
1672:E
1668:X
1648:+
1645:X
1615:T
1606:T
1585:)
1579:,
1575:E
1571:(
1568:L
1548:)
1542:,
1538:E
1534:(
1501:T
1497:E
1483:)
1477:,
1473:E
1469:(
1442:T
1433:T
1405:L
1385:i
1377:d
1373:X
1353:X
1347:L
1321:)
1315:)
1312:Y
1309:(
1306:i
1297:)
1294:X
1291:(
1288:i
1285:(
1282:d
1277:2
1274:1
1261:Y
1255:L
1241:X
1235:L
1229:+
1226:]
1223:Y
1220:,
1217:X
1214:[
1211:=
1208:]
1202:+
1199:Y
1196:,
1190:+
1187:X
1184:[
1123:1
1108:L
1090:.
1087:L
1073:,
1050:0
1047:=
1033:,
1017:N
1005:L
1001:;
989:0
986:=
978:L
970:L
951:L
934:}
931:)
925:+
922:X
919:(
914:1
906:=
903:)
897:+
894:X
891:(
886:J
878::
871:C
864:)
854:T
845:T
841:(
832:+
829:X
826:{
823:=
820:L
796:C
789:)
779:T
770:T
766:(
746:L
734:-
720:1
688:.
679:+
676:Y
673:,
667:+
664:X
658:=
652:)
646:+
643:Y
640:(
635:J
630:,
627:)
621:+
618:X
615:(
610:J
582:,
577:d
574:I
566:=
561:2
555:J
522:T
513:T
499:T
490:T
486::
481:J
445:.
442:)
439:)
436:X
433:(
427:+
424:)
421:Y
418:(
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407:2
404:1
399:=
390:+
387:Y
384:,
378:+
375:X
351:η
347:Ο
343:Y
339:X
335:N
331:N
329:(
299:T
290:T
257:M
249:M
245:T
241:M
233:M
225:T
217:M
209:M
201:T
197:M
189:M
184:N
77:)
71:(
66:)
62:(
48:.
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