1037:, proved in increasing generality, that Gromov–Witten and Donaldson–Thomas theories of algebraic three-folds are actually equivalent. More concretely, their generating functions are equal after an appropriate change of variables. For Calabi–Yau threefolds, the Donaldson–Thomas invariants can be formulated as weighted Euler characteristic on the moduli space. There have also been recent connections between these invariants, the motivic Hall algebra, and the ring of functions on the quantum torus.
39:
1539:
1720:
1325:
1016:
to a smooth target. The moduli stack of all such maps admits a virtual fundamental class, and intersection theory on this stack yields numerical invariants that can often contain enumerative information. In similar spirit, the approach of
Donaldson–Thomas theory is to study curves in an algebraic
1020:
Whereas in Gromov–Witten theory, maps are allowed to be multiple covers and collapsed components of the domain curve, Donaldson–Thomas theory allows for nilpotent information contained in the sheaves, however, these are integer valued invariants. There are deep conjectures due to
1570:
1534:{\displaystyle {\begin{aligned}T_{}{\mathcal {M}}^{\sigma }(Y,\alpha )&\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}})\\{\text{Ob}}_{}({\mathcal {M}}^{\sigma }(Y,\alpha ))&\cong {\text{Ext}}^{2}({\mathcal {E}},{\mathcal {E}})\end{aligned}}}
1319:-stable sheaves. These moduli stacks have much nicer properties, such as being separated of finite type. The only technical difficulty is they can have bad singularities due to the existence of obstructions of deformations of a fixed sheaf. In particular
2024:
the invariant defined above does not change. At the outset researchers chose the
Gieseker stability condition, but other DT-invariants in recent years have been studied based on other stability conditions, leading to wall-crossing formulas.
1847:
1017:
three-fold by their equations. More accurately, by studying ideal sheaves on a space. This moduli space also admits a virtual fundamental class and yields certain numerical invariants that are enumerative.
1715:{\displaystyle {\text{Ext}}^{2}({\mathcal {E}},{\mathcal {E}})\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}}\otimes \omega _{Y})^{\vee }\cong {\text{Ext}}^{1}({\mathcal {E}},{\mathcal {E}})^{\vee }}
994:
1129:
1330:
1253:
1207:
1168:
940:
1045:
is a discrete set of 2875 points. The virtual number of points is the actual number of points, and hence the
Donaldson–Thomas invariant of this moduli space is the integer 2875.
1952:
1277:
2241:
942:
of the moduli spaces being studied. Essentially, these stability conditions correspond to points in the Kahler moduli space of a Calabi-Yau manifold, as considered in
887:
1995:
1975:
1317:
1297:
2022:
2384:
1870:
1562:
1209:. In general, this is a non-separated Artin stack of infinite type which is difficult to define numerical invariants upon it. Instead, there are open substacks
1079:
251:
1731:
811:
851:
threefold, its
Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual
2344:
2217:
2304:
Kontsevich, Maxim; Soibelman, Yan (2008-11-16). "Stability structures, motivic
Donaldson-Thomas invariants and cluster transformations".
185:
2331:(1998), "Gauge theory in higher dimensions", in Huggett, S. A.; Mason, L. J.; Tod, K. P.; Tsou, S. T.; Woodhouse, N. M. J. (eds.),
949:
1084:
190:
2328:
1725:
which gives a perfect obstruction theory of dimension 0. In particular, this implies the associated virtual fundamental class
899:
868:
557:
2145:
Maulik, D.; Nekrasov, N.; Okounkov, A.; Pandharipande, R. (2006). "Gromov–Witten theory and
Donaldson–Thomas theory, I".
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804:
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17:
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1173:
1134:
780:
602:
522:
337:
2194:. Proceedings of Symposia in Pure Mathematics. Vol. 93. American Mathematical Society. pp. 363–396.
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652:
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2257:"A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $ K3$ fibrations"
607:
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1855:
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30:
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38:
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597:
477:
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382:
241:
2256:
2124:
Bridgeland, Tom (2006-02-08). "Stability conditions on triangulated categories".
1842:{\displaystyle ^{vir}\in H_{0}({\mathcal {M}}^{\sigma }(Y,\alpha ),\mathbb {Z} )}
2209:
757:
727:
707:
562:
517:
472:
437:
387:
2168:
702:
637:
572:
264:
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1012:
is to probe the geometry of a space by studying pseudoholomorphic maps from
612:
402:
342:
128:
123:
118:
138:
133:
68:
2273:
2159:
2130:
2064:
Instead of moduli spaces of sheaves, one considers moduli spaces of
2200:
2310:
2190:
Szendrői, Balázs (2016). "Cohomological
Donaldson–Thomas theory".
1049:
63:
1048:
Similarly, the
Donaldson–Thomas invariant of the moduli space of
879:
of algebraic three-folds and the theory of stable pairs due to
1897:
1802:
1741:
1697:
1687:
1639:
1629:
1601:
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1519:
1509:
1456:
1440:
1416:
1406:
1356:
1343:
1264:
1219:
1185:
1140:
978:
955:
924:
989:{\displaystyle {\mathcal {P}}\subset D^{b}({\mathcal {M}})}
886:
Donaldson–Thomas theory is physically motivated by certain
1124:{\displaystyle \alpha \in H^{\text{even}}(Y,\mathbb {Q} )}
875:). Donaldson–Thomas invariants have close connections to
2003:
1983:
1963:
1885:
1858:
1734:
1573:
1550:
1328:
1305:
1285:
1261:
1215:
1176:
1137:
1087:
1067:
952:
908:
898:. This is due to the fact the invariants depend on a
2075:
Instead of integer valued invariants, one considers
2046:. The weight function associates to every point in
996:
is the category of BPS states for the corresponding
2034:The Donaldson–Thomas invariant of the moduli space
1997:. It was proved by Thomas that for a smooth family
1248:{\displaystyle {\mathcal {M}}^{\sigma }(Y,\alpha )}
2016:
1989:
1969:
1946:
1864:
1841:
1714:
1556:
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1311:
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1271:
1247:
1201:
1162:
1123:
1073:
988:
934:
18:Maulik–Nekrasov–Okounkov–Pandharipande conjecture
2072:that count stable pairs of a Calabi–Yau 3-fold.
872:
805:
8:
2240:: CS1 maint: DOI inactive as of June 2024 (
1957:which depends upon the stability condition
2383:: CS1 maint: location missing publisher (
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26:
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1224:
1218:
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1214:
1202:{\displaystyle c({\mathcal {E}})=\alpha }
1184:
1183:
1175:
1170:of coherent sheaves with Chern character
1163:{\displaystyle {\mathcal {M}}(Y,\alpha )}
1139:
1138:
1136:
1114:
1113:
1098:
1086:
1066:
977:
976:
967:
954:
953:
951:
923:
922:
913:
907:
2116:
855:. The Donaldson–Thomas invariant is a
177:
151:
110:
76:
45:
29:
2376:
2233:
2333:The geometric universe (Oxford, 1996)
935:{\displaystyle D^{b}({\mathcal {M}})}
863:. The invariants were introduced by
7:
1564:is Calabi-Yau, Serre duality implies
1255:parametrizing such coherent sheaves
1131:there is an associated moduli stack
255:= 4 supersymmetric Yang–Mills theory
2373:, Mathematische Arbeitstagung, Bonn
186:Geometric Langlands correspondence
25:
1279:which have a stability condition
1041:The moduli space of lines on the
2261:Journal of Differential Geometry
946:, and the resulting subcategory
2070:Pandharipande–Thomas invariants
1947:{\displaystyle \int _{^{vir}}1}
1924:
1920:
1908:
1891:
1836:
1825:
1813:
1796:
1768:
1764:
1752:
1735:
1703:
1682:
1658:
1624:
1606:
1586:
1524:
1504:
1482:
1479:
1467:
1450:
1445:
1435:
1421:
1401:
1379:
1367:
1348:
1338:
1272:{\displaystyle {\mathcal {E}}}
1242:
1230:
1190:
1180:
1157:
1145:
1118:
1104:
983:
973:
929:
919:
1:
1081:and a fixed cohomology class
824:In mathematics, specifically
2054:of a hyperplane singularity.
2367:Donaldson–Thomas invariants
1061:For a Calabi-Yau threefold
834:Donaldson–Thomas invariants
2423:
2169:10.1112/S0010437X06002302
2038:is equal to the weighted
1977:and the cohomology class
1872:. We can then define the
1852:is in homological degree
1052:on the quintic is 609250.
2068:objects. That gives the
1299:imposed upon them, i.e.
1010:Gromov–Witten invariants
902:on the derived category
877:Gromov–Witten invariants
111:Non-perturbative results
2337:Oxford University Press
2212:(inactive 2024-06-23).
2210:10.1090/pspum/093/01589
2095:Gromov–Witten invariant
1990:{\displaystyle \alpha }
1970:{\displaystyle \sigma }
1312:{\displaystyle \sigma }
1292:{\displaystyle \sigma }
1004:Definition and examples
830:Donaldson–Thomas theory
2283:10.4310/jdg/1214341649
2255:Thomas, R. P. (2000).
2147:Compositio Mathematica
2018:
1991:
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1075:
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227:Conformal field theory
144:AdS/CFT correspondence
2019:
2017:{\displaystyle Y_{t}}
1992:
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270:Holographic principle
247:Twistor string theory
2090:Enumerative geometry
2040:Euler characteristic
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222:Theory of everything
2325:Donaldson, Simon K.
2050:an analogue of the
1035:Rahul Pandharipande
900:stability condition
881:Rahul Pandharipande
865:Simon Donaldson
260:Kaluza–Klein theory
196:Monstrous moonshine
77:Perturbative theory
46:Fundamental objects
2402:Algebraic geometry
2339:, pp. 31–47,
2329:Thomas, Richard P.
2105:Quantum cohomology
2014:
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1008:The basic idea of
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932:
826:algebraic geometry
2362:Kontsevich, Maxim
2346:978-0-19-850059-9
2219:978-1-4704-1992-9
1865:{\displaystyle 0}
1674:
1616:
1578:
1557:{\displaystyle Y}
1496:
1432:
1393:
1101:
1074:{\displaystyle Y}
1043:quintic threefold
853:fundamental class
832:is the theory of
822:
821:
553:van Nieuwenhuizen
16:(Redirected from
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2192:String-Math 2014
2187:
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2153:(5): 1263–1285.
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2066:derived category
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1128:
1127:
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1117:
1103:
1102:
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1080:
1078:
1077:
1072:
1014:Riemann surfaces
995:
993:
992:
987:
982:
981:
972:
971:
959:
958:
941:
939:
938:
933:
928:
927:
918:
917:
861:Casson invariant
859:analogue of the
814:
807:
800:
216:Related concepts
41:
27:
21:
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2236:cite conference
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2059:Generalizations
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1133:
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1094:
1083:
1082:
1063:
1062:
1059:
1031:Nikita Nekrasov
1027:Andrei Okounkov
1006:
963:
948:
947:
944:mirror symmetry
909:
904:
903:
818:
773:
772:
283:
275:
274:
232:Quantum gravity
217:
191:Mirror symmetry
23:
22:
15:
12:
11:
5:
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2404:
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2317:
2296:
2267:(2): 367–438.
2247:
2218:
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2137:
2115:
2114:
2112:
2109:
2108:
2107:
2102:
2100:Hilbert scheme
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916:
912:
890:that occur in
869:Richard Thomas
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405:
400:
395:
390:
385:
380:
375:
370:
365:
360:
355:
350:
345:
340:
335:
330:
325:
320:
315:
310:
305:
300:
295:
290:
284:
281:
280:
277:
276:
273:
272:
267:
262:
257:
249:
244:
239:
234:
229:
224:
218:
215:
214:
211:
210:
209:
208:
203:
201:Vertex algebra
198:
193:
188:
180:
179:
175:
174:
173:
172:
167:
162:
154:
153:
149:
148:
147:
146:
141:
136:
131:
126:
121:
113:
112:
108:
107:
106:
105:
87:
79:
78:
74:
73:
72:
71:
66:
61:
56:
48:
47:
43:
42:
34:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2419:
2408:
2407:String theory
2405:
2403:
2400:
2399:
2397:
2386:
2380:
2369:
2368:
2363:
2359:
2356:
2352:
2348:
2342:
2338:
2334:
2330:
2326:
2322:
2321:
2312:
2307:
2300:
2297:
2292:
2288:
2284:
2280:
2275:
2270:
2266:
2262:
2258:
2251:
2248:
2243:
2237:
2229:
2225:
2221:
2215:
2211:
2207:
2202:
2197:
2193:
2186:
2183:
2178:
2174:
2170:
2166:
2161:
2156:
2152:
2148:
2141:
2138:
2132:
2127:
2120:
2117:
2110:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2087:
2083:
2078:
2074:
2071:
2067:
2063:
2062:
2058:
2053:
2052:Milnor number
2049:
2045:
2041:
2037:
2033:
2032:
2028:
2026:
2009:
2005:
1984:
1964:
1954:
1941:
1934:
1931:
1928:
1917:
1914:
1911:
1903:
1887:
1877:
1875:
1859:
1849:
1828:
1822:
1819:
1816:
1808:
1791:
1787:
1783:
1778:
1775:
1772:
1761:
1758:
1755:
1747:
1726:
1722:
1707:
1692:
1677:
1667:
1662:
1652:
1648:
1644:
1634:
1619:
1609:
1596:
1581:
1565:
1551:
1541:
1514:
1499:
1489:
1487:
1476:
1473:
1470:
1462:
1411:
1396:
1386:
1384:
1376:
1373:
1370:
1362:
1334:
1320:
1306:
1286:
1239:
1236:
1233:
1225:
1196:
1193:
1177:
1154:
1151:
1148:
1110:
1107:
1095:
1091:
1088:
1068:
1056:
1051:
1047:
1044:
1040:
1039:
1038:
1036:
1032:
1028:
1024:
1023:Davesh Maulik
1018:
1015:
1011:
1003:
1001:
999:
968:
964:
960:
945:
914:
910:
901:
897:
893:
889:
884:
883:and Thomas.
882:
878:
874:
870:
867: and
866:
862:
858:
854:
850:
846:
842:
839:
835:
831:
827:
815:
810:
808:
803:
801:
796:
795:
793:
792:
787:
784:
782:
779:
778:
777:
776:
769:
766:
764:
761:
759:
756:
754:
753:Zamolodchikov
751:
749:
748:Zamolodchikov
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
614:
611:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
539:
536:
534:
531:
529:
526:
524:
521:
519:
516:
514:
511:
509:
506:
504:
501:
499:
496:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
424:
421:
419:
416:
414:
411:
409:
406:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
349:
346:
344:
341:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
285:
279:
278:
271:
268:
266:
263:
261:
258:
256:
254:
250:
248:
245:
243:
240:
238:
237:Supersymmetry
235:
233:
230:
228:
225:
223:
220:
219:
213:
212:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
183:
182:
181:
176:
171:
168:
166:
163:
161:
160:Phenomenology
158:
157:
156:
155:
152:Phenomenology
150:
145:
142:
140:
137:
135:
132:
130:
127:
125:
122:
120:
117:
116:
115:
114:
109:
103:
99:
95:
91:
88:
86:
83:
82:
81:
80:
75:
70:
67:
65:
62:
60:
59:Cosmic string
57:
55:
52:
51:
50:
49:
44:
40:
36:
35:
32:
31:String theory
28:
19:
2366:
2332:
2299:
2274:math/9806111
2264:
2260:
2250:
2191:
2185:
2160:math/0312059
2150:
2146:
2140:
2131:math/0212237
2119:
2047:
2043:
2035:
1956:
1879:
1874:DT invariant
1873:
1851:
1728:
1724:
1567:
1544:Now because
1543:
1322:
1060:
1019:
1007:
896:gauge theory
885:
841:moduli space
836:. Given a
833:
829:
823:
293:Arkani-Hamed
252:
242:Supergravity
2079:invariants.
857:holomorphic
653:Silverstein
178:Mathematics
90:Superstring
2396:Categories
2201:1503.07349
2111:References
1057:Definition
888:BPS states
849:Calabi–Yau
673:Strominger
668:Steinhardt
663:Staudacher
578:Polchinski
528:Nanopoulos
488:Mandelstam
468:Kontsevich
308:Berenstein
265:Multiverse
2311:0811.2435
1985:α
1965:σ
1918:α
1904:σ
1888:∫
1823:α
1809:σ
1784:∈
1762:α
1748:σ
1708:∨
1668:≅
1663:∨
1649:ω
1645:⊗
1610:≅
1490:≅
1477:α
1463:σ
1387:≅
1377:α
1363:σ
1307:σ
1287:σ
1240:α
1226:σ
1197:α
1155:α
1092:∈
1089:α
961:⊂
713:Veneziano
588:Rajaraman
483:Maldacena
373:Gopakumar
323:Dijkgraaf
318:Curtright
282:Theorists
170:Landscape
165:Cosmology
129:U-duality
124:T-duality
119:S-duality
102:Heterotic
2379:citation
2364:(2007),
2084:See also
786:Glossary
768:Zwiebach
723:Verlinde
718:Verlinde
693:Townsend
688:'t Hooft
683:Susskind
618:Sagnotti
583:Polyakov
538:Nekrasov
503:Minwalla
498:Martinec
463:Knizhnik
458:Klebanov
453:Kapustin
423:Horowitz
353:Fischler
288:Aganagić
206:K-theory
139:F-theory
134:M-theory
2355:1634503
2291:1818182
2228:3526001
2177:5760317
2077:motivic
871: (
845:sheaves
838:compact
781:History
698:Trivedi
678:Sundrum
643:Shenker
633:Seiberg
628:Schwarz
598:Randall
558:Novikov
548:Nielsen
533:Năstase
443:Kallosh
428:Gibbons
368:Gliozzi
358:Friedan
348:Ferrara
333:Douglas
328:Distler
98:Type II
85:Bosonic
69:D-brane
2353:
2343:
2289:
2226:
2216:
2175:
1050:conics
892:string
763:Zumino
758:Zaslow
743:Yoneya
733:Witten
648:Siegel
623:Scherk
593:Ramond
568:Ooguri
493:Marolf
448:Kaluza
433:Kachru
418:Hořava
413:Harvey
408:Hanson
393:Gubser
383:Greene
313:Bousso
298:Atiyah
94:Type I
54:String
2371:(PDF)
2306:arXiv
2269:arXiv
2196:arXiv
2173:S2CID
2155:arXiv
2126:arXiv
2029:Facts
847:on a
703:Turok
608:Roček
573:Ovrut
563:Olive
543:Neveu
523:Myers
518:Mukhi
508:Moore
478:Linde
473:Klein
398:Gukov
388:Gross
378:Green
363:Gates
343:Dvali
303:Banks
64:Brane
2385:link
2341:ISBN
2242:link
2214:ISBN
1100:even
1033:and
998:SCFT
894:and
873:1998
728:Wess
708:Vafa
613:Rohm
513:Motl
438:Kaku
403:Guth
338:Duff
2279:doi
2206:doi
2165:doi
2151:142
2042:of
1673:Ext
1615:Ext
1577:Ext
1495:Ext
1392:Ext
843:of
738:Yau
658:Sơn
638:Sen
2398::
2381:}}
2377:{{
2351:MR
2349:,
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2327:;
2287:MR
2285:.
2277:.
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2259:.
2238:}}
2234:{{
2224:MR
2222:.
2204:.
2171:.
2163:.
2149:.
1876:as
1431:Ob
1029:,
1025:,
1000:.
828:,
100:,
96:,
2387:)
2314:.
2308::
2293:.
2281::
2271::
2244:)
2230:.
2208::
2198::
2179:.
2167::
2157::
2134:.
2128::
2048:M
2044:M
2036:M
2010:t
2006:Y
1942:1
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1932:i
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1921:)
1915:,
1912:Y
1909:(
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1860:0
1837:)
1833:Z
1829:,
1826:)
1820:,
1817:Y
1814:(
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1797:(
1792:0
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1765:)
1759:,
1756:Y
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1742:M
1736:[
1704:)
1698:E
1693:,
1688:E
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1587:(
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1471:Y
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974:(
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