39:
1890:
2790:. Here there are six spacetime dimensions, which constitute a symplectic manifold, and it turns out that the worldsheets are necessarily parametrized by pseudoholomorphic curves, whose moduli spaces are only finite-dimensional. GW invariants, as integrals over these moduli spaces, are then path integrals of the theory. In particular, the
2436:
2731:
The Gromov-Witten invariants of smooth projective varieties can be defined entirely within algebraic geometry. The classical enumerative geometry of plane curves and of rational curves in homogeneous spaces are both captured by GW invariants. However, the major advantage that GW invariants have over
1634:
2732:
the classical enumerative counts is that they are invariant under deformations of the complex structure of the target. The GW invariants also furnish deformations of the product structure in the cohomology ring of a symplectic or projective manifold; they can be organized to construct the
2728:, which are meant to give an underlying integer count to the typically rational Gromov-Witten theory. The Gopakumar-Vafa invariants do not presently have a rigorous mathematical definition, and this is one of the major problems in the subject.
1480:
2255:
2740:, which is a deformation of the ordinary cohomology. The associativity of the deformed product is essentially a consequence of the self-similar nature of the moduli space of stable maps that are used to define the invariants.
1885:{\displaystyle {\begin{cases}\mathrm {ev} :{\overline {\mathcal {M}}}_{g,n}(X,A)\to Y\\\mathrm {ev} (C,x_{1},\ldots ,x_{n},f)=\left(\operatorname {st} (C,x_{1},\ldots ,x_{n}),f(x_{1}),\ldots ,f(x_{n})\right).\end{cases}}}
2683:
to one or more other spaces whose GW invariants are more easily computed. Of course, one must first understand how the invariants behave under the surgeries. For such applications one often uses the more elaborate
1562:
2053:
2235:
1958:
1152:
1072:
2133:
1372:
1003:
2179:
1223:
2596:
2771:. As a string travels through spacetime it traces out a surface, called the worldsheet of the string. Unfortunately, the moduli space of such parametrized surfaces, at least
1622:
2510:
3005:
2459:
2431:{\displaystyle GW_{g,n}^{X,A}(\beta ,\alpha _{1},\ldots ,\alpha _{n}):=GW_{g,n}^{X,A}\cdot \beta \cdot \alpha _{1}\cdots \alpha _{n}\in H_{0}(Y,\mathbb {Q} ),}
1391:
251:
2523:. However, due to the "virtual" nature of the count, it need not be a natural number, as one might expect a count to be. For the space of stable maps is an
1006:
2860:
Piunikhin, Sergey; Salamon, Dietmar & Schwarz, Matthias (1996). "Symplectic Floer–Donaldson theory and quantum cohomology". In Thomas, C. B. (ed.).
3428:
876:. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed
2625:
and thus a moduli space of pseudoholomorphic curves that is larger than expected. Loosely speaking, one corrects for this effect by forming from the
2720:) are equivalent to the Seiberg–Witten invariants. For algebraic threefolds, they are conjectured to contain the same information as integer valued
811:
3157:
3520:
3117:
2998:
2913:
895:
article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important.
2665:
2856:
An analytically flavoured overview of Gromov–Witten invariants and quantum cohomology for symplectic manifolds, very technically complete
2530:
There are numerous variations on this construction, in which cohomology is used instead of homology, integration replaces intersection,
3586:
3208:
3107:
1503:
185:
3576:
2881:
2849:
3286:
2991:
190:
2897:
2721:
1978:
891:
The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the
3433:
3344:
557:
3354:
3281:
2188:
1906:
1100:
1020:
3031:
3251:
2725:
2705:
3611:
3147:
2717:
2676:, to reduce, or localize, the computation of a GW invariant to an integration over the fixed-point locus of the action.
582:
3510:
2709:
2610:
with special properties, such as nongeneric symmetries or integrability. Indeed, computations are often carried out on
3616:
3173:
3086:
2096:
1279:
966:
804:
2138:
3484:
3122:
2869:
2473:
for the given classes. This number gives a "virtual" count of the number of pseudoholomorphic curves (in the class
881:
3530:
2787:
2780:
785:
417:
377:
3443:
3423:
3359:
3276:
3137:
3178:
1164:
780:
602:
522:
337:
3142:
2559:
482:
259:
3334:
2543:
2083:
1087:
853:
747:
552:
169:
101:
3626:
3621:
3127:
877:
861:
797:
507:
226:
143:
3505:
3241:
657:
3041:
2791:
2776:
362:
302:
269:
246:
97:
84:
3203:
3152:
532:
3581:
3453:
3364:
3112:
2947:
2925:
752:
221:
159:
93:
3418:
2664:, meaning that it is acted upon by a complex torus, or at least locally toric. Then one can use the
2542:
Gromov–Witten invariants are generally difficult to compute. While they are defined for any generic
1643:
3296:
3261:
3218:
3198:
2802:
2795:
2764:
2701:
2646:
2645:
of the obstruction bundle. Making this idea precise requires significant technical arguments using
2462:
914:
857:
829:
662:
547:
195:
1574:
3548:
3132:
2937:
2756:
2733:
2488:
873:
833:
712:
627:
527:
487:
367:
332:
205:
89:
3339:
3319:
3291:
2611:
2246:
687:
2873:
2700:
The GW invariants are closely related to a number of other concepts in geometry, including the
3448:
3395:
3266:
3081:
3076:
2909:
2898:
Moduli Spaces of Genus-One Stable Maps, Virtual
Classes and an Exercise of Intersection Theory
2877:
2845:
2824:
2818:
2760:
1900:
672:
577:
412:
322:
292:
3438:
3324:
3301:
2955:
2861:
2768:
2444:
682:
617:
587:
467:
407:
372:
317:
307:
287:
164:
53:
2755:
GW invariants are of interest in string theory, a branch of physics that attempts to unify
2688:, which count curves with prescribed tangency conditions along a symplectic submanifold of
3553:
3369:
3311:
3213:
3036:
3015:
2713:
911:
849:
767:
722:
667:
652:
642:
537:
502:
327:
231:
2966:
607:
3236:
2951:
1475:{\displaystyle \mathrm {st} (C,x_{1},\ldots ,x_{n})\in {\overline {\mathcal {M}}}_{g,n}}
3061:
3046:
3023:
2924:, treats extensively the case of projective spaces using the basics in the language of
2744:
2669:
2661:
742:
737:
697:
632:
622:
542:
462:
452:
447:
442:
357:
352:
347:
312:
297:
200:
3605:
3568:
3349:
3329:
3256:
3051:
2862:
2634:
2550:
2093:
To interpret the Gromov–Witten invariant geometrically, let β be a homology class in
885:
732:
717:
692:
677:
647:
592:
567:
512:
497:
492:
457:
432:
422:
392:
236:
58:
30:
38:
17:
3515:
3489:
3479:
3469:
3271:
3091:
2921:
2906:
An
Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves
762:
597:
477:
427:
397:
382:
241:
3390:
3228:
2959:
2837:
2642:
2531:
869:
825:
757:
727:
707:
562:
517:
472:
437:
387:
3385:
2673:
2599:
2527:, whose points of isotropy can contribute noninteger values to the invariant.
1490:
1075:
892:
865:
702:
637:
572:
264:
2983:
3246:
612:
402:
342:
128:
123:
118:
2978:
2932:
Vakil, Ravi (2006). "The Moduli Space of Curves and Gromov–Witten Theory".
2515:
Put simply, a GW invariant counts how many curves there are that intersect
2626:
2524:
138:
133:
2743:
The quantum cohomology ring is known to be isomorphic to the symplectic
3558:
3543:
2920:
A nice introduction with history and exercises to the formal notion of
947:
Now we define the Gromov–Witten invariants associated to the 4-tuple: (
68:
3538:
2942:
2534:
pulled back from the
Deligne–Mumford space are also integrated, etc.
2602:, they must actually be computed with respect to a specific, chosen
2712:
in the algebraic category. For compact symplectic four-manifolds,
63:
2763:. In this theory, everything in the universe, beginning with the
2481:, with domain in the β-part of the Deligne–Mumford space) whose
1557:{\displaystyle Y:={\overline {\mathcal {M}}}_{g,n}\times X^{n},}
2987:
2934:
Enumerative
Invariants in Algebraic Geometry and String Theory
2679:
Another approach is to employ symplectic surgeries to relate
2641:, and then realizing the GW invariant as the integral of the
2104:
1914:
1661:
1517:
1451:
1108:
1028:
974:
2086:
of the symplectic isotopy class of the symplectic manifold
1878:
1269:
is pseudoholomorphic. The moduli space has real dimension
2844:. American Mathematical Society colloquium publications.
2048:{\displaystyle GW_{g,n}^{X,A}\in H_{d}(Y,\mathbb {Q} ).}
2979:
Gromov-Witten theory of schemes in mixed characteristic
2230:{\displaystyle \beta ,\alpha _{1},\ldots ,\alpha _{n}}
2562:
2491:
2447:
2258:
2191:
2141:
2099:
1981:
1953:{\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)}
1909:
1637:
1577:
1506:
1394:
1282:
1167:
1147:{\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)}
1103:
1097:
compatible with its symplectic form. The elements of
1067:{\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)}
1023:
969:
2485:
marked points are mapped to cycles representing the
3567:
3529:
3498:
3462:
3411:
3404:
3378:
3310:
3227:
3191:
3166:
3100:
3069:
3060:
3022:
2590:
2504:
2453:
2430:
2229:
2173:
2127:
2047:
1952:
1884:
1616:
1556:
1474:
1366:
1217:
1146:
1066:
997:
868:class in an appropriate space, or as the deformed
2786:The situation improves in the variation known as
2716:showed that a variant of the GW invariants (see
2128:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
1367:{\displaystyle d:=2c_{1}^{X}(A)+(2k-6)(1-g)+2n.}
998:{\displaystyle {\overline {\mathcal {M}}}_{g,n}}
2174:{\displaystyle \alpha _{1},\ldots ,\alpha _{n}}
2936:. Vol. 1947. Springer. pp. 143–198.
2999:
805:
8:
2842:J-Holomorphic Curves and Symplectic Topology
2724:. Physical considerations also give rise to
2614:using the techniques of algebraic geometry.
2967:Notes on stable maps and quantum cohomology
2904:Kock, Joachim; Vainsencher, Israel (2007).
2185:, such that the sum of the codimensions of
3408:
3066:
3006:
2992:
2984:
2783:of the theory lack a rigorous definition.
2775:, is infinite-dimensional; no appropriate
2696:Related invariants and other constructions
812:
798:
37:
26:
2941:
2576:
2565:
2564:
2561:
2496:
2490:
2446:
2418:
2417:
2402:
2389:
2376:
2351:
2340:
2321:
2302:
2277:
2266:
2257:
2221:
2202:
2190:
2165:
2146:
2140:
2113:
2103:
2101:
2098:
2035:
2034:
2019:
2000:
1989:
1980:
1923:
1913:
1911:
1908:
1858:
1830:
1808:
1789:
1747:
1728:
1707:
1670:
1660:
1658:
1646:
1638:
1636:
1576:
1545:
1526:
1516:
1514:
1505:
1460:
1450:
1448:
1435:
1416:
1395:
1393:
1301:
1296:
1281:
1237:is a (not necessarily stable) curve with
1218:{\displaystyle (C,x_{1},\ldots ,x_{n},f)}
1200:
1181:
1166:
1117:
1107:
1105:
1102:
1037:
1027:
1025:
1022:
983:
973:
971:
968:
860:. The GW invariants may be packaged as a
856:meeting prescribed conditions in a given
2591:{\displaystyle {\bar {\partial }}_{j,J}}
1964:-dimensional rational homology class in
2062:In a sense, this homology class is the
177:
151:
110:
76:
45:
29:
3429:Clifford's theorem on special divisors
1007:Deligne–Mumford moduli space of curves
2779:on this space is known, and thus the
7:
2652:The main computational technique is
927:: a 2-dimensional homology class in
255:= 4 supersymmetric Yang–Mills theory
2241:. These induce homology classes in
852:that, in certain situations, count
3587:Vector bundles on algebraic curves
3521:Weber's theorem (Algebraic curves)
3118:Hasse's theorem on elliptic curves
3108:Counting points on elliptic curves
2606:. It is most convenient to choose
2567:
1711:
1708:
1650:
1647:
1399:
1396:
186:Geometric Langlands correspondence
25:
2469:. This is a rational number, the
2747:with its pair-of-pants product.
2708:in the symplectic category, and
3209:Hurwitz's automorphisms theorem
2864:Contact and Symplectic Geometry
2840:& Salamon, Dietmar (2004).
2666:Atiyah–Bott fixed-point theorem
3434:Gonality of an algebraic curve
3345:Differential of the first kind
2570:
2422:
2408:
2327:
2289:
2039:
2025:
1947:
1935:
1864:
1851:
1836:
1823:
1814:
1776:
1759:
1715:
1697:
1694:
1682:
1608:
1596:
1441:
1403:
1349:
1337:
1334:
1319:
1313:
1307:
1212:
1168:
1141:
1129:
1061:
1049:
1:
3577:Birkhoff–Grothendieck theorem
3287:Nagata's conjecture on curves
3158:Schoof–Elkies–Atkin algorithm
3032:Five points determine a conic
1899:The evaluation map sends the
1624:. There is an evaluation map
3148:Supersingular elliptic curve
2465:in the rational homology of
2108:
1918:
1665:
1617:{\displaystyle 6g-6+2(k+1)n}
1521:
1455:
1112:
1032:
978:
3355:Riemann's existence theorem
3282:Hilbert's sixteenth problem
3174:Elliptic curve cryptography
3087:Fundamental pair of periods
2960:10.1007/978-3-540-79814-9_4
2722:Donaldson–Thomas invariants
2621:may induce a nonsurjective
2505:{\displaystyle \alpha _{i}}
1074:denote the moduli space of
3643:
3485:Moduli of algebraic curves
2870:Cambridge University Press
2726:Gopakumar–Vafa invariants
2718:Taubes's Gromov invariant
2706:Seiberg–Witten invariants
2692:of real codimension two.
1571:which has real dimension
943:: a non-negative integer.
937:: a non-negative integer,
3252:Cayley–Bacharach theorem
3179:Elliptic curve primality
2821:– for deformation theory
2544:almost complex structure
2538:Computational techniques
1088:almost complex structure
903:Consider the following:
854:pseudoholomorphic curves
111:Non-perturbative results
3511:Riemann–Hurwitz formula
3475:Gromov–Witten invariant
3335:Compact Riemann surface
3123:Mazur's torsion theorem
2710:Donaldson–Thomas theory
2519:chosen submanifolds of
2471:Gromov–Witten invariant
2064:Gromov–Witten invariant
880:. They are named after
3128:Modular elliptic curve
2908:. New York: Springer.
2751:Application in physics
2686:relative GW invariants
2592:
2506:
2455:
2454:{\displaystyle \cdot }
2432:
2231:
2175:
2129:
2049:
1954:
1886:
1618:
1558:
1476:
1368:
1219:
1148:
1068:
999:
878:type IIA string theory
227:Conformal field theory
144:AdS/CFT correspondence
3042:Rational normal curve
2736:ring of the manifold
2593:
2507:
2456:
2433:
2232:
2176:
2130:
2050:
1955:
1887:
1619:
1559:
1477:
1369:
1220:
1149:
1069:
1000:
270:Holographic principle
247:Twistor string theory
3582:Stable vector bundle
3454:Weil reciprocity law
3444:Riemann–Roch theorem
3424:Brill–Noether theory
3360:Riemann–Roch theorem
3277:Genus–degree formula
3138:Mordell–Weil theorem
3113:Division polynomials
2765:elementary particles
2702:Donaldson invariants
2656:. This applies when
2647:Kuranishi structures
2560:
2489:
2463:intersection product
2445:
2256:
2189:
2181:homology classes in
2139:
2097:
1979:
1907:
1635:
1575:
1504:
1392:
1280:
1165:
1101:
1021:
967:
222:Theory of everything
18:Gromov–Witten theory
3612:Symplectic topology
3405:Structure of curves
3297:Quartic plane curve
3219:Hyperelliptic curve
3199:De Franchis theorem
3143:Nagell–Lutz theorem
2952:2006math......2347V
2803:generating function
2617:However, a special
2362:
2288:
2011:
1306:
915:symplectic manifold
858:symplectic manifold
830:symplectic topology
260:Kaluza–Klein theory
196:Monstrous moonshine
77:Perturbative theory
46:Fundamental objects
3617:Algebraic geometry
3412:Divisors on curves
3204:Faltings's theorem
3153:Schoof's algorithm
3133:Modularity theorem
2794:of the A-model at
2767:, is made of tiny
2757:general relativity
2734:quantum cohomology
2639:obstruction bundle
2588:
2502:
2451:
2428:
2336:
2262:
2227:
2171:
2125:
2045:
1985:
1950:
1882:
1877:
1614:
1554:
1493:of the curve. Let
1472:
1364:
1292:
1215:
1154:are of the form:
1144:
1086:, for some chosen
1064:
1017:marked points and
995:
874:quantum cohomology
834:algebraic geometry
828:, specifically in
3599:
3598:
3595:
3594:
3506:Hasse–Witt matrix
3449:Weierstrass point
3396:Smooth completion
3365:Teichmüller space
3267:Cubic plane curve
3187:
3186:
3101:Arithmetic theory
3082:Elliptic integral
3077:Elliptic function
2973:Research articles
2915:978-0-8176-4456-7
2825:Schubert calculus
2819:Cotangent complex
2761:quantum mechanics
2573:
2111:
1921:
1901:fundamental class
1668:
1524:
1458:
1115:
1035:
981:
822:
821:
553:van Nieuwenhuizen
16:(Redirected from
3634:
3439:Jacobian variety
3409:
3312:Riemann surfaces
3302:Real plane curve
3262:Cramer's paradox
3242:Bézout's theorem
3067:
3016:algebraic curves
3008:
3001:
2994:
2985:
2963:
2945:
2919:
2900:- Andrea Tirelli
2887:
2867:
2855:
2612:Kähler manifolds
2597:
2595:
2594:
2589:
2587:
2586:
2575:
2574:
2566:
2549:, for which the
2511:
2509:
2508:
2503:
2501:
2500:
2460:
2458:
2457:
2452:
2437:
2435:
2434:
2429:
2421:
2407:
2406:
2394:
2393:
2381:
2380:
2361:
2350:
2326:
2325:
2307:
2306:
2287:
2276:
2236:
2234:
2233:
2228:
2226:
2225:
2207:
2206:
2180:
2178:
2177:
2172:
2170:
2169:
2151:
2150:
2134:
2132:
2131:
2126:
2124:
2123:
2112:
2107:
2102:
2054:
2052:
2051:
2046:
2038:
2024:
2023:
2010:
1999:
1959:
1957:
1956:
1951:
1934:
1933:
1922:
1917:
1912:
1891:
1889:
1888:
1883:
1881:
1880:
1871:
1867:
1863:
1862:
1835:
1834:
1813:
1812:
1794:
1793:
1752:
1751:
1733:
1732:
1714:
1681:
1680:
1669:
1664:
1659:
1653:
1623:
1621:
1620:
1615:
1563:
1561:
1560:
1555:
1550:
1549:
1537:
1536:
1525:
1520:
1515:
1481:
1479:
1478:
1473:
1471:
1470:
1459:
1454:
1449:
1440:
1439:
1421:
1420:
1402:
1373:
1371:
1370:
1365:
1305:
1300:
1224:
1222:
1221:
1216:
1205:
1204:
1186:
1185:
1153:
1151:
1150:
1145:
1128:
1127:
1116:
1111:
1106:
1073:
1071:
1070:
1065:
1048:
1047:
1036:
1031:
1026:
1004:
1002:
1001:
996:
994:
993:
982:
977:
972:
850:rational numbers
814:
807:
800:
216:Related concepts
41:
27:
21:
3642:
3641:
3637:
3636:
3635:
3633:
3632:
3631:
3602:
3601:
3600:
3591:
3563:
3554:Delta invariant
3525:
3494:
3458:
3419:Abel–Jacobi map
3400:
3374:
3370:Torelli theorem
3340:Dessin d'enfant
3320:Belyi's theorem
3306:
3292:Plücker formula
3223:
3214:Hurwitz surface
3183:
3162:
3096:
3070:Analytic theory
3062:Elliptic curves
3056:
3037:Projective line
3024:Rational curves
3018:
3012:
2975:
2931:
2916:
2903:
2894:
2892:Further reading
2884:
2859:
2852:
2836:
2833:
2815:
2809:GW invariants.
2753:
2714:Clifford Taubes
2698:
2563:
2558:
2557:
2540:
2492:
2487:
2486:
2443:
2442:
2398:
2385:
2372:
2317:
2298:
2254:
2253:
2247:Künneth formula
2217:
2198:
2187:
2186:
2161:
2142:
2137:
2136:
2100:
2095:
2094:
2015:
1977:
1976:
1910:
1905:
1904:
1876:
1875:
1854:
1826:
1804:
1785:
1769:
1765:
1743:
1724:
1704:
1703:
1657:
1639:
1633:
1632:
1573:
1572:
1541:
1513:
1502:
1501:
1447:
1431:
1412:
1390:
1389:
1278:
1277:
1256:
1247:
1196:
1177:
1163:
1162:
1104:
1099:
1098:
1024:
1019:
1018:
970:
965:
964:
901:
818:
773:
772:
283:
275:
274:
232:Quantum gravity
217:
191:Mirror symmetry
23:
22:
15:
12:
11:
5:
3640:
3638:
3630:
3629:
3624:
3619:
3614:
3604:
3603:
3597:
3596:
3593:
3592:
3590:
3589:
3584:
3579:
3573:
3571:
3569:Vector bundles
3565:
3564:
3562:
3561:
3556:
3551:
3546:
3541:
3535:
3533:
3527:
3526:
3524:
3523:
3518:
3513:
3508:
3502:
3500:
3496:
3495:
3493:
3492:
3487:
3482:
3477:
3472:
3466:
3464:
3460:
3459:
3457:
3456:
3451:
3446:
3441:
3436:
3431:
3426:
3421:
3415:
3413:
3406:
3402:
3401:
3399:
3398:
3393:
3388:
3382:
3380:
3376:
3375:
3373:
3372:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3316:
3314:
3308:
3307:
3305:
3304:
3299:
3294:
3289:
3284:
3279:
3274:
3269:
3264:
3259:
3254:
3249:
3244:
3239:
3233:
3231:
3225:
3224:
3222:
3221:
3216:
3211:
3206:
3201:
3195:
3193:
3189:
3188:
3185:
3184:
3182:
3181:
3176:
3170:
3168:
3164:
3163:
3161:
3160:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3104:
3102:
3098:
3097:
3095:
3094:
3089:
3084:
3079:
3073:
3071:
3064:
3058:
3057:
3055:
3054:
3049:
3047:Riemann sphere
3044:
3039:
3034:
3028:
3026:
3020:
3019:
3013:
3011:
3010:
3003:
2996:
2988:
2982:
2981:
2974:
2971:
2970:
2969:
2964:
2929:
2914:
2901:
2893:
2890:
2889:
2888:
2882:
2857:
2850:
2832:
2829:
2828:
2827:
2822:
2814:
2811:
2788:closed A-model
2781:path integrals
2752:
2749:
2745:Floer homology
2697:
2694:
2670:Michael Atiyah
2585:
2582:
2579:
2572:
2569:
2539:
2536:
2499:
2495:
2450:
2439:
2438:
2427:
2424:
2420:
2416:
2413:
2410:
2405:
2401:
2397:
2392:
2388:
2384:
2379:
2375:
2371:
2368:
2365:
2360:
2357:
2354:
2349:
2346:
2343:
2339:
2335:
2332:
2329:
2324:
2320:
2316:
2313:
2310:
2305:
2301:
2297:
2294:
2291:
2286:
2283:
2280:
2275:
2272:
2269:
2265:
2261:
2224:
2220:
2216:
2213:
2210:
2205:
2201:
2197:
2194:
2168:
2164:
2160:
2157:
2154:
2149:
2145:
2122:
2119:
2116:
2110:
2106:
2060:
2059:
2058:
2057:
2056:
2055:
2044:
2041:
2037:
2033:
2030:
2027:
2022:
2018:
2014:
2009:
2006:
2003:
1998:
1995:
1992:
1988:
1984:
1949:
1946:
1943:
1940:
1937:
1932:
1929:
1926:
1920:
1916:
1897:
1896:
1895:
1894:
1893:
1892:
1879:
1874:
1870:
1866:
1861:
1857:
1853:
1850:
1847:
1844:
1841:
1838:
1833:
1829:
1825:
1822:
1819:
1816:
1811:
1807:
1803:
1800:
1797:
1792:
1788:
1784:
1781:
1778:
1775:
1772:
1768:
1764:
1761:
1758:
1755:
1750:
1746:
1742:
1739:
1736:
1731:
1727:
1723:
1720:
1717:
1713:
1710:
1706:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1679:
1676:
1673:
1667:
1663:
1656:
1652:
1649:
1645:
1644:
1642:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1569:
1568:
1567:
1566:
1565:
1564:
1553:
1548:
1544:
1540:
1535:
1532:
1529:
1523:
1519:
1512:
1509:
1487:
1486:
1485:
1484:
1483:
1482:
1469:
1466:
1463:
1457:
1453:
1446:
1443:
1438:
1434:
1430:
1427:
1424:
1419:
1415:
1411:
1408:
1405:
1401:
1398:
1379:
1378:
1377:
1376:
1375:
1374:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1304:
1299:
1295:
1291:
1288:
1285:
1252:
1245:
1241:marked points
1231:
1230:
1229:
1228:
1227:
1226:
1214:
1211:
1208:
1203:
1199:
1195:
1192:
1189:
1184:
1180:
1176:
1173:
1170:
1143:
1140:
1137:
1134:
1131:
1126:
1123:
1120:
1114:
1110:
1063:
1060:
1057:
1054:
1051:
1046:
1043:
1040:
1034:
1030:
992:
989:
986:
980:
976:
945:
944:
938:
932:
922:
917:of dimension 2
900:
897:
882:Mikhail Gromov
820:
819:
817:
816:
809:
802:
794:
791:
790:
789:
788:
783:
775:
774:
771:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
615:
610:
605:
603:Randjbar-Daemi
600:
595:
590:
585:
580:
575:
570:
565:
560:
555:
550:
545:
540:
535:
530:
525:
520:
515:
510:
505:
500:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
445:
440:
435:
430:
425:
420:
415:
410:
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:
3639:
3628:
3627:Moduli theory
3625:
3623:
3622:String theory
3620:
3618:
3615:
3613:
3610:
3609:
3607:
3588:
3585:
3583:
3580:
3578:
3575:
3574:
3572:
3570:
3566:
3560:
3557:
3555:
3552:
3550:
3547:
3545:
3542:
3540:
3537:
3536:
3534:
3532:
3531:Singularities
3528:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3503:
3501:
3497:
3491:
3488:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3467:
3465:
3461:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3416:
3414:
3410:
3407:
3403:
3397:
3394:
3392:
3389:
3387:
3384:
3383:
3381:
3379:Constructions
3377:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3350:Klein quartic
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3330:Bolza surface
3328:
3326:
3325:Bring's curve
3323:
3321:
3318:
3317:
3315:
3313:
3309:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3257:Conic section
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3238:
3237:AF+BG theorem
3235:
3234:
3232:
3230:
3226:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3196:
3194:
3190:
3180:
3177:
3175:
3172:
3171:
3169:
3165:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3105:
3103:
3099:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3074:
3072:
3068:
3065:
3063:
3059:
3053:
3052:Twisted cubic
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3029:
3027:
3025:
3021:
3017:
3009:
3004:
3002:
2997:
2995:
2990:
2989:
2986:
2980:
2977:
2976:
2972:
2968:
2965:
2961:
2957:
2953:
2949:
2944:
2939:
2935:
2930:
2927:
2923:
2917:
2911:
2907:
2902:
2899:
2896:
2895:
2891:
2885:
2883:0-521-57086-7
2879:
2875:
2871:
2866:
2865:
2858:
2853:
2851:0-8218-3485-1
2847:
2843:
2839:
2835:
2834:
2830:
2826:
2823:
2820:
2817:
2816:
2812:
2810:
2808:
2805:of the genus
2804:
2800:
2797:
2793:
2789:
2784:
2782:
2778:
2774:
2770:
2766:
2762:
2758:
2750:
2748:
2746:
2741:
2739:
2735:
2729:
2727:
2723:
2719:
2715:
2711:
2707:
2703:
2695:
2693:
2691:
2687:
2682:
2677:
2675:
2671:
2667:
2663:
2659:
2655:
2650:
2648:
2644:
2640:
2637:, called the
2636:
2635:vector bundle
2632:
2628:
2624:
2620:
2615:
2613:
2609:
2605:
2601:
2583:
2580:
2577:
2555:
2552:
2551:linearization
2548:
2545:
2537:
2535:
2533:
2532:Chern classes
2528:
2526:
2522:
2518:
2513:
2497:
2493:
2484:
2480:
2476:
2472:
2468:
2464:
2448:
2425:
2414:
2411:
2403:
2399:
2395:
2390:
2386:
2382:
2377:
2373:
2369:
2366:
2363:
2358:
2355:
2352:
2347:
2344:
2341:
2337:
2333:
2330:
2322:
2318:
2314:
2311:
2308:
2303:
2299:
2295:
2292:
2284:
2281:
2278:
2273:
2270:
2267:
2263:
2259:
2252:
2251:
2250:
2248:
2244:
2240:
2222:
2218:
2214:
2211:
2208:
2203:
2199:
2195:
2192:
2184:
2166:
2162:
2158:
2155:
2152:
2147:
2143:
2120:
2117:
2114:
2091:
2089:
2085:
2081:
2077:
2073:
2070:for the data
2069:
2065:
2042:
2031:
2028:
2020:
2016:
2012:
2007:
2004:
2001:
1996:
1993:
1990:
1986:
1982:
1975:
1974:
1973:
1972:
1971:
1970:
1969:
1967:
1963:
1944:
1941:
1938:
1930:
1927:
1924:
1902:
1872:
1868:
1859:
1855:
1848:
1845:
1842:
1839:
1831:
1827:
1820:
1817:
1809:
1805:
1801:
1798:
1795:
1790:
1786:
1782:
1779:
1773:
1770:
1766:
1762:
1756:
1753:
1748:
1744:
1740:
1737:
1734:
1729:
1725:
1721:
1718:
1700:
1691:
1688:
1685:
1677:
1674:
1671:
1654:
1640:
1631:
1630:
1629:
1628:
1627:
1626:
1625:
1611:
1605:
1602:
1599:
1593:
1590:
1587:
1584:
1581:
1578:
1551:
1546:
1542:
1538:
1533:
1530:
1527:
1510:
1507:
1500:
1499:
1498:
1497:
1496:
1495:
1494:
1492:
1491:stabilization
1467:
1464:
1461:
1444:
1436:
1432:
1428:
1425:
1422:
1417:
1413:
1409:
1406:
1388:
1387:
1386:
1385:
1384:
1383:
1382:
1361:
1358:
1355:
1352:
1346:
1343:
1340:
1331:
1328:
1325:
1322:
1316:
1310:
1302:
1297:
1293:
1289:
1286:
1283:
1276:
1275:
1274:
1273:
1272:
1271:
1270:
1268:
1264:
1260:
1255:
1251:
1244:
1240:
1236:
1209:
1206:
1201:
1197:
1193:
1190:
1187:
1182:
1178:
1174:
1171:
1161:
1160:
1159:
1158:
1157:
1156:
1155:
1138:
1135:
1132:
1124:
1121:
1118:
1096:
1092:
1089:
1085:
1081:
1077:
1058:
1055:
1052:
1044:
1041:
1038:
1016:
1012:
1008:
990:
987:
984:
962:
958:
954:
950:
942:
939:
936:
933:
930:
926:
923:
920:
916:
913:
909:
906:
905:
904:
898:
896:
894:
889:
887:
886:Edward Witten
883:
879:
875:
871:
867:
863:
859:
855:
851:
847:
843:
839:
838:Gromov–Witten
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:
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237:Supersymmetry
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160:Phenomenology
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152:Phenomenology
150:
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59:Cosmic string
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50:
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40:
36:
35:
32:
31:String theory
28:
19:
3516:Prym variety
3490:Stable curve
3480:Hodge bundle
3474:
3470:ELSV formula
3272:Fermat curve
3229:Plane curves
3192:Higher genus
3167:Applications
3092:Modular form
2943:math/0602347
2933:
2922:moduli space
2905:
2863:
2841:
2838:McDuff, Dusa
2806:
2798:
2785:
2772:
2754:
2742:
2737:
2730:
2699:
2689:
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2678:
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2654:localization
2653:
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2607:
2603:
2598:operator is
2553:
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2238:
2182:
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1965:
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924:
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907:
902:
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841:
837:
823:
293:Arkani-Hamed
252:
242:Supergravity
3391:Polar curve
2872:. pp.
2792:free energy
2643:Euler class
2477:, of genus
2082:. It is an
1489:denote the
1076:stable maps
870:cup product
826:mathematics
653:Silverstein
178:Mathematics
90:Superstring
3606:Categories
3386:Dual curve
3014:Topics in
2831:References
2674:Raoul Bott
2600:surjective
1968:, denoted
899:Definition
893:stable map
866:cohomology
846:invariants
673:Strominger
668:Steinhardt
663:Staudacher
578:Polchinski
528:Nanopoulos
488:Mandelstam
468:Kontsevich
308:Berenstein
265:Multiverse
3499:Morphisms
3247:Bitangent
2571:¯
2568:∂
2494:α
2449:⋅
2396:∈
2387:α
2383:⋯
2374:α
2370:⋅
2367:β
2364:⋅
2319:α
2312:…
2300:α
2293:β
2219:α
2212:…
2200:α
2193:β
2163:α
2156:…
2144:α
2109:¯
2084:invariant
2013:∈
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1843:…
1799:…
1774:
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1445:∈
1426:…
1344:−
1329:−
1191:…
1113:¯
1082:of class
1033:¯
1009:of genus
979:¯
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
2813:See also
2773:a priori
2627:cokernel
2525:orbifold
1261: :
862:homology
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
3559:Tacnode
3544:Crunode
2948:Bibcode
2926:schemes
2801:is the
2777:measure
2769:strings
2556:of the
2245:by the
2237:equals
1248:, ...,
1005:be the
963:). Let
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
3539:Acnode
3463:Moduli
2912:
2880:
2876:–200.
2848:
2441:where
2249:. Let
2078:, and
1233:where
912:closed
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
2938:arXiv
2796:genus
2668:, of
2662:toric
1960:to a
1078:into
1013:with
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
3549:Cusp
2910:ISBN
2878:ISBN
2846:ISBN
2759:and
2704:and
2672:and
2135:and
1381:Let
1257:and
910:: a
884:and
848:are
832:and
728:Wess
708:Vafa
613:Rohm
513:Motl
438:Kaku
403:Guth
338:Duff
2956:doi
2874:171
2660:is
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2066:of
1903:of
1093:on
872:of
864:or
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658:Sơn
638:Sen
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840:(
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