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Connected with each hole in the Calabi–Yau space is a group of low-energy string vibrational patterns. Since string theory states that our familiar elementary particles correspond to low-energy string vibrations, the presence of multiple holes causes the string patterns to fall into multiple groups,
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give examples of complex manifolds that have Ricci-flat metrics, but their canonical bundles are not trivial, so they are Calabi–Yau manifolds according to the second but not the first definition above. On the other hand, their double covers are Calabi–Yau manifolds for both definitions (in fact, K3
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have shown that the masses of particles depend on the manner of the intersection of the various holes in a Calabi–Yau. In other words, the positions of the holes relative to one another and to the substance of the Calabi–Yau space was found by
Strominger and Witten to affect the masses of particles
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have first Chern class that vanishes as an element of the real cohomology group, but not as an element of the integral cohomology group, so Yau's theorem about the existence of a Ricci-flat metric still applies to them but they are sometimes not considered to be Calabi–Yau manifolds. Abelian
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In three complex dimensions, classification of the possible Calabi–Yau manifolds is an open problem, although Yau suspects that there is a finite number of families (albeit a much bigger number than his estimate from 20 years ago). In turn, it has also been conjectured by
827:, which implies that a compact Kähler manifold with a vanishing first real Chern class has a Kähler metric in the same class with vanishing Ricci curvature. (The class of a Kähler metric is the cohomology class of its associated 2-form.) Calabi showed such a metric is unique.
2466:. Essentially, Calabi–Yau manifolds are shapes that satisfy the requirement of space for the six "unseen" spatial dimensions of string theory, which may be smaller than our currently observable lengths as they have not yet been detected. A popular alternative known as
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Most definitions assume the manifold is non-singular, but some allow mild singularities. While the Chern class fails to be well-defined for singular Calabi–Yau's, the canonical bundle and canonical class may still be defined if all the singularities are
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2558:. Although the following statement has been simplified, it conveys the logic of the argument: if the Calabi–Yau has three holes, then three families of vibrational patterns and thus three families of particles will be observed experimentally.
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There are many other definitions of a Calabi–Yau manifold used by different authors, some inequivalent. This section summarizes some of the more common definitions and the relations between them.
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Logically, since strings vibrate through all the dimensions, the shape of the curled-up ones will affect their vibrations and thus the properties of the elementary particles observed. For example,
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of a Calabi–Yau manifold, such as demanding that it be finite or trivial. Any Calabi–Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi–Yau manifold.
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the following conditions are equivalent to each other, but are weaker than the conditions above, though they are sometimes used as the definition of a Calabi–Yau manifold:
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By far the hardest part of proving the equivalences between the various properties above is proving the existence of Ricci-flat metrics. This follows from Yau's proof of the
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that the number of topological types of Calabi–Yau 3-folds is infinite, and that they can all be transformed continuously (through certain mild singularizations such as
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on a projective space which one can restrict to the algebraic variety. By definition, if ω is the Kähler metric on the algebraic variety X and the canonical bundle K
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Most definitions assert that Calabi–Yau manifolds are compact, but some allow them to be non-compact. In the generalization to non-compact manifolds, the difference
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or 2 supercharges in a compactification of type I. When fluxes are included the supersymmetry condition instead implies that the compactification manifold be a
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compactifications on various Calabi–Yau four-folds provide physicists with a method to find a large number of classical solutions in the so-called
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is an interactive reference which describes many examples and classes of Calabi–Yau manifolds and also the physical theories in which they appear.
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There are many other inequivalent definitions of Calabi–Yau manifolds that are sometimes used, which differ in the following ways (among others):
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1999:
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actions are also Calabi–Yau and have received a lot of attention in the literature. One of these is related to the original quintic by
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Most definitions assume that a Calabi–Yau manifold has a
Riemannian metric, but some treat them as complex manifolds without a metric.
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2509:, compactification on a Calabi–Yau 3-fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken if the
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surfaces are sometimes excluded from the classification of being Calabi–Yau, as their holonomy (again the trivial group) is a
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is trivial, then X is Calabi–Yau. Moreover, there is unique Kähler metric ω on X such that = ∈
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If a compact Kähler manifold is simply connected, then the weak definition above is equivalent to the stronger definition.
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Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I",
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are canonically associated with the fibers of the vector bundle. Using this, we can use the relative cotangent sequence
1107:, and so may be used to extend the definition of a smooth Calabi–Yau manifold to a possibly singular Calabi–Yau variety.
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furnish the only compact simply connected Calabi–Yau manifolds. These can be constructed as quartic surfaces in
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are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of
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2182:{\displaystyle \omega _{V}=\bigwedge ^{3}\Omega _{V}\cong f^{*}\omega _{C}\otimes \bigwedge ^{2}\Omega _{V/C}}
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models, is that the Calabi–Yau is large but we are confined to a small subset on which it intersects a
2646:, Dr. Mossman describes teleporters as working via a 'String-based' technology using 'the Calabi-Yau model.'
2524:) leaves 2 of the original supersymmetry unbroken, corresponding to 2 supercharges in a compactification of
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3100:, Surveys in Differential Geometry, vol. 13, Somerville, Massachusetts: Int. Press, pp. 277–318,
2478:. Further extensions into higher dimensions are currently being explored with additional ramifications for
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More generally, Calabi–Yau varieties/orbifolds can be found as weighted complete intersections in a
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and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used.
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Other examples can be constructed as elliptic fibrations, as quotients of abelian surfaces, or as
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vanishes. Nevertheless, the converse is not true. The simplest examples where this happens are
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Geometry, analysis, and algebraic geometry: forty years of the
Journal of Differential Geometry
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Tian, Gang; Yau, Shing-Tung (1991), "Complete Kähler manifolds with zero Ricci curvature, II",
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Tian, Gang; Yau, Shing-Tung (1990), "Complete Kähler manifolds with zero Ricci curvature, I",
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3154:, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Berlin, New York:
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in order to illustrate the high-dimensional abilities of the San-Ti alien civilization.
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2264:{\displaystyle \Omega _{V/C}\cong {\mathcal {L}}_{1}^{*}\oplus {\mathcal {L}}_{2}^{*}}
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and observing the only tangent vectors in the fiber which are not in the pre-image of
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can. One example of a three-dimensional Calabi–Yau manifold is a non-singular
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is the trivial group SU(1). A one-dimensional Calabi–Yau manifold is a complex
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Calabi, Eugenio (1957), "On Kähler manifolds with vanishing canonical class", in
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Schedler, Travis (2019). "Deformations of algebras in noncommutative geometry".
1096:(in fact trivial) so are not Calabi–Yau manifolds according to such definitions.
1064:. Abelian surfaces have a Ricci flat metric with holonomy strictly smaller than
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a quasi-projective Calabi-Yau threefold can be constructed as the total space
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Candelas, Philip; Horowitz, Gary; Strominger, Andrew; Witten, Edward (1985),
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1729:{\displaystyle {\mathcal {L}}_{1}\otimes {\mathcal {L}}_{2}\cong \omega _{C}}
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Propp, Oron Y. (2019-05-22). "Constructing explicit K3 spectra". p. 4.
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In the most conventional superstring models, ten conjectural dimensions in
1668:{\displaystyle V={\text{Tot}}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}
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are supposed to come as four of which we are aware, carrying some kind of
2072:{\displaystyle 0\to p^{*}\Omega _{C}\to \Omega _{V}\to \Omega _{V/C}\to 0}
1225:, such as the complex algebraic variety defined by the vanishing locus of
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The first Chern class may vanish as an integral class or as a real class.
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The Calabi–Yau
Landscape: From Geometry, to Physics, to Machine Learning
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Szendroi, Balazs (2016-04-27). "Cohomological
Donaldson-Thomas theory".
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Algebraic geometry and topology. A symposium in honor of S. Lefschetz
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Desingularizations of Calabi-Yau 3-folds with a conical singularity
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2447:{\displaystyle {\text{Tot}}({\mathcal {O}}_{\mathbb {P} ^{2}}(-3))}
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by Andrew J. Hanson with additional contributions by Jeff Bryant,
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1860:{\displaystyle p^{*}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}
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The Calabi-Yau manifold was the subject of a paper coauthored by
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1540:, in the homogeneous coordinates of the complex projective space
19:"Calabi–Yau" redirects here. For the play by Susanna Speier, see
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Some definitions require that the holonomy be exactly equal to
895:{\displaystyle (\Omega \wedge {\bar {\Omega }}-\omega ^{n}/n!)}
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3096:
Yau, Shing-Tung (2009a), "A survey of Calabi–Yau manifolds",
2501:-folds are important because they leave some of the original
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of SU(2), instead of being isomorphic to SU(2). However, the
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with a vanishing first Chern class, that is also Ricci flat.
88:
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subset do not conform entirely to the SU(2) subgroup in the
152:) who first conjectured that such surfaces might exist, and
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in a certain way. This is true of all particle properties.
528:
These conditions imply that the first integral Chern class
234:-fold or Calabi–Yau manifold of (complex) dimension
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1313:{\displaystyle x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}=0}
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Using a similar argument as for curves, the total space
1946:{\displaystyle 0\to T_{V/C}\to T_{V}\to p^{*}T_{C}\to 0}
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consisting of all of the zeros of a homogeneous quintic
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In one complex dimension, the only compact examples are
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satisfying one of the following equivalent conditions:
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to transport the geometry of a Calabi–Yau manifold to
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1560: = 1 describes an elliptic curve, while for
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rather than a subgroup of it, which implies that the
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is the Kähler form associated with the Kähler metric,
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has a finite cover that is a product of a torus and a
3347:
Calabi–Yau
Manifolds: a Bestiary for Physicists
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has a finite cover that has trivial canonical bundle.
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1518:. Some discrete quotients of the quintic by various
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2516:More generally, a flux-free compactification on an
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has a Kähler metric with vanishing Ricci curvature.
3330:, Switzerland: Springer International Publishing,
2982:(2003), "Generalized Calabi–Yau manifolds",
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2629:Imagery based on Calabi-Yau manifolds was used in
2446:
2391:forms a Calabi-Yau threefold. A simple example is
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2082:together with the properties of wedge powers that
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3448:Yau, S. T. (2009b), "Calabi–Yau manifold",
3280:Calabi–Yau manifolds and related geometries
2761: = 0 may nevertheless be irreducible".
1124:is a Kähler manifold, because there is a natural
4756:
2837:Ginzburg, Victor (2007). "Calabi-Yau algebras".
2916:, Princeton Mathematical Series, vol. 12,
2889:, vol. 2, pp. 206–207, archived from
1571:. The main tool for finding such spaces is the
92:A 2D slice of a 6D Calabi–Yau quintic manifold.
3183:Diffeomorphism Classes of Calabi–Yau Varieties
3069:Communications on Pure and Applied Mathematics
1159:
3575:
3539:An overview of Calabi-Yau Elliptic fibrations
2505:unbroken. More precisely, in the absence of
2462:Calabi–Yau manifolds are important in
1514:. Another example is a smooth model of the
1450:equipped with a complex manifold structure.
2711:Bulletin of the London Mathematical Society
2709:Szymik, Markus (2020-02-12). "K3 spectra".
1417:Non simply-connected examples are given by
183:). They were originally defined as compact
3582:
3568:
2757:(1987). "The Moduli space of 3-folds with
2620:in the episode 2 of the seventh season in
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2337:{\displaystyle {\text{Tot}}(\omega _{S})}
1564: = 2 one obtains a K3 surface.
1430:
1387:
1205:
945:Some definitions put restrictions on the
76:Learn how and when to remove this message
3121:
2944:"Vacuum configurations for superstrings"
2887:Proc. Internat. Congress Math. Amsterdam
2857:
2836:
2797:
2663:
1768:we can find the relative tangent bundle
1187:
1116:The fundamental fact is that any smooth
87:
39:This article includes a list of general
3392:Compact Manifolds with Special Holonomy
3029:
2978:
2675:
2533:
1544:, of a non-singular homogeneous degree
811:manifold with trivial canonical bundle.
16:Riemannian manifold with SU(n) holonomy
4757:
3496:Spinning Calabi–Yau Space video.
3122:Yau, Shing-Tung; Nadis, Steve (2010),
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1510:in the homogeneous coordinates of the
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3242:String theory on Calabi–Yau manifolds
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654:has vanishing first real Chern class.
199:The motivational definition given by
3980:Bogomol'nyi–Prasad–Sommerfield bound
3532:
3476:
3447:
3095:
2984:The Quarterly Journal of Mathematics
2753:
1867:using the relative tangent sequence
1400:{\displaystyle \in \mathbb {P} ^{3}}
25:
3065:
2820:"The Shape of Curled-Up Dimensions"
2302:Constructed from algebraic surfaces
1144:), a fact which was conjectured by
157:
13:
3282:, Universitext, Berlin, New York:
3140:
2458:Applications in superstring theory
2409:
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902:must vanish asymptotically. Here,
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254:is sometimes defined as a compact
45:it lacks sufficient corresponding
14:
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2588:noncommutative algebraic geometry
1586:Constructed from algebraic curves
1488:) one into another—much as
1057:{\displaystyle 0<i<\dim(M)}
1474:
1443:{\displaystyle \mathbb {T} ^{4}}
1218:{\displaystyle \mathbb {P} ^{3}}
486:has a Kähler metric with global
30:
4177:Eleven-dimensional supergravity
1736:. For the canonical projection
1192:In two complex dimensions, the
698:has a Kähler metric with local
167:Calabi–Yau manifolds are
3507:Wolfram Demonstrations Project
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179:(i.e. any even number of real
116:which has properties, such as
1:
3625:Second superstring revolution
3550:Fibrations in CICY Threefolds
2883:"The space of Kähler metrics"
2650:
2536:. These models are known as
1548: + 2 polynomial in
611:-dimensional Kähler manifold
274:-dimensional Kähler manifold
4119:Generalized complex manifold
3620:First superstring revolution
2968:10.1016/0550-3213(85)90602-9
2530:generalized Calabi–Yau
364:-form that vanishes nowhere.
171:that are generalizations of
7:
3274:Gross, M.; Huybrechts, D.;
2593:
2520:-manifold with holonomy SU(
2364:{\displaystyle \omega _{S}}
2291:{\displaystyle \omega _{V}}
1532:For every positive integer
1421:, which are real four tori
1160:Calabi–Yau algebraic curves
1111:
136:. Their name was coined by
120:, yielding applications in
10:
4801:
3717:Non-critical string theory
3107:10.4310/SDG.2008.v13.n1.a9
2918:Princeton University Press
2493:with fiber dimension six.
1986:{\displaystyle p^{*}T_{C}}
1582:are Calabi–Yau manifolds.
128:, the extra dimensions of
112:, is a particular type of
18:
4261:
4238:
4215:
4162:
4047:
3955:
3897:
3829:
3778:
3745:
3640:
3597:
3490:Calabi–Yau Homepage
3471:10.4249/scholarpedia.6524
3293:10.1007/978-3-642-19004-9
2532:, a notion introduced by
2371:for an algebraic surface
2271:giving the triviality of
1569:weighted projective space
106:Calabi–Yau manifold
4253:Introduction to M-theory
3947:Wess–Zumino–Witten model
3889:Hanany–Witten transition
3615:History of string theory
3518:"Calabi–Yau Space"
3344:Hübsch, Tristan (1994),
3124:The Shape of Inner Space
2470:, which often occurs in
1761:{\displaystyle p:V\to C}
739:A positive power of the
557:{\displaystyle c_{1}(M)}
3932:Vertex operator algebra
3632:String theory landscape
3396:Oxford University Press
3350:, Singapore, New York:
3211:Chan, Yat-Ming (2004),
2548:string theory landscape
2454:over projective space.
2344:of the canonical sheaf
1796:{\displaystyle T_{V/C}}
1590:For an algebraic curve
1469:String theory landscape
1013:{\displaystyle h^{i,0}}
915:{\displaystyle \omega }
187:with a vanishing first
60:more precise citations.
4230:AdS/CFT correspondence
3985:Exceptional Lie groups
3927:Superconformal algebra
3899:Conformal field theory
3770:Montonen–Olive duality
3722:Non-linear sigma model
3502:Calabi–Yau Space
3082:10.1002/cpa.3160310304
2826:on September 13, 2006.
2664:Yau & Nadis (2010)
2538:flux compactifications
2468:large extra dimensions
2448:
2385:
2365:
2338:
2292:
2265:
2190:
2183:
2080:
2073:
1987:
1954:
1947:
1861:
1797:
1762:
1730:
1669:
1604:
1580:hyper-Kähler manifolds
1456:hyperelliptic surfaces
1444:
1412:complete intersections
1408:
1401:
1314:
1219:
1090:
1058:
1014:
978:
936:
916:
896:
801:
779:
757:
729:
692:
670:
648:
625:
605:
586:hyperelliptic surfaces
578:
558:
517:
480:
454:
418:
389:
358:
338:
316:
288:
268:
248:
228:
138:Candelas et al. (1985)
110:Calabi–Yau space
93:
4770:Differential geometry
4225:Holographic principle
4192:Type IIB supergravity
4187:Type IIA supergravity
4039:-form electrodynamics
3658:Bosonic string theory
3554:complete intersection
3180:Bini; Iacono (2016),
2764:Mathematische Annalen
2676:Tian & Yau (1991)
2526:type IIA supergravity
2449:
2386:
2366:
2339:
2293:
2266:
2184:
2084:
2074:
1995:
1988:
1948:
1869:
1862:
1798:
1763:
1731:
1670:
1605:
1445:
1402:
1315:
1227:
1220:
1188:CY algebraic surfaces
1180:, and in particular,
1091:
1089:{\displaystyle SU(2)}
1059:
1015:
979:
977:{\displaystyle SU(n)}
937:
917:
897:
802:
780:
758:
730:
728:{\displaystyle SU(n)}
693:
671:
649:
626:
606:
579:
559:
518:
516:{\displaystyle SU(n)}
481:
462:special unitary group
455:
453:{\displaystyle SU(n)}
419:
390:
359:
339:
317:
289:
269:
249:
229:
102:differential geometry
91:
4775:Mathematical physics
4144:Hořava–Witten theory
4091:Hyperkähler manifold
3779:Particles and fields
3727:Tachyon condensation
3712:Matrix string theory
3545:Calabi-Yau Landscape
3326:He, Yang-Hu (2021),
3016:10.1093/qmath/hag025
2974:on December 20, 2012
2497:on Calabi–Yau
2395:
2375:
2348:
2310:
2275:
2196:
2089:
2000:
1960:
1874:
1807:
1772:
1740:
1679:
1614:
1594:
1425:
1324:
1232:
1200:
1068:
1024:
991:
956:
926:
906:
841:
791:
769:
747:
707:
682:
660:
638:
615:
595:
568:
532:
495:
470:
432:
417:{\displaystyle U(n)}
399:
395:can be reduced from
379:
348:
328:
306:
278:
258:
238:
218:
4182:Type I supergravity
4086:Calabi–Yau manifold
4081:Ricci-flat manifold
4060:Kaluza–Klein theory
3801:Ramond–Ramond field
3707:String field theory
3462:2009SchpJ...4.6524Y
3422:J. Amer. Math. Soc.
3260:1997hep.th....2155G
3231:2004math.....10260C
3205:2016arXiv161204311B
3044:1991InMat.106...27T
2960:1985NuPhB.258...46C
2733:10.1112/blms/bdp106
2513:is the full SU(3).
2260:
2238:
1516:Barth–Nieto quintic
1303:
1285:
1267:
1249:
1126:Fubini–Study metric
214:A Calabi–Yau
122:theoretical physics
4765:Algebraic geometry
4149:K-theory (physics)
4026:ADE classification
3663:Superstring theory
3515:Weisstein, Eric W.
3152:Einstein manifolds
3052:10.1007/BF01243902
2920:, pp. 78–89,
2904:Spencer, Donald C.
2777:10.1007/bf01458074
2582:was introduced by
2580:Calabi–Yau algebra
2574:Calabi-Yau algebra
2480:general relativity
2464:superstring theory
2444:
2381:
2361:
2334:
2288:
2261:
2242:
2220:
2179:
2069:
1983:
1943:
1857:
1793:
1758:
1726:
1665:
1600:
1573:adjunction formula
1440:
1397:
1310:
1289:
1271:
1253:
1235:
1215:
1086:
1054:
1010:
974:
932:
912:
892:
797:
775:
753:
725:
688:
666:
644:
621:
601:
574:
554:
513:
476:
450:
414:
385:
354:
344:has a holomorphic
334:
312:
284:
264:
244:
224:
177:complex dimensions
154:Shing-Tung Yau
142:Eugenio Calabi
126:superstring theory
124:. Particularly in
108:, also known as a
94:
4785:Complex manifolds
4752:
4751:
4534:van Nieuwenhuizen
4070:Why 10 dimensions
3975:Chern–Simons form
3942:Kac–Moody algebra
3922:Conformal algebra
3917:Conformal anomaly
3811:Magnetic monopole
3806:Kalb–Ramond field
3648:Nambu–Goto action
3533:Beginner articles
3405:978-0-19-850601-0
3361:978-981-02-1927-7
3337:978-3-030-77562-9
3303:978-3-540-44059-8
3165:978-3-540-15279-8
3133:978-0-465-02023-2
2948:Nuclear Physics B
2908:Tucker, Albert W.
2633:of the TV series
2600:Quintic threefold
2563:Andrew Strominger
2401:
2384:{\displaystyle S}
2316:
2160:
2114:
1626:
1603:{\displaystyle C}
1556:-fold. The case
1504:algebraic variety
1494:quintic threefold
1452:Enriques surfaces
1154:Calabi conjecture
1118:algebraic variety
947:fundamental group
935:{\displaystyle g}
862:
825:Calabi conjecture
817:Enriques surfaces
800:{\displaystyle M}
778:{\displaystyle M}
756:{\displaystyle M}
691:{\displaystyle M}
669:{\displaystyle M}
647:{\displaystyle M}
624:{\displaystyle M}
604:{\displaystyle n}
577:{\displaystyle M}
479:{\displaystyle M}
388:{\displaystyle M}
357:{\displaystyle n}
337:{\displaystyle M}
315:{\displaystyle M}
287:{\displaystyle M}
267:{\displaystyle n}
247:{\displaystyle n}
227:{\displaystyle n}
175:in any number of
169:complex manifolds
162:Calabi conjecture
160:) who proved the
86:
85:
78:
21:Calabi-Yau (play)
4792:
4262:String theorists
4202:Lie superalgebra
4154:Twisted K-theory
4112:Spin(7)-manifold
4065:Compactification
3907:Virasoro algebra
3690:Heterotic string
3584:
3577:
3570:
3561:
3560:
3543:Lectures on the
3528:
3527:
3474:
3473:
3444:
3416:
3383:
3382:
3381:
3372:, archived from
3352:World Scientific
3340:
3322:
3295:
3270:
3253:
3251:hep-th/9702155v1
3233:
3224:
3207:
3198:
3188:
3176:
3148:Besse, Arthur L.
3136:
3118:
3109:
3092:
3062:
3026:
3009:
2999:
2975:
2970:, archived from
2938:
2894:
2870:
2869:
2867:
2855:
2849:
2848:
2846:
2834:
2828:
2827:
2822:. Archived from
2816:
2810:
2809:
2807:
2795:
2789:
2788:
2771:(1–4): 329–334.
2751:
2745:
2744:
2726:
2706:
2700:
2699:
2697:
2685:
2679:
2673:
2667:
2661:
2495:Compactification
2453:
2451:
2450:
2445:
2428:
2427:
2426:
2425:
2420:
2413:
2412:
2402:
2399:
2390:
2388:
2387:
2382:
2370:
2368:
2367:
2362:
2360:
2359:
2343:
2341:
2340:
2335:
2330:
2329:
2317:
2314:
2297:
2295:
2294:
2289:
2287:
2286:
2270:
2268:
2267:
2262:
2259:
2254:
2249:
2248:
2237:
2232:
2227:
2226:
2216:
2215:
2211:
2188:
2186:
2185:
2180:
2178:
2177:
2173:
2159:
2151:
2147:
2146:
2137:
2136:
2124:
2123:
2113:
2105:
2101:
2100:
2078:
2076:
2075:
2070:
2062:
2061:
2057:
2041:
2040:
2028:
2027:
2018:
2017:
1992:
1990:
1989:
1984:
1982:
1981:
1972:
1971:
1952:
1950:
1949:
1944:
1936:
1935:
1926:
1925:
1913:
1912:
1900:
1899:
1895:
1866:
1864:
1863:
1858:
1853:
1852:
1847:
1846:
1836:
1835:
1830:
1829:
1819:
1818:
1802:
1800:
1799:
1794:
1792:
1791:
1787:
1767:
1765:
1764:
1759:
1735:
1733:
1732:
1727:
1725:
1724:
1712:
1711:
1706:
1705:
1695:
1694:
1689:
1688:
1674:
1672:
1671:
1666:
1661:
1660:
1655:
1654:
1644:
1643:
1638:
1637:
1627:
1624:
1609:
1607:
1606:
1601:
1490:Riemann surfaces
1465:Enriques surface
1449:
1447:
1446:
1441:
1439:
1438:
1433:
1419:abelian surfaces
1406:
1404:
1403:
1398:
1396:
1395:
1390:
1378:
1377:
1365:
1364:
1352:
1351:
1339:
1338:
1319:
1317:
1316:
1311:
1302:
1297:
1284:
1279:
1266:
1261:
1248:
1243:
1224:
1222:
1221:
1216:
1214:
1213:
1208:
1122:projective space
1095:
1093:
1092:
1087:
1063:
1061:
1060:
1055:
1019:
1017:
1016:
1011:
1009:
1008:
983:
981:
980:
975:
941:
939:
938:
933:
921:
919:
918:
913:
901:
899:
898:
893:
882:
877:
876:
864:
863:
855:
809:simply connected
806:
804:
803:
798:
784:
782:
781:
776:
762:
760:
759:
754:
741:canonical bundle
734:
732:
731:
726:
697:
695:
694:
689:
675:
673:
672:
667:
653:
651:
650:
645:
630:
628:
627:
622:
610:
608:
607:
602:
583:
581:
580:
575:
563:
561:
560:
555:
544:
543:
522:
520:
519:
514:
485:
483:
482:
477:
459:
457:
456:
451:
423:
421:
420:
415:
394:
392:
391:
386:
363:
361:
360:
355:
343:
341:
340:
335:
321:
319:
318:
313:
300:canonical bundle
293:
291:
290:
285:
273:
271:
270:
265:
253:
251:
250:
245:
233:
231:
230:
225:
203:is of a compact
185:Kähler manifolds
81:
74:
70:
67:
61:
56:this article by
47:inline citations
34:
33:
26:
4800:
4799:
4795:
4794:
4793:
4791:
4790:
4789:
4755:
4754:
4753:
4748:
4257:
4234:
4211:
4158:
4106:
4076:Kähler manifold
4043:
4020:
4013:
4006:
3999:
3992:
3951:
3912:Mirror symmetry
3893:
3879:Brane cosmology
3825:
3774:
3741:
3697:N=2 superstring
3683:Type IIB string
3678:Type IIA string
3653:Polyakov action
3636:
3593:
3588:
3535:
3486:
3434:10.2307/1990928
3419:
3406:
3386:
3379:
3377:
3362:
3343:
3338:
3325:
3304:
3284:Springer-Verlag
3273:
3236:
3210:
3186:
3179:
3166:
3156:Springer-Verlag
3146:
3143:
3141:Further reading
3134:
3126:, Basic Books,
3007:10.1.1.237.8935
2997:math.DG/0209099
2928:
2879:Calabi, Eugenio
2874:
2873:
2856:
2852:
2835:
2831:
2818:
2817:
2813:
2796:
2792:
2752:
2748:
2707:
2703:
2686:
2682:
2674:
2670:
2662:
2658:
2653:
2613:
2596:
2584:Victor Ginzburg
2576:
2460:
2421:
2416:
2415:
2414:
2408:
2407:
2406:
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2393:
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2276:
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2272:
2255:
2250:
2244:
2243:
2233:
2228:
2222:
2221:
2207:
2203:
2199:
2197:
2194:
2193:
2169:
2165:
2161:
2155:
2142:
2138:
2132:
2128:
2119:
2115:
2109:
2096:
2092:
2090:
2087:
2086:
2053:
2049:
2045:
2036:
2032:
2023:
2019:
2013:
2009:
2001:
1998:
1997:
1977:
1973:
1967:
1963:
1961:
1958:
1957:
1931:
1927:
1921:
1917:
1908:
1904:
1891:
1887:
1883:
1875:
1872:
1871:
1848:
1842:
1841:
1840:
1831:
1825:
1824:
1823:
1814:
1810:
1808:
1805:
1804:
1783:
1779:
1775:
1773:
1770:
1769:
1741:
1738:
1737:
1720:
1716:
1707:
1701:
1700:
1699:
1690:
1684:
1683:
1682:
1680:
1677:
1676:
1656:
1650:
1649:
1648:
1639:
1633:
1632:
1631:
1623:
1615:
1612:
1611:
1595:
1592:
1591:
1588:
1527:mirror symmetry
1524:
1502:, which is the
1477:
1461:proper subgroup
1434:
1429:
1428:
1426:
1423:
1422:
1391:
1386:
1385:
1373:
1369:
1360:
1356:
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1330:
1325:
1322:
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1298:
1293:
1280:
1275:
1262:
1257:
1244:
1239:
1233:
1230:
1229:
1209:
1204:
1203:
1201:
1198:
1197:
1190:
1162:
1131:
1114:
1069:
1066:
1065:
1025:
1022:
1021:
998:
994:
992:
989:
988:
957:
954:
953:
927:
924:
923:
907:
904:
903:
878:
872:
868:
854:
853:
842:
839:
838:
792:
789:
788:
770:
767:
766:
748:
745:
744:
708:
705:
704:
683:
680:
679:
661:
658:
657:
639:
636:
635:
616:
613:
612:
596:
593:
592:
569:
566:
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539:
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529:
496:
493:
492:
471:
468:
467:
433:
430:
429:
400:
397:
396:
380:
377:
376:
369:structure group
349:
346:
345:
329:
326:
325:
307:
304:
303:
279:
276:
275:
259:
256:
255:
239:
236:
235:
219:
216:
215:
205:Kähler manifold
197:
134:mirror symmetry
82:
71:
65:
62:
52:Please help to
51:
35:
31:
24:
17:
12:
11:
5:
4798:
4788:
4787:
4782:
4777:
4772:
4767:
4750:
4749:
4747:
4746:
4741:
4736:
4731:
4726:
4721:
4716:
4711:
4706:
4701:
4696:
4691:
4686:
4681:
4676:
4671:
4666:
4661:
4656:
4651:
4646:
4641:
4636:
4631:
4626:
4621:
4616:
4611:
4606:
4601:
4596:
4591:
4586:
4584:Randjbar-Daemi
4581:
4576:
4571:
4566:
4561:
4556:
4551:
4546:
4541:
4536:
4531:
4526:
4521:
4516:
4511:
4506:
4501:
4496:
4491:
4486:
4481:
4476:
4471:
4466:
4461:
4456:
4451:
4446:
4441:
4436:
4431:
4426:
4421:
4416:
4411:
4406:
4401:
4396:
4391:
4386:
4381:
4376:
4371:
4366:
4361:
4356:
4351:
4346:
4341:
4336:
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4296:
4291:
4286:
4281:
4276:
4271:
4265:
4263:
4259:
4258:
4256:
4255:
4250:
4244:
4242:
4236:
4235:
4233:
4232:
4227:
4221:
4219:
4213:
4212:
4210:
4209:
4207:Lie supergroup
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4168:
4166:
4160:
4159:
4157:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4115:
4114:
4109:
4104:
4100:
4099:
4098:
4088:
4078:
4073:
4067:
4062:
4057:
4051:
4049:
4045:
4044:
4042:
4041:
4033:
4028:
4023:
4018:
4011:
4004:
3997:
3990:
3982:
3977:
3972:
3967:
3961:
3959:
3953:
3952:
3950:
3949:
3944:
3939:
3934:
3929:
3924:
3919:
3914:
3909:
3903:
3901:
3895:
3894:
3892:
3891:
3886:
3884:Quiver diagram
3881:
3876:
3871:
3866:
3861:
3856:
3851:
3846:
3841:
3835:
3833:
3827:
3826:
3824:
3823:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3782:
3780:
3776:
3775:
3773:
3772:
3767:
3762:
3757:
3751:
3749:
3747:String duality
3743:
3742:
3740:
3739:
3734:
3729:
3724:
3719:
3714:
3709:
3704:
3699:
3694:
3693:
3692:
3687:
3686:
3685:
3680:
3673:Type II string
3670:
3660:
3655:
3650:
3644:
3642:
3638:
3637:
3635:
3634:
3629:
3628:
3627:
3622:
3612:
3610:Cosmic strings
3607:
3601:
3599:
3595:
3594:
3587:
3586:
3579:
3572:
3564:
3558:
3557:
3547:
3541:
3534:
3531:
3530:
3529:
3510:
3498:
3493:
3485:
3484:External links
3482:
3481:
3480:
3445:
3428:(3): 579–609,
3417:
3404:
3388:Joyce, Dominic
3384:
3360:
3341:
3336:
3323:
3302:
3276:Joyce, Dominic
3271:
3234:
3208:
3177:
3164:
3142:
3139:
3138:
3137:
3132:
3119:
3093:
3076:(3): 339–411,
3063:
3027:
2990:(3): 281–308,
2980:Hitchin, Nigel
2976:
2939:
2926:
2895:
2872:
2871:
2850:
2829:
2811:
2790:
2746:
2701:
2680:
2668:
2655:
2654:
2652:
2649:
2648:
2647:
2640:
2636:3 Body Problem
2627:
2618:Sheldon Cooper
2612:
2611:In Pop culture
2609:
2608:
2607:
2602:
2595:
2592:
2575:
2572:
2534:Hitchin (2003)
2459:
2456:
2443:
2440:
2437:
2434:
2431:
2424:
2419:
2411:
2405:
2380:
2358:
2354:
2333:
2328:
2324:
2320:
2303:
2300:
2285:
2281:
2258:
2253:
2247:
2241:
2236:
2231:
2225:
2219:
2214:
2210:
2206:
2202:
2176:
2172:
2168:
2164:
2158:
2154:
2150:
2145:
2141:
2135:
2131:
2127:
2122:
2118:
2112:
2108:
2104:
2099:
2095:
2068:
2065:
2060:
2056:
2052:
2048:
2044:
2039:
2035:
2031:
2026:
2022:
2016:
2012:
2008:
2005:
1980:
1976:
1970:
1966:
1942:
1939:
1934:
1930:
1924:
1920:
1916:
1911:
1907:
1903:
1898:
1894:
1890:
1886:
1882:
1879:
1856:
1851:
1845:
1839:
1834:
1828:
1822:
1817:
1813:
1790:
1786:
1782:
1778:
1757:
1754:
1751:
1748:
1745:
1723:
1719:
1715:
1710:
1704:
1698:
1693:
1687:
1664:
1659:
1653:
1647:
1642:
1636:
1630:
1622:
1619:
1599:
1587:
1584:
1522:
1476:
1473:
1437:
1432:
1394:
1389:
1384:
1381:
1376:
1372:
1368:
1363:
1359:
1355:
1350:
1346:
1342:
1337:
1333:
1329:
1309:
1306:
1301:
1296:
1292:
1288:
1283:
1278:
1274:
1270:
1265:
1260:
1256:
1252:
1247:
1242:
1238:
1212:
1207:
1189:
1186:
1178:elliptic curve
1172:, so that the
1161:
1158:
1150:Shing-Tung Yau
1148:and proved by
1146:Eugenio Calabi
1129:
1120:embedded in a
1113:
1110:
1109:
1108:
1100:
1097:
1085:
1082:
1079:
1076:
1073:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1007:
1004:
1001:
997:
973:
970:
967:
964:
961:
950:
943:
931:
911:
891:
888:
885:
881:
875:
871:
867:
861:
858:
852:
849:
846:
835:
813:
812:
796:
786:
774:
764:
752:
737:
724:
721:
718:
715:
712:
687:
677:
665:
655:
643:
620:
600:
591:For a compact
573:
553:
550:
547:
542:
538:
526:
525:
512:
509:
506:
503:
500:
475:
465:
449:
446:
443:
440:
437:
413:
410:
407:
404:
384:
373:tangent bundle
365:
353:
333:
323:
311:
283:
263:
243:
223:
201:Shing-Tung Yau
196:
193:
118:Ricci flatness
84:
83:
38:
36:
29:
15:
9:
6:
4:
3:
2:
4797:
4786:
4783:
4781:
4780:String theory
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4762:
4760:
4745:
4742:
4740:
4737:
4735:
4732:
4730:
4729:Zamolodchikov
4727:
4725:
4724:Zamolodchikov
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4547:
4545:
4542:
4540:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4422:
4420:
4417:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4285:
4282:
4280:
4277:
4275:
4272:
4270:
4267:
4266:
4264:
4260:
4254:
4251:
4249:
4248:Matrix theory
4246:
4245:
4243:
4241:
4237:
4231:
4228:
4226:
4223:
4222:
4220:
4218:
4214:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4169:
4167:
4165:
4164:Supersymmetry
4161:
4155:
4152:
4150:
4147:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4113:
4110:
4108:
4101:
4097:
4094:
4093:
4092:
4089:
4087:
4084:
4083:
4082:
4079:
4077:
4074:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4052:
4050:
4046:
4040:
4038:
4034:
4032:
4029:
4027:
4024:
4021:
4014:
4007:
4000:
3993:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3966:
3963:
3962:
3960:
3958:
3954:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3923:
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3904:
3902:
3900:
3896:
3890:
3887:
3885:
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3836:
3834:
3832:
3828:
3822:
3819:
3817:
3816:Dual graviton
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3783:
3781:
3777:
3771:
3768:
3766:
3763:
3761:
3758:
3756:
3753:
3752:
3750:
3748:
3744:
3738:
3735:
3733:
3732:RNS formalism
3730:
3728:
3725:
3723:
3720:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3691:
3688:
3684:
3681:
3679:
3676:
3675:
3674:
3671:
3669:
3668:Type I string
3666:
3665:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3645:
3643:
3639:
3633:
3630:
3626:
3623:
3621:
3618:
3617:
3616:
3613:
3611:
3608:
3606:
3603:
3602:
3600:
3596:
3592:
3591:String theory
3585:
3580:
3578:
3573:
3571:
3566:
3565:
3562:
3555:
3551:
3548:
3546:
3542:
3540:
3537:
3536:
3525:
3524:
3519:
3516:
3511:
3508:
3504:
3503:
3499:
3497:
3494:
3491:
3488:
3487:
3478:
3475:(similar to (
3472:
3467:
3463:
3459:
3455:
3451:
3446:
3443:
3439:
3435:
3431:
3427:
3423:
3418:
3415:
3411:
3407:
3401:
3397:
3393:
3389:
3385:
3376:on 2010-01-13
3375:
3371:
3367:
3363:
3357:
3353:
3349:
3348:
3342:
3339:
3333:
3329:
3324:
3321:
3317:
3313:
3309:
3305:
3299:
3294:
3289:
3285:
3281:
3277:
3272:
3269:
3265:
3261:
3257:
3252:
3247:
3243:
3239:
3238:Greene, Brian
3235:
3232:
3228:
3223:
3218:
3214:
3209:
3206:
3202:
3197:
3192:
3185:
3184:
3178:
3175:
3171:
3167:
3161:
3157:
3153:
3149:
3145:
3144:
3135:
3129:
3125:
3120:
3117:
3113:
3108:
3103:
3099:
3094:
3091:
3087:
3083:
3079:
3075:
3071:
3070:
3064:
3061:
3057:
3053:
3049:
3045:
3041:
3037:
3033:
3032:Invent. Math.
3028:
3025:
3021:
3017:
3013:
3008:
3003:
2998:
2993:
2989:
2985:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2953:
2949:
2945:
2940:
2937:
2933:
2929:
2927:9780691079073
2923:
2919:
2915:
2914:
2909:
2905:
2901:
2900:Fox, Ralph H.
2896:
2893:on 2011-07-17
2892:
2888:
2884:
2880:
2876:
2875:
2866:
2861:
2854:
2845:
2840:
2833:
2825:
2821:
2815:
2806:
2801:
2794:
2786:
2782:
2778:
2774:
2770:
2766:
2765:
2760:
2756:
2750:
2742:
2738:
2734:
2730:
2725:
2720:
2716:
2712:
2705:
2696:
2691:
2684:
2677:
2672:
2665:
2660:
2656:
2645:
2641:
2638:
2637:
2632:
2628:
2625:
2624:
2623:Young Sheldon
2619:
2615:
2614:
2606:
2603:
2601:
2598:
2597:
2591:
2589:
2585:
2581:
2571:
2568:
2567:Edward Witten
2564:
2559:
2557:
2551:
2549:
2545:
2541:
2539:
2535:
2531:
2527:
2523:
2519:
2514:
2512:
2508:
2504:
2503:supersymmetry
2500:
2496:
2492:
2488:
2487:string theory
2483:
2481:
2477:
2473:
2469:
2465:
2455:
2435:
2432:
2422:
2378:
2356:
2352:
2326:
2322:
2299:
2283:
2279:
2256:
2251:
2239:
2234:
2229:
2217:
2212:
2208:
2204:
2189:
2174:
2170:
2166:
2156:
2152:
2148:
2143:
2139:
2133:
2129:
2125:
2120:
2110:
2106:
2102:
2097:
2093:
2083:
2079:
2066:
2058:
2054:
2050:
2037:
2024:
2014:
2010:
2003:
1994:
1978:
1974:
1968:
1964:
1953:
1940:
1932:
1928:
1922:
1918:
1909:
1905:
1896:
1892:
1888:
1884:
1877:
1868:
1849:
1837:
1832:
1815:
1811:
1788:
1784:
1780:
1776:
1755:
1749:
1746:
1743:
1721:
1717:
1713:
1708:
1696:
1691:
1657:
1645:
1640:
1620:
1617:
1597:
1583:
1581:
1576:
1574:
1570:
1565:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1530:
1528:
1521:
1517:
1513:
1509:
1505:
1501:
1500:
1495:
1491:
1487:
1483:
1475:CY threefolds
1472:
1470:
1466:
1462:
1457:
1453:
1435:
1420:
1415:
1413:
1407:
1392:
1382:
1374:
1370:
1366:
1361:
1357:
1353:
1348:
1344:
1340:
1335:
1331:
1307:
1304:
1299:
1294:
1290:
1286:
1281:
1276:
1272:
1268:
1263:
1258:
1254:
1250:
1245:
1240:
1236:
1226:
1210:
1195:
1185:
1183:
1179:
1175:
1171:
1167:
1157:
1155:
1151:
1147:
1143:
1139:
1135:
1127:
1123:
1119:
1106:
1101:
1098:
1080:
1074:
1071:
1048:
1042:
1039:
1036:
1033:
1030:
1027:
1005:
1002:
999:
995:
987:
986:Hodge numbers
968:
962:
959:
951:
948:
944:
929:
909:
886:
883:
879:
873:
869:
865:
850:
836:
833:
832:
831:
828:
826:
821:
818:
810:
794:
787:
772:
765:
750:
742:
738:
735:
719:
713:
710:
702:contained in
701:
685:
678:
663:
656:
641:
634:
633:
632:
618:
598:
589:
587:
571:
548:
540:
536:
523:
507:
501:
498:
490:contained in
489:
473:
466:
463:
444:
438:
435:
427:
426:unitary group
408:
402:
382:
374:
370:
366:
351:
331:
324:
309:
301:
297:
296:
295:
281:
261:
241:
221:
212:
208:
206:
202:
192:
190:
186:
182:
178:
174:
170:
165:
163:
159:
155:
151:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
107:
103:
99:
90:
80:
77:
69:
59:
55:
49:
48:
42:
37:
28:
27:
22:
4274:Arkani-Hamed
4172:Supergravity
4139:Moduli space
4085:
4036:
4031:Dirac string
3957:Gauge theory
3937:Loop algebra
3874:Black string
3737:GS formalism
3521:
3500:
3453:
3450:Scholarpedia
3449:
3425:
3421:
3391:
3378:, retrieved
3374:the original
3346:
3327:
3279:
3241:
3222:math/0410260
3212:
3182:
3151:
3123:
3097:
3073:
3067:
3038:(1): 27–60,
3035:
3031:
2987:
2983:
2972:the original
2951:
2947:
2912:
2891:the original
2886:
2853:
2844:math/0612139
2832:
2824:the original
2814:
2793:
2768:
2762:
2758:
2749:
2714:
2710:
2704:
2683:
2671:
2659:
2634:
2621:
2579:
2577:
2560:
2552:
2542:
2521:
2517:
2515:
2498:
2484:
2461:
2305:
2191:
2085:
2081:
1996:
1955:
1870:
1589:
1577:
1566:
1561:
1557:
1553:
1549:
1545:
1541:
1533:
1531:
1519:
1511:
1498:
1478:
1416:
1409:
1228:
1191:
1163:
1141:
1137:
1133:
1115:
829:
822:
814:
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3456:(8): 6524,
2755:Reid, Miles
2717:: 137–148.
2644:Half-Life 2
2605:G2 manifold
1194:K3 surfaces
1170:flat metric
1020:vanish for
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763:is trivial.
322:is trivial.
195:Definitions
189:Chern class
173:K3 surfaces
58:introducing
4759:Categories
4654:Strominger
4649:Steinhardt
4644:Staudacher
4559:Polchinski
4509:Nanopoulos
4469:Mandelstam
4449:Kontsevich
4289:Berenstein
4217:Holography
4197:Superspace
4096:K3 surface
4055:Worldsheet
3970:Instantons
3598:Background
3380:2009-02-04
3196:1612.04311
2805:1503.07349
2724:2002.04879
2695:1810.08953
2651:References
2472:braneworld
1508:polynomial
1482:Miles Reid
1105:Gorenstein
181:dimensions
41:references
4689:Veneziano
4569:Rajaraman
4464:Maldacena
4354:Gopakumar
4304:Dijkgraaf
4299:Curtright
3965:Anomalies
3844:NS5-brane
3765:U-duality
3760:S-duality
3755:T-duality
3523:MathWorld
3477:Yau 2009a
3060:122638262
3002:CiteSeerX
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130:spacetime
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66:July 2018
4744:Zwiebach
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4694:Verlinde
4669:Townsend
4664:Susskind
4599:Sagnotti
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4519:Nekrasov
4484:Minwalla
4479:Martinec
4444:Knizhnik
4439:Klebanov
4434:Kapustin
4399:'t Hooft
4334:Fischler
4269:Aganagić
4240:M-theory
4129:Conifold
4124:Orbifold
4107:manifold
4048:Geometry
3854:M5-brane
3849:M2-brane
3786:Graviton
3702:F-theory
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3370:34989218
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3174:13793300
3150:(1987),
2910:(eds.),
2881:(1954),
2594:See also
2556:families
2544:F-theory
2511:holonomy
1538:zero set
1174:holonomy
1112:Examples
700:holonomy
488:holonomy
140:, after
114:manifold
4674:Trivedi
4659:Sundrum
4624:Shenker
4614:Seiberg
4609:Schwarz
4579:Randall
4539:Novikov
4529:Nielsen
4514:Năstase
4424:Kallosh
4409:Gibbons
4349:Gliozzi
4339:Friedan
4329:Ferrara
4314:Douglas
4309:Distler
3859:S-brane
3839:D-brane
3796:Tachyon
3791:Dilaton
3605:Strings
3458:Bibcode
3442:1990928
3312:1963559
3268:1479700
3256:Bibcode
3227:Bibcode
3201:Bibcode
3116:2537089
3090:0480350
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2476:D-brane
371:of the
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3191:arXiv
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