Showing papers on "Calabi–Yau manifold published in 2011"
TL;DR: In this article, potential modularity theorems for l-adic representations of any dimension were proved for all elliptic curves with nonintegral j -invariant defined over a real field.
Abstract: We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j -invariant defined over a totally real field.
459 citations
TL;DR: In this article, a real ane manifold with singularities (a tropical manifold) is constructed from a Calabi-Yau manifold, and an explicit and canonical order-by-order description of the degeneration via families of tropical trees is given.
Abstract: We construct from a real ane manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods.
291 citations
TL;DR: In this paper, a systematic search over complete intersection Calabi-Yau manifolds with less than six Kahler parameters leads to over 200 such models which are constructed by compactifying on smooth CalabiYau three-folds in the presence of purely Abelian internal gauge fields.
Abstract: We construct heterotic standard models by compactifying on smooth Calabi-Yau three-folds in the presence of purely Abelian internal gauge fields. A systematic search over complete intersection Calabi-Yau manifolds with less than six K\"ahler parameters leads to over 200 such models which we present. Each of these models has precisely the matter spectrum of the minimal supersymmetric standard model, at least one pair of Higgs doublets, the standard model gauge group, and no exotics. For about 100 of these models there are four additional $U(1)$ symmetries which are Green-Schwarz anomalous and, hence, massive. In the remaining cases, three $U(1)$ symmetries are anomalous, while the fourth, massless one can be spontaneously broken by singlet vacuum expectation values. The presence of additional global $U(1)$ symmetries, together with the possibility of switching on singlet vacuum expectation values, leads to a rich phenomenology which is illustrated for a particular example. Our database of standard models, which can be further enlarged by simply extending the computer-based search, allows for a detailed and systematic phenomenological analysis of string standard models, covering issues such as the structure of Yukawa couplings, $R$-parity violation, proton stability, and neutrino masses.
261 citations
TL;DR: In this paper, the resolution procedure of Esole and Yau was used to study G-flux and the emergence of spectral covers from elliptically fibered Calabi-Yau's with a surface of A 4 singularities.
Abstract: We use the resolution procedure of Esole and Yau [1] to study Yukawa couplings, G-flux, and the emergence of spectral covers from elliptically fibered Calabi-Yau’s with a surface of A
4 singularities. We provide a global description of the Esole-Yau resolution and use it to explicitly compute Chern classes of the resolved 4-fold, proving the conjecture of [2] for the Euler character in the process. We comment on the physical implications of the surprising singular fibers in codimension 2 and 3 in [1] and emphasize a group theoretic interpretation based on the A
4 weight lattice. We then construct explicit G-fluxes by brute force in one of the 6 birationally equivalent Esole-Yau resolutions, quantize them explicitly using our result for the second Chern class, and compute the spectrum and flux-induced 3-brane charges, finding agreement with results and conjectures of local models in all cases. Finally, we provide a precise description of the spectral divisor formalism in this setting and sharpen the procedure described in [3] in order to explicitly demonstrate how the Higgs bundle spectral cover of the local model emerges from the resolved Calabi-Yau geometry. Along the way, we demonstrate explicitly how the quantization rules for fluxes in the local and global models are related.
187 citations
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TL;DR: In this article, the authors give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curve on Y meeting D in a single point.
Abstract: We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anti-canonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga's 1981 cusp conjecture.
185 citations
TL;DR: In this paper, the authors define and investigate deformed n-Calabi-Yau completions of homologically smooth dierential graded (dg) categories and show that they are compatible with derived equivalences and with localizations.
Abstract: We define and investigate deformed n-Calabi-Yau completions of homolog- ically smooth dierential graded (=dg) categories. Important examples are: deformed preprojective algebras of connected non Dynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that deformed Calabi-Yau com- pletions do have the Calabi-Yau property and that their construction is compatible with derived equivalences and with localizations. In particular, Ginzburg dg algebras have the Calabi-Yau property. We show that deformed 3-Calabi-Yau completions of algebras of global dimension at most 2 are quasi-isomorphic to Ginzburg dg algebras and apply this to the study of cluster-tilted algebras and to the construction of derived equivalences as- sociated to mutations of quivers with potentials. In the appendix, Michel Van den Bergh uses non commutative dierential geometry to give an alternative proof of the fact that Ginzburg dg algebras have the Calabi-Yau property.
174 citations
TL;DR: In this article, all automorphisms of the product of projective spaces that descend to a free action on the Calabi-Yau manifold are identified and classified by means of a computer search.
Abstract: In order to find novel examples of non-simply connected Calabi-Yau threefolds, free quotients of complete intersections in products of projective spaces are classified by means of a computer search. More precisely, all automorphisms of the product of projective spaces that descend to a free action on the Calabi-Yau manifold are identified.
124 citations
TL;DR: In this paper, the authors show that the presence of gauge fields in heterotic Calabi-Yau compactifications can cause the stabilization of some, or all, of the complex structure moduli while maintaining a Minkowski vacuum.
Abstract: In this paper, we show that the presence of gauge fields in heterotic Calabi-Yau compactifications can cause the stabilization of some, or all, of the complex structure moduli while maintaining a Minkowski vacuum. Certain deformations of the Calabi-Yau complex structure, with all other moduli held fixed, can lead to the gauge bundle becoming non-holomorphic and, hence, non-supersymmetric. This is manifested by a positive F-term potential which stabilizes the corresponding complex structure moduli. We use 10-and 4-dimensional field theory arguments as well as a derivation based purely on algebraic geometry to show that this picture is indeed correct. An explicit example is presented in which a large subset of complex structure moduli is fixed. We demonstrate that this type of theory can serve as the hidden sector in heterotic vacua with realistic particle physics.
114 citations
TL;DR: In this article, the tools necessary to use holomorphic bundles as a mechanism for moduli stabilization are systematically developed, including the Atiyah class, which determines the deformations of the complex structure for which the gauge bundle becomes non-holomorphic and hence, non-supersymmetric.
Abstract: Holomorphic gauge fields in N = 1 supersymmetri cheterotic compactifications can constrain the complex structure moduli of a Calabi-Yau manifold. In this paper, the tools necessary to use holomorphic bundles as a mechanism for moduli stabilization are systematically developed. We review the requisite deformation theory — including the Atiyah class, which determines the deformations of the complex structure for which the gauge bundle becomes non-holomorphic and, hence, non-supersymmetric. In addition, two equivalent approaches to this mechanism of moduli stabilization are presented. The first isan efficient computational algorithm for determining the supersymmetric moduli space, while the second is an F-term potential in the four-dimensional theory associated with vector bundle holomorphy. These three methods are proven to be rigorously equivalent. We present explicit examples in which large numbers of complex structure moduli are stabilized. Finally, higher-order corrections to the moduli space are discussed.
101 citations
TL;DR: In this paper, the rationality/unirationality of moduli spaces of (1, d)-polarized Abelian surfaces with canonical level structure for small values of d is investigated.
Abstract: We describe birational models and decide the rationality/unirationality of moduli spaces $\cal A$d (and $\cal A$levd) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.
99 citations
TL;DR: In this article, the Ricci-flat Kahler metrics on projective Calabi-Yau manifolds are shown to collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit.
Abstract: We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit. In this setting we prove an analog of a result of Gross-Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.
01 Jan 2011
TL;DR: In this paper, the authors investigated mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture and obtained an enumerative meaning for the (inverse) mirror maps, and gave a geometric reason for why their Taylor series expansions in terms of the Kahler parameters of the complex manifold $X$ have integral coefficients.
Abstract: We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the Kahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.
TL;DR: In this article, the convergence of Ricci-flat Kahler metrics on Calabi-Yau manifolds along a smoothing is established, which can be of independent interest.
Abstract: In this paper, we study the behavior of Ricci-flat Kahler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat Kahler metrics on Calabi-Yau manifolds along a smoothing is established, which can be of independent interest.
01 Jun 2011
TL;DR: In this article, Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations were generalized for algebraic transformations of Calabi-Yau differential equations.
Abstract: We provide certain unusual generalizations of Clausen's and Orr's theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi-Yau differential equations.
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TL;DR: In this article, the convergence for Gromov-Witten invariant of elliptic orbifolds was established for all the three types of the above three types with type (3, 3, 3), (4, 4, 2) and (6, 3 2 ).
Abstract: In this paper, we establish the convergence for Gromov-Witten invariant of elliptic orbifold $\mathbb{P}^1$ with type $(3,3,3), (4,4,2)$ and $(6,3,2)$. We also prove the mirror theorems of Gromov-Witten theory for those orbifolds and FJRW theory of elliptic singularities. Using T.Milanov and Y. Ruan's work, we prove the Landau-Ginzburg/Calabi-Yau correspondence of all genera for the above three types of elliptic orbifold $\mathbb{P}^1$.
TL;DR: In this article, it was shown that all representation-finite finite algebras are twisted fractionally Calabi-Yau, and that for any constant value of n ≥ 0, twisted k = n(n(k)-1/n(n)-1) 1/n/k-1/k) twisted k-calabi-yau finite algesbras of global dimension at most k = k(n) are representation-feasible.
Abstract: In this short paper, we study $n$-representation-finite algebras from the viewpoint of the fractionally Calabi-Yau property. We shall show that all $n$-representation-finite algebras are twisted fractionally Calabi-Yau. We also show that for any $\ell>0$, twisted $\frac{n(\ell-1)}{\ell}$-Calabi-Yau algebras of global dimension at most $n$ are $n$-representation-finite. As an application, we give a construction of $n$-representation-finite algebras using the tensor product.
TL;DR: In this article, the authors consider heterotic string solutions based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold, preserving two supercharges.
Abstract: We consider heterotic string solutions based on a warped product of a four-dimensional domain wall and a six-dimensional internal manifold, preserving two supercharges. The constraints on the internal manifolds with SU(3) structure are derived. They are found to be generalized half-flat manifolds with a particular pattern of torsion classes and they include half-flat manifolds and Strominger’s complex non-Kahler manifolds as special cases. We also verify that previous heterotic compactifications on half-flat mirror manifolds are based on this class of solutions.
TL;DR: In this paper, wall-crossing for toric Calabi-Yau manifolds without compact four-cycles is described in terms of free fermions, vertex operators, and crystal melting.
Abstract: We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non-commutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field through each two-cycle in the manifold. BPS partition functions in non-trivial chambers are given by the expectation values of these operators. Thirdly, we discuss crystal interpretation of such correlators for this whole class of manifolds. We describe evolution of these crystals upon a change of the moduli, and find crystal interpretation of the flop transition and the DT/PT transition. The crystals which we find generalize and unify various other Calabi-Yau crystal models which appeared in literature in recent years.
TL;DR: In this paper, the authors use the resolution procedure of Esole and Yau arXiv:1107.0733 to study Yukawa couplings, G-flux, and the emergence of spectral covers from elliptically fibered Calabi-Yau's with a surface of A_4 singularities.
Abstract: We use the resolution procedure of Esole and Yau arXiv:1107.0733 to study Yukawa couplings, G-flux, and the emergence of spectral covers from elliptically fibered Calabi-Yau's with a surface of A_4 singularities. We provide a global description of the Esole-Yau resolution and use it to explicitly compute Chern classes of the resolved 4-fold, proving the conjecture of arXiv:0908.1784 for the Euler character in the process. We comment on the physical implications of the surprising singular fibers in codimension 2 and 3 in arXiv:1107.0733 and emphasize a group theoretic interpretation based on the A_4 weight lattice. We then construct explicit G-fluxes by brute force in one of the 6 birationally equivalent Esole-Yau resolutions, quantize them explicitly using our result for the second Chern class, and compute the spectrum and flux-induced 3-brane charges, finding agreement with results and conjectures of local models in all cases. Finally, we provide a precise description of the spectral divisor formalism in this setting and sharpen the procedure described in arXiv:1107.1718 in order to explicitly demonstrate how the Higgs bundle spectral cover of the local model emerges from the resolved Calabi-Yau geometry. Along the way, we demonstrate explicitly how the quantization rules for fluxes in the local and global models are related.
TL;DR: A survey of curve counting invariants on Calabi-Yau 3-folds is given in this paper, with a focus on the rationality conjecture of the generating series of PT invariants and its conjectural Gopakumar-Vafa form.
Abstract: The purpose of this paper is twofold: first we give a survey on the recent developments of curve counting invariants on Calabi-Yau 3-folds, e.g. Gromov-Witten theory, Donaldson-Thomas theory and Pandharipande-Thomas theory. Next we focus on the proof of the rationality conjecture of the generating series of PT invariants, and discuss its conjectural Gopakumar-Vafa form.
TL;DR: In this paper, the mirror conjecture of Aganagic-vafa and klemm-Vafa on disk enumeration in toric Calabi-Yau 3-folds was shown to hold for all smooth semi-projective toric calabi-yau 3folds.
Abstract: We present a proof of the mirror conjecture of Aganagic-Vafa [arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] on disk enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary framing. In particular, we recover previous results on the conjecture for (i) an inner brane at zero framing in the total space of the canonical line bundle of the projective plane (Graber-Zaslow [arXiv:hep-th/0109075]), (ii) an outer brane at arbitrary framing in the resolved conifold (Zhou [arXiv:1001.0447]), and (iii) an outer brane at zero framing in the total space of the canonical line bundle of the projective plane (Brini [arXiv:1102.0281, Section 5.3]).
TL;DR: In this paper, the convergence of Calabi-Yau manifolds under Kahler degeneration to orbifold singularities and complex degeneration of the conifolds to canonical singularities was studied.
TL;DR: The proof of Serre's conjecture on Galois representations over finite fields has been proved in this article, showing that all rigid Calabi-Yau threefolds defined over Q ǫ are modular.
TL;DR: In this paper, the authors proved homological mirror symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic threefold).
Abstract: We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.
TL;DR: In the Hermitian case, this paper showed that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also for Hermitians.
Abstract: We show that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also in the Hermitian case.
TL;DR: The Ricci-flat ALE Kahler spaces have been studied in this paper, where it was shown that every 3-dimensional Gorenstein toric Kahler cone admits a crepant resolution for which the above theorem applies.
Abstract: Previously the author has proved that a crepant resolution π : Y → X of a Ricci-flat Kahler cone X admits a complete Ricci-flat Kahler metric asymptotic to the cone metric in every Kahler class in \({H^2_c(Y,\mathbb R)}\). These manifolds can be considered to be generalizations of the Ricci-flat ALE Kahler spaces known by the work of P. Kronheimer, D. Joyce and others. This article considers further the problem of constructing examples. We show that every 3-dimensional Gorenstein toric Kahler cone admits a crepant resolution for which the above theorem applies. This gives infinitely many examples of asymptotically conical Ricci-flat manifolds. Then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit Ricci-flat Kahler cone metrics by the work of C. Boyer, K. Galicki, J. Kollar, and others. We concentrate on 3-dimensional examples. Two families of hypersurface examples are given which are distinguished by the condition b3(Y) = 0 or b3(Y) ≠ 0.
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TL;DR: In this paper, the authors investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds and the "matter" representation of the associated Lie algebra.
Abstract: We investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the "matter" representation of the associated Lie algebra. The main tool is the analysis and the appropriate interpretation of the anomaly formula for six-dimensional supersymmetric theories. We find that this anomaly formula is geometrically captured by a relation among codimension two cycles on the base of the elliptic fibration, and that this relation holds for elliptic fibrations of any dimension. We introduce a "Tate cycle" which efficiently describes this relationship, and which is remarkably easy to calculate explicitly from the Weierstrass equation of the fibration. We check the anomaly cancellation formula in a number of situations and show how this formula constrains the geometry (and in particular the Euler characteristic) of the Calabi-Yau threefold.
01 Jun 2011
TL;DR: The Strominger system can be viewed as a canonical structure on non-Kahler Calabi-Yau three-folds with balanced metrics as mentioned in this paper, and the existence of balanced metrics is studied in this paper.
Abstract: The solution of the Strominger system can be viewed as a canonical structure on non-Kahler Calabi-Yau threefolds with balanced metrics. In this talk, we review the existence of balanced metrics on non-Kahler complex manifolds and the existence of solutions to the Strominger system.
TL;DR: In this paper, it was shown that the Ricci-flat Kahler manifold decomposes into a product of complex tori, irreducible holomorphic symplectic manifolds, and Calabi-Yau manifolds.
Abstract: After a finite etale cover, any Ricci-flat Kahler manifold decomposes into a product of complex tori, irreducible holomorphic symplectic manifolds, and Calabi–Yau manifolds. We present results indicating that this decomposition is an invariant of the derived category. The main idea to distinguish the derived category of an irreducible holomorphic symplectic manifold from that of a Calabi–Yau manifold is that point sheaves do not deform in certain (non-commutative) deformations of the former, whereas they do for the latter. On the way, we prove a conjecture of Caldararu on the module structure of the Hochschild–Kostant–Rosenberg isomorphism for manifolds with trivial canonical bundle as a direct consequence of recent work by Calaque, van den Bergh, and Ramadoss.
TL;DR: In this paper, Kustin-Miller unprojections between Calabi-Yau 3folds with Picard group of rank one were studied and described by Pfaffian equations in weighted projective spaces.