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Showing papers on "Mirror symmetry published in 2005"


Journal ArticleDOI
TL;DR: The N = 1 effective action for generic type IIA Calabi-Yau orientifolds in the presence of background fluxes is computed from a Kaluza-Klein reduction as discussed by the authors.

391 citations


Journal ArticleDOI
TL;DR: In this paper, the conditions to have supersymmetric D-branes on general N = 1 backgrounds with Ramond-Ramond fluxes were studied, and the results preserve the generalised mirror symmetry relating the type IIA and IIB backgrounds considered.
Abstract: We study the conditions to have supersymmetric D-branes on general {\cal N}=1 backgrounds with Ramond-Ramond fluxes. These conditions can be written in terms of the two pure spinors associated to the SU(3)\times SU(3) structure on T_M\oplus T^\star_M, and can be split into two parts each involving a different pure spinor. The first involves the integrable pure spinor and requires the D-brane to wrap a generalised complex submanifold with respect to the generalised complex structure associated to it. The second contains the non-integrable pure spinor and is related to the stability of the brane. The two conditions can be rephrased as a generalised calibration condition for the brane. The results preserve the generalised mirror symmetry relating the type IIA and IIB backgrounds considered, giving further evidence for this duality.

254 citations


Journal ArticleDOI
TL;DR: In this article, the conditions to have supersymmetric D-branes on general = 1 backgrounds with Ramond-Ramond fluxes were studied. But the results preserve the generalised mirror symmetry relating the type IIA and IIB backgrounds considered.
Abstract: We study the conditions to have supersymmetric D-branes on general = 1 backgrounds with Ramond-Ramond fluxes. These conditions can be written in terms of the two pure spinors associated to the SU(3) × SU(3) structure on TM⊕TM, and can be split into two parts each involving a different pure spinor. The first involves the integrable pure spinor and requires the D-brane to wrap a generalised complex submanifold with respect to the generalised complex structure associated to it. The second contains the non-integrable pure spinor and is related to the stability of the brane. The two conditions can be rephrased as a generalised calibration condition for the brane. The results preserve the generalised mirror symmetry relating the type IIA and IIB backgrounds considered, giving further evidence for this duality.

253 citations


Posted Content
TL;DR: In this paper, the authors used mirror symmetry to show that the dimer graph is a mirror to the D6-branes at the singular point, and geometrically encoded the same quiver theory on their world volume.
Abstract: Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume.

245 citations


Journal ArticleDOI
TL;DR: In this paper, the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form eiJ and the holomorphic form O.
Abstract: We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form eiJ and the holomorphic form O. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: eiJ is closed under the action of the twisted exterior derivative in IIA theory, and similarly O is closed in IIB. This means that supersymmetric SU(3)-structure manifolds are always complex in IIB while they are twisted symplectic in IIA. Modulo a different action of the B-field, these are all generalized Calabi-Yau manifolds, as defined by Hitchin.

182 citations


Journal ArticleDOI
TL;DR: In this article, a general class of T3-fibered geometries admitting SU(3) structure was considered, and an exchange of pure spinors (eiJ and Ω) was found in dual geometry under fiberwise T-duality, and the transformations of the NS flux and the components of intrinsic torsion.
Abstract: When string theory is compactified on a six-dimensional manifold with a nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely geometrical composite NS form associated with lack of integrability of the complex structure on the mirror side. Considering a general class of T3-fibered geometries admitting SU(3) structure, we find an exchange of pure spinors (eiJ and Ω) in dual geometries under fiberwise T–duality, and study the transformations of the NS flux and the components of intrinsic torsion. A complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with NS flux.

141 citations


Journal ArticleDOI
TL;DR: In this article, a simple class of type IIA string compactifications on Calabi-Yau manifolds where background fluxes generate a potential for the complex structure moduli, the dilaton, and the Kaehler moduli is described.
Abstract: We describe a simple class of type IIA string compactifications on Calabi-Yau manifolds where background fluxes generate a potential for the complex structure moduli, the dilaton, and the Kaehler moduli. This class of models corresponds to gauged {Nu} = 2 supergravities, and the potential is completely determined by a choice of gauging and by data of the {Nu} = 2 Calabi-Yau model--the prepotential for vector multiplets and the quaternionic metric on the hypermultiplet moduli space. Using mirror symmetry, one can determine many (though not all) of the quantum corrections which are relevant in these models.

124 citations


Proceedings ArticleDOI
01 Jul 2005
TL;DR: In this paper, the authors provide a self-contained guide to the derived category approach to B-branes and the idea of Pi-stability and argue that this mathematical machinery is hard to avoid for a proper understanding of Bbranes.
Abstract: In this review we study BPS D-branes on Calabi-Yau threefolds. Such D-branes naturally divide into two sets called A-branes and B-branes which are most easily understood from topological field theory. The main aim of this paper is to provide a self-contained guide to the derived category approach to B-branes and the idea of Pi-stability. We argue that this mathematical machinery is hard to avoid for a proper understanding of B-branes. A-branes and B-branes are related in a very complicated and interesting way which ties in with the ``homological mirror symmetry'' conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3-fold, flops and orbifolds are discussed at some length. In the latter case we describe the role of McKay quivers in the context of D-branes. These notes are to be submitted to the proceedings of TASI03.

123 citations


Journal ArticleDOI
TL;DR: In this article, the N = 1 effective action for generic type IIA and type IIB Calabi-Yau orientifolds in the presence of background fluxes was derived by using a Kaluza-Klein reduction.
Abstract: This article first reviews the calculation of the N = 1 effective action for generic type IIA and type IIB Calabi-Yau orientifolds in the presence of background fluxes by using a Kaluza-Klein reduction. The Kahler potential, the gauge kinetic functions and the flux-induced superpotential are determined in terms of geometrical data of the Calabi-Yau orientifold and the background fluxes. As a new result, it is shown that the chiral description directly relates to Hitchin's generalized geometry encoded by special odd and even forms on a threefold, whereas a dual formulation with several linear multiplets makes contact to the underlying N = 2 special geometry. In type IIB setups, the flux-potentials can be expressed in terms of superpotentials, D-terms and, generically, a massive linear multiplet. The type IIA superpotential depends on all geometric moduli of the theory. It is reviewed, how type IIA orientifolds arise as a special limit of M-theory compactified on specific G2 manifolds by matching the effective actions. In a similar spirit type IIB orientifolds are shown to descend from F-theory on a specific class of Calabi-Yau fourfolds. In addition, mirror symmetry for Calabi-Yau orientifolds is briefly discussed and it is shown that the N = 1 chiral coordinates linearize the appropriate instanton actions.

119 citations


Journal ArticleDOI
TL;DR: In this article, the authors give an introductory review of topological strings and their application to various aspects of superstrings and supersymmetric gauge theories, including the notions of Calabi-Yau manifold and toric geometry, as well as physical methods developed for solving them.
Abstract: We give an introductory review of topological strings and their application to various aspects of superstrings and supersymmetric gauge theories. This review includes developing the necessary mathematical background for topological strings, such as the notions of Calabi-Yau manifold and toric geometry, as well as physical methods developed for solving them, such as mirror symmetry, large N dualities, the topological vertex and quantum foam. In addition, we discuss applications of topological strings to N=1,2 supersymmetric gauge theories in 4 dimensions as well as to BPS black hole entropy in 4 and 5 dimensions. (These are notes from lectures given by the second author at the 2004 Simons Workshop in Mathematics and Physics.)

115 citations


Journal ArticleDOI
TL;DR: In this paper, the second-order nonlinearity is dominated by the interface where the symmetry is broken, and directional and polarization selection rules for second-harmonic generation from partially asymmetric smmd nanostructures are provided.
Abstract: Second-harmonic generation sSHGd from individual nanoscopic metal tips has been investigated. Compared to both planar interfaces, as well as spherical or ellipsoidal nanoparticles, very different polarization selection rules and SH-emission directions result. As a partially asymmetric nanostructure the tip allows for the distinction of otherwise inseparable local surface and nonlocal bulk second-harmonic polarizations. This provides opportunities for second-harmonic investigations of nanoparticles and in scattering-type near field microscopy. The optics of media of dimensions small compared to the optical wavelength is characterized by distinctive phenomena such as optical field confinement and structural resonances. With a strong focus in research on the linear optical processes of surface nanostructures and colloids, the nonlinear optical properties have remained largely unexplored. The nonlinear response, however, is expected to differ fundamentally due to its high symmetry selectivity. Specifically, for media with inversion symmetry the second-order nonlinearity is dominated by the interface where the symmetry is broken. 1 Second-harmonic generation sSHGd has thus become a well-established technique for the investigation of planar surfaces and interfaces. 2 Many applications, however, would call for an expansion of SHG to also address molecular adsorption and surface electronic and geometric structure on the nanoscale. Here, the problem of SHG becomes particularly intriguing: Although at the surface of the nanostructure the inversion symmetry is broken locally, depending on its dimension and macroscopic symmetry, emission can be highly restricted due to the destructive interference of the induced local surface second-harmonic sSHd polarizations. 3,4 In this paper we address the different contributions to the nonlinear source polarization and provide general directional and polarization selection rules for second-harmonic generation from partially asymmetric smmd nanostructures. Metal wire tips with a nanometer-sized apex represent a model geometry, with the mirror symmetry being broken along the tip axis but conserved in all other directions. This structure allows for the direct separation between local surface and nonlocal higher-order bulk contributions to the SH response—a

Journal ArticleDOI
TL;DR: In this article, the authors give geometric explanations and proofs of various mirror symmetry conjectures for T -invariant Calabi-Yau manifolds when instanton corrections are absent.
Abstract: We give geometric explanations and proofs of various mirror symmetry conjectures for T -invariant Calabi-Yau manifolds when instanton corrections are absent. This uses a fiberwise Fourier transformation together with a base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex structures and complexified symplectic structures, H’s, Yukawa couplings; (ii) sl (2)× sl (2)-actions; (iii) holomorphic and symplectic automorphisms and (iv) Aand B-connections, supersymmetric Aand B-cycles, correlation functions. We also study (ii) for T -invariant hyperkahler manifolds.

Posted Content
TL;DR: In this paper, the authors introduced aspects of the authors' work relating mirror symmetry and integral variations of Hodge structure, which can underly families of Calabi-Yau threefolds over the thrice-punctured sphere with b^3 = 4, or equivalently h^{2,1} = 1.
Abstract: This proceedings note introduces aspects of the authors' work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly families of Calabi-Yau threefolds over the thrice-punctured sphere with b^3 = 4, or equivalently h^{2,1} = 1, and the related issues of geometric realization of these variations. The presentation parallels that of the first author's talk at the BIRS workshop.

Journal ArticleDOI
R. Paul Horja1
TL;DR: Inspired by the homological mirror symmetry conjecture of Kontsevich [30], this paper constructed new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi-projective variety.
Abstract: Inspired by the homological mirror symmetry conjecture of Kontsevich [30], we construct new classes of automorphisms of the bounded derived category of coherent sheaves on a smooth quasi– projective variety. MSC (2000): 18E30; 14J32.

Journal ArticleDOI
TL;DR: In this paper, the authors discuss the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks, and check that physical predictions of those measured models exactly match the corresponding stacks.
Abstract: In this paper we will discuss gauged linear sigma model descriptions of toric stacks. Toric stacks have a simple description in terms of (symplectic, GIT) ${\bf C}^{\times}$ quotients of homogeneous coordinates, in exactly the same form as toric varieties. We describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks, and check that physical predictions of those gauged linear sigma models exactly match the corresponding stacks. We also check in examples that when a given toric stack has multiple presentations in a form accessible as a gauged linear sigma model, that the IR physics of those different presentations matches, so that the IR physics is presentation-independent, making it reasonable to associate CFT's to stacks, not just presentations of stacks. We discuss mirror symmetry for stacks, using Morrison-Plesser-Hori-Vafa techniques to compute mirrors explicitly, and also find a natural generalization of Batyrev's mirror conjecture. In the process of studying mirror symmetry, we find some new abstract CFT's, involving fields valued in roots of unity.

Journal ArticleDOI
TL;DR: In this paper, the authors show how to construct half-flat topological mirrors to Calabi-Yau manifolds with NS fluxes, which is the topological complement of previous differential-geometric mirror rules.
Abstract: Motivated by SU(3) structure compactifications, we show explicitly how to construct half-flat topological mirrors to Calabi-Yau manifolds with NS fluxes. Units of flux are exchanged with torsion factors in the cohomology of the mirror; this is the topological complement of previous differential-geometric mirror rules. The construction modifies explicit SYZ fibrations for compact Calabi-Yaus. The results are of independent interest for SU(3) compactifications. For example one can exhibit explicitly which massive forms should be used for Kaluza-Klein reduction, proving previous conjectures. Formality shows that these forms carry no topological information; this is also confirmed by infrared limits and old classification theorems.

Journal ArticleDOI
TL;DR: In this article, the Hitchin sigma model is extended with branes and a detailed study of the boundary conditions obeyed by the world sheet fields is provided, and it is shown that when branes are present, the classical Batalin-Vilkovisky cohomology contains an extra sector that is related non trivially to a novel cohomological associated with the branes as generalized complex submanifolds.
Abstract: Hitchin's generalized complex geometry has been shown to be relevant in compactifications of superstring theory with fluxes and is expected to lead to a deeper understanding of mirror symmetry. Gualtieri's notion of generalized complex submanifold seems to be a natural candidate for the description of branes in this context. Recently, we introduced a Batalin–Vilkovisky field theoretic realization of generalized complex geometry, the Hitchin sigma model, extending the well known Poisson sigma model. In this paper, exploiting Gualtieri's formalism, we incorporate branes into the model. A detailed study of the boundary conditions obeyed by the world sheet fields is provided. Finally, it is found that, when branes are present, the classical Batalin–Vilkovisky cohomology contains an extra sector that is related non trivially to a novel cohomology associated with the branes as generalized complex submanifolds.

Journal ArticleDOI
TL;DR: In this article, a sequence of Lagrangian submanifolds with boundary on a level set of the Landau-Ginzburg mirror of a smooth toric variety X and an ample line bundle O(1) was constructed.
Abstract: Given a smooth toric variety X and an ample line bundle O(1), we construct a sequence of Lagrangian submanifolds of (C^*)^n with boundary on a level set of the Landau-Ginzburg mirror of X The corresponding Floer homology groups form a graded algebra under the cup product which is canonically isomorphic to the homogeneous coordinate ring of X

Posted Content
TL;DR: The integral cohomology groups for all examples of Calabi-Yau 3-folds obtained from hypersurfaces in 4-dimensional Gorenstein toric Fano varieties were derived in this paper.
Abstract: In this paper, we compute the integral cohomology groups for all examples of Calabi-Yau 3-folds obtained from hypersurfaces in 4-dimensional Gorenstein toric Fano varieties. Among 473 800 776 families of Calabi-Yau 3-folds $X$ corresponding to 4-dimensional reflexive polytopes there exist exactly 32 families having non-trivial torsion in $H^*(X, \Z)$. We came to an interesting observation that the torsion subgroups in $H^2$ and $H^3$ are exchanged by the mirror symmetry involution, i.e. the torsion subgroup in the Picard group of $X$ is isomorphic to the Brauer group of the mirror $X^*$

Journal ArticleDOI
TL;DR: In this paper, the authors consider a special case of the Batyrev-Borisov construction for complete intersections in toric varieties and give a topological description of the Strominger-Yau-Zaslow fibrations on complete intersections.
Abstract: This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-Borisov construction for complete intersections in toric varieties. Given a pair of reflexive polytopes with nef decompositions dual to each other, and given polarizations on the corresponding toric varieties, we construct examples of toric degenerations and their dual intersection complexes, which are affine manifolds with singularities. We show these affine manifolds are related by the discrete Legendre transform, thus showing that the Batyrev-Borisov construction is a special case of our more general construction. The description of the dual intersection complexes in terms of the combinatoricsof the setup generalises work of Haase and Zharkov in the toric hypersurface case, and similar work of Ruan. In particular, this gives a topological description of the Strominger-Yau-Zaslow fibrations on complete intersections in toric varieties.

Posted Content
TL;DR: In this paper, the authors describe new autoequivalences of derived categories of coherent sheaves arising from what they call $\mathbb P^n$-objects of the category.
Abstract: We describe new autoequivalences of derived categories of coherent sheaves arising from what we call $\mathbb P^n$-objects of the category. Standard examples arise from holomorphic symplectic manifolds. Under mirror symmetry these autoequivalences should be mirror to Seidel's Dehn twists about lagrangian $\mathbb P^n$ submanifolds. We give various connections to spherical objects and spherical twists, and include a simple description of Atiyah and Kodaira-Spencer classes in an appendix.

Book ChapterDOI
TL;DR: In this article, the mirror symmetry conjecture for rank 1 Fano 3-folds of Picard rank 1 is investigated and the modularity conjecture for Fano 4-fold is discussed.
Abstract: Introduction In this paper we make precise, in the case of rank 1 Fano 3-folds, the following programme: Given a classification problem in algebraic geometry, use mirror duality to translate it into a problem in differential equations; solve this problem and translate the result back into geometry. The paper is based on the notes of the lecture series the author gave at the University of Cambridge in 2003. It expands the announcement, providing the background for and discussion of the modularity conjecture. We start with basic material on mirror symmetry for Fano varieties. The quantum D –module and the regularized quantum D –module are introduced in Section 1. We state the mirror symmetry conjecture for Fano varieties. We give more conjectures implying, or implied by, the mirror symmetry conjecture. We review the algebraic Mellin transform of Loeser and Sabbah and define hypergeometric D –modules on tori. In Section 2 we consider Fano 3-folds of Picard rank 1 and review Iskovskikh's classification into 17 algebraic deformation families. We apply the basic setup to Fano 3-folds to obtain the so called counting differential equations of type D 3. We introduce DN equations as generalizations of these, discuss their properties and take a brief look at their singularities. In Section 3, motivated by the Dolgachev-Nikulin-Pinkham picture of mirror symmetry for K 3 surfaces, we introduce ( N, d )-modular families; these are pencils of K 3 surfaces whose Picard-Fuchs equations are the counting D 3 equations of rank 1 Fano 3-folds.

Journal ArticleDOI
TL;DR: In this paper, an extended set of differential operators for local mirror symmetry was proposed, and a conjecture for intersection theory for such a set of operators was uncovered, along with operators on several examples of type X =KS through similar techniques.
Abstract: We propose an extended set of differential operators for local mirror symmetry. If X is Calabi-Yau such that dimH4(X,Z)=0, then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection theory for such X is uncovered. We also find operators on several examples of type X=KS through similar techniques. In addition, open string Picard-Fuchs systems are considered.

Book ChapterDOI
TL;DR: The mirror symmetry conjecture of Hausel-Rodriguez-Villegas as discussed by the authors is related to the mirror symmetry conjectures of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan.
Abstract: The paper surveys the mirror symmetry conjectures of Hausel-Thaddeus and Hausel-Rodriguez-Villegas concerning the equality of certain Hodge numbers of SL(n, ℂ) vs. PGL(n, ℂ) flat connections and character varieties for curves, respectively. Several new results and conjectures and their relations to works of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan are explained. These use the representation theory of finite groups of Lie-type via the arithmetic of character varieties and lead to an unexpected conjecture for a Hard Lefschetz theorem for their cohomology.

Journal ArticleDOI
TL;DR: In this paper, the super-Landau-Ginzburg mirror of the A-twisted topological sigma model on a twistor superspace was obtained, which is a Calabi-Yau supermanifold.

Journal ArticleDOI
TL;DR: In this article, the super Landau-Ginzburg mirrors of the weighted projective superspace WCP3|2 have been studied in the topological B-model.
Abstract: We study super Landau–Ginzburg mirrors of the weighted projective superspace WCP3|2 which is a Calabi–Yau supermanifold and appeared in hep-th/0312171 in the topological B-model. One of them is an elliptic fibration over the complex plane whose coordinate is given in terms of two bosonic and two fermionic variables as well as Kahler parameter of WCP3|2. The other is some patch of a degree-3 Calabi–Yau hypersurface in CP2 fibered by the complex plane whose coordinate depends on both the above four variables and Kahler parameter but its dependence behaves quite differently.

Journal ArticleDOI
TL;DR: In this paper, the authors studied mirror symmetry of supermanifolds constructed as fermionic extensions of compact toric varieties and showed that there is a relation between the super-Calabi-Yau conditions of the A-model and quasi-homogeneity of the B-model.
Abstract: We study mirror symmetry of supermanifolds constructed as fermionic extensions of compact toric varieties. We mainly discuss the case where the linear sigma A-model contains as many fermionic fields as there are U(1) factors in the gauge group. In the mirror super-Landau-Ginzburg B-model, focus is on the bosonic structure obtained after integrating out all the fermions. Our key observation is that there is a relation between the super-Calabi-Yau conditions of the A-model and quasi-homogeneity of the B-model, and that the degree of the associated superpotential in the B-model is given in terms of the determinant of the fermion charge matrix of the A-model.

Journal ArticleDOI
TL;DR: In this article, it was shown that morphisms between coisotropic A-branes can be equated with a fundamental representation of the non-commutatively deformed algebra of functions on the intersection.
Abstract: We study coisotropic A-branes in the sigma model on a four-torus by explicitly constructing examples. We find that morphisms between coisotropic branes can be equated with a fundamental representation of the noncommutatively deformed algebra of functions on the intersection. The noncommutativity parameter is expressed in terms of the bundles on the branes. We conjecture these findings hold in general. To check mirror symmetry, we verify that the dimensions of morphism spaces are equal to the corresponding dimensions of morphisms between mirror objects.

Journal ArticleDOI
TL;DR: In this article, the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces was derived as an effect of coordinate change of the virtual Gauss-Manin system.
Abstract: In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss–Manin system.

Journal ArticleDOI
TL;DR: In this paper, it was shown that morphisms between coisotropic A-branes can be equated with a fundamental representation of the non-commutatively deformed algebra of functions on the intersection.
Abstract: We study coisotropic A-branes in the sigma model on a four-torus by explicitly constructing examples. We find that morphisms between coisotropic branes can be equated with a fundamental representation of the noncommutatively deformed algebra of functions on the intersection. The noncommutativity parameter is expressed in terms of the bundles on the branes. We conjecture these findings hold in general. To check mirror symmetry, we verify that the dimensions of morphism spaces are equal to the corresponding dimensions of morphisms between mirror objects.