Showing papers on "Mirror symmetry published in 2000"

D-Branes And Mirror Symmetry

Abstract: We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in nonlinear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around � �

Mirror symmetry, D-branes and counting holomorphic discs

Abstract: We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions.

D-branes on the quintic

Abstract: We study D-branes on the quintic CY by combining results from several directions: general results on holomorphic curves and vector bundles, stringy geometry and mirror symmetry, and the boundary states in Gepner models recently constructed by Recknagel and Schomerus, to begin sketching a picture of D-branes in the stringy regime. We also make first steps towards computing superpotentials on the D-brane world-volumes.

Gauge theory and calibrated geometry, I

Abstract: The geometry of submanifolds is intimately related to the theory of functions and vector bundles It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems A typical example of this relation is that the Picard group of line bundles on an algebraic manifold is isomorphic to the group of divisors, which is generated by holomorphic hypersurfaces modulo linear equivalence A similar correspondence can be made between the K-group of sheaves and the Chow ring of holomorphic cycles There are two more very recent examples of such a relation The mirror symmetry in string theory has revealed a deeper phenomenon involving special Lagrangian cycles (cf [SYZ]) On the other hand, C Taubes has shown that the Seiberg-Witten invariant coincides with the Gromov-Witten invariant on any symplectic 4-manifolds In this paper, we will show another natural interaction between Yang-Mills connections, which are critical points of a Yang-Mills action associated to a vector bundle, and minimal submanifolds, which have been studied extensively for years in classical differential geometry and the calculus of variations

Large Complex Structure Limits of K3 Surfaces

Abstract: Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we make a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kahler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibrations associated with the large complex structure limit, namely an n-sphere, and that the metric on this Sn is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the limit discriminant locus of the fibrations. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkahler rotation trick, we reduce the problem to that of studying Kahler degenerations of elliptic K3 surfaces, with the Kahler class approaching the wall of the Kahler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric: if the elliptic fibres are of area ∊ > 0, then the error is O(e−C/∊) for some constant C > 0. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small ∊, this is a sufficiently good approximation that the above conjecture is then an easy consequence.

Fractional branes and boundary states in orbifold theories

Abstract: We study the D-brane spectrum of = 2 string orbifold theories using the boundary state formalism. The construction is carried out for orbifolds with isolated singularities, non-isolated singularities and orbifolds with discrete torsion. Our results agree with the corresponding K-theoretic predictions when they are available and generalize them when they are not. This suggests that the classification of boundary states provides a sort of ``quantum K-theory'' just as chiral rings in CFT provide ``quantum'' generalizations of cohomology. We discuss the identification of these states with D-branes wrapping holomorphic cycles in the large radius limit of the CFT moduli space. The example 3/3 is worked out in full detail using local mirror symmetry techniques. We find a precise correspondence between fractional branes at the orbifold point and configurations of D-branes described by vector bundles on the exceptional 2 cycle.

Large Complex Structure Limits of K3 Surfaces

Abstract: Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kahler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibration associated with the large complex structure limit, namely an n-sphere, and that the metric on this S^n is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the discriminant locus of the fibration. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkahler rotation trick, we reduce the problem to that of studying Kahler degenerations of elliptic K3 surfaces, with the Kahler class approaching the wall of the Kahler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric -- if the elliptic fibres are of area $\epsilon >0$, then the error is $O(e^{-C/\epsilon})$ for some constant $C>0$. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small $\epsilon$, this is a sufficiently good approximation that the above conjecture is then an easy consequence.

Mirror symmetry for open strings

Abstract: The authors discuss the generation of superpotentials in d = 4, N = 1 supersymmetric field theories arising from type IIA D6-branes wrapped on supersymmetric three-cycles of a Calabi-Yau threefold. In general, nontrivial superpotentials arise from sums over disc instantons. They then find several examples of special Lagrangian three-cycles with nontrivial topology which are mirror to obstructed rational curves, conclusively demonstrating the existence of such instanton effects. In addition, they present explicit examples of disc instantons ending on the relevant three-cycles. Finally, they give a preliminary construction of a mirror map for the open string moduli, in a large-radius limit of the type IIA compactification.

The Pfaffian Calabi-Yau, its Mirror, and their link to the Grassmannian G(2,7)

Abstract: The rank 4 locus of a general skew-symmetric 7 × 7 matrix gives the Pfaffian variety in P20 which is not defined as a complete intersection. Intersecting this with a general P6 gives a Calabi–Yau manifold. An orbifold construction seems to give the 1-parameter mirror-family of this. However, corresponding to two points in the 1-parameter family of complex structures, both with maximally unipotent monodromy, are two different mirror-maps: one corresponding to the general Pfaffian section, the other to a general intersection of G(2,7) ⊂ P20 with a P13. Apparently, the Pfaffian and G(2,7) sections constitute different parts of the A-model (Kahler structure related) moduli space, and, thus, represent different parts of the same conformal field theory moduli space.

D-branes on Stringy Calabi-Yau Manifolds

Abstract: We argue that D-branes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau-Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states with coherent sheaves, and to calculate their K-theory classes in the large volume limit, without needing to invoke mirror symmetry. We check this identification against the mirror symmetry results for the example of the Calabi-Yau hypersurface in $\WP^{1,1,2,2,2}$.

Mirror symmetry by O3-planes

Abstract: We construct the three dimensional mirror theory of SO(2k) and SO(2k+1) gauge groups by using O3-planes. An essential ingredient in constructing the mirror is the splitting of a physical brane (NS-brane or D5-brane) on O3-planes. In particular, matching the dimensions of moduli spaces of mirror pair (for example, the SO(2k+1) and its mirror) there is a D3-brane creation or annihilation accompanying the splitting. This novel dynamical process gives a nontrivial prediction for strongly coupled field theories, which will be very interesting to check by Seiberg-Witten curves. Furthermore, applying the same idea, we revisit the mirror theory of SP(k) gauge group and find new mirrors which differ from previously known results. Our new result for SP(k) gives another example to a previously observed fact, which shows that different theories can be mirror to the same theory. We also discussed the phenomena such as ``hidden FI-parameters'' when the number of flavors and the rank of the gauge group satisfy certain relations, ``incomplete higgsing'' for the mirror of SO(2k+1) and the ``hidden global symmetry''. After discussing the mirror for a single SP or SO gauge group, we extend the study to a product of two gauge groups in two different models, namely the elliptic and the non-elliptic models.

Gauge theory and calibrated geometry, I

Abstract: The geometry of submanifolds is intimately related to the theory of functions and vector bundles. It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems. A typical example of this relation is that the Picard group of line bundles on an algebraic manifold is isomorphic to the group of divisors, which is generated by holomorphic hypersurfaces modulo linear equivalence. A similar correspondence can be made between the K-group of sheaves and the Chow ring of holomorphic cycles. There are two more very recent examples of such a relation. The mirror symmetry in string theory has revealed a deeper phenomenon involving special Lagrangian cycles (cf. [SYZ]). On the other hand, C. Taubes has shown that the Seiberg-Witten invariant coincides with the Gromov-Witten invariant on any symplectic 4-manifolds. In this paper, we will show another natural interaction between Yang-Mills connections, which are critical points of a Yang-Mills action associated to a vector bundle, and minimal submanifolds, which have been studied extensively for years in classical differential geometry and the calculus of variations.

Mirror symmetry and toric geometry in three-dimensional gauge theories

Abstract: We study three dimensional gauge theories with = 2 supersymmetry. We show that the Coulomb branches of such theories may be rendered compact by the dynamical generation of Chern-Simons terms and present a new class of mirror symmetric theories in which both Coulomb and Higgs branches have a natural description in terms of toric geometry.

Calabi-Yau manifolds over finite fields. 1.

Abstract: We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.

Local mirror symmetry and type IIA monodromy of Calabi-Yau manifolds

Abstract: We propose a monodromy invariant pairing Khol(X) H3(X _ ;Z) ! Q for a mirror pair of Calabi-Yau manifolds, (X; X _ ). This pairing is utilized implicitly in the previous calculations of the prepotentials for Gromov-Witten invariants. After identifying the pairing explicitly we interpret some hypergeometric series from the viewpoint of homo- logical mirror symmetry due to Kontsevich. Also we consider the local mirror symmetry limit to del Pezzo surfaces in Calabi-Yau 3-folds.

Special Lagrangian m-folds in C m with symmetries

Abstract: This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct, G-invariant special Lagrangian cones on T^{m-1} in C^m. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models will be needed to understand Mirror Symmetry and the SYZ conjecture.

Fourier-Mukai Transform and Mirror Symmetry for D-Branes on Elliptic Calabi-Yau

Abstract: Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau three-fold $X$ is seen to have an expected adiabatic form for its induced cohomology operation only when an appropriately twisted operation resp twisted charge is defined Some differences with the case of $K3$ as well as connections with the spectral cover construction for bundles on $X$ are pointed out In the context of mirror symmetry Kontsevich's association of line bundle twists (resp a certain 'diagonal' operation) with monodromies (esp the conifold monodromy) is made explicit and checked for two example models Interpreting this association as a relation between FM transforms and monodromies, we express the fibrewise FM transform through known monodromies The operation of fibrewise duality as well as the notion of a certain index relevant to the computation of the moduli space of the bundle is transported to the sLag side Finally the moduli space for D4-branes and its behaviour under the FM transform is considered with an application to the spectral cover

Picard–Fuchs Uniformization and Modularity¶of the Mirror Map

Abstract: Arithmetic properties of mirror symmetry (type IIA-IIB string duality) are studied. We give criteria for the mirror map q-series of certain families of Calabi–Yau manifolds to be automorphic functions. For families of elliptic curves and lattice polarized K3 surfaces with surjective period mappings, global Torelli theorems allow one to present these criteria in terms of the ramification behavior of natural algebraic invariants – the functional and generalized functional invariants respectively. In particular, when applied to one parameter families of rank 19 lattice polarized K3 surfaces, our criterion demystifies the Mirror-Moonshine phenomenon of Lian and Yau and highlights its non-monstrous nature. The lack of global Torelli theorems and presence of instanton corrections makes Calabi–Yau threefold families more complicated. Via the constraints of special geometry, the Picard–Fuchs equations for one parameter families of Calabi–Yau threefolds imply a differential equation criterion for automorphicity of the mirror map in terms of the Yukawa coupling. In the absence of instanton corrections, the projective periods map to a twisted cubic space curve. A hierarchy of “algebraic” instanton corrections correlated with the differential Galois group of the Picard–Fuchs equation is proposed.

The Arithmetic mirror symmetry and Calabi-Yau manifolds

Abstract: We extend our variant of mirror symmetry for K3 surfaces [GN3] and clarify its relation with mirror symmetry for Calabi–Yau manifolds. We introduce two classes (for the models A and B) of Calabi–Yau manifolds fibrated by K3 surfaces with some special Picard lattices. These two classes are related with automorphic forms on IV type domains which we studied in our papers [GN1]–[GN6]. Conjecturally these automorphic forms take part in the quantum intersection pairing for model A, Yukawa coupling for model B and mirror symmetry between these two classes of Calabi–Yau manifolds. Recently there were several papers by physicists where it was shown on some examples. We propose a problem of classification of introduced Calabi–Yau manifolds. Our papers [GN1]–[GN6] and [N3]–[N14] give hope that this is possible. They describe possible Picard or transcendental lattices of general K3 fibers of the Calabi–Yau manifolds.

Log mirror symmetry and local mirror symmetry

Abstract: We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dxdy/xy over 2-chains whose boundaries lie on B_\phi where {B_\phi} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for projective plane in a Calabi-Yau 3-fold by Chiang-Klemm-Yau-Zaslow.

Quantum tori, mirror symmetry and deformation theory

Abstract: We suggest to compactify the universal covering of the moduli space of complex structures by noncommutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of Abelian varieties, this approach gives quantum tori as a noncommutative boundary of the moduli space. Relations to mirror symmetry, modular forms and deformation theory are discussed.

Lagrangian torus fibration and mirror symmetry of Calabi-Yau hypersurface in toric variety

Abstract: In this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres)

Mirror symmetry and actions of braid groups on derived categories

Abstract: Talk given at Harvard, January 1999, published in the Proceedings of the Harvard Winter School on mirror symmetry, vector bundles and lagrangian cycles, 1999, International Press. Surveys the joint work [ST, KS] with Paul Seidel and Mikhail Khovanov.