Mirror symmetry

About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.

Generalized periods and mirror symmetry in dimensions n>3

Abstract: The predictions of the Mirror Symmetry are extended in dimensions n>3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov-Witten invariants of such variety can be expressed in terms of certain invariants of a new generalization of variation of Hodge structures attached to the dual variety. To formulate the general principles of Mirror Symmetry in arbitrary dimension it is necessary to introduce the ``extended moduli space of complex structures'' M. An analog M\to H*(X,C)[n] of the classical period map is described and is shown to be a local isomorphism. The invariants of the generalized variations of Hodge structures are introduced. It is proven that their generating function satisfies the system of WDVV-equations exactly as in the case of Gromov-Witten invariants. The basic technical tool utilized is the Deformation theory.

Conifold singularities, resumming instantons and non-perturbative mirror symmetry

Abstract: We determine the instanton corrected hypermultiplet moduli space in type IIB compactifications near a Calabi-Yau conifold point where the size of a two-cycle shrinks to zero. We show that D1-instantons resolve the conifold singularity caused by worldsheet instantons. Furthermore, by resumming the instanton series, we reproduce exactly the results obtained by Ooguri and Vafa on the type IIA side, where membrane instantons correct the hypermultiplet moduli space. Our calculations therefore establish that mirror symmetry holds non-perturbatively in the string coupling.

Mirror symmetry, Langlands duality, and commuting elements of Lie groups

Abstract: . By normalizing a component of the space of commuting pairs of elements ina reductive Lie group G, and the corresponding space for the Langlands dual group,we construct pairs of hyperk¨ahler orbifolds which satisfy the conditions to be mirrorpartners in the sense of Strominger-Yau-Zaslow. The same holds true for commutingquadruples in a compact Lie group. The Hodge numbers of the mirror partners, ormore precisely their orbifold E-polynomials, are shown to agree, as predicted by mirrorsymmetry. These polynomials are explicitly calculated when G is a quotient of SL(n). Mirror symmetry made its first appearance in 1990 as an equivalence between two linearsigma-models in superstring theory [10, 21]. The targets were Calabi-Yau 3-folds, so mirrorsymmetry predicted that these should come in pairs, M and Mˆ , satisfying h p,q (M) =H p,3−q (Mˆ).Although many examples were known, the physics did not immediately provide anygeneral construction of a mathematical nature for the mirror. Since then, however, twomathematical constructions have emerged: that of Batyrev [1] and Batyrev-Borisov [2],generalizing the original idea of Greene-Plesser [21], and that of Strominger-Yau-Zaslow [46]with which this paper is concerned.Ofthe two, Batyrev’s construction has theadvantageofbeing precise, andmoreamenableto explicit calculations. One can prove, for example, that the Hodge numbers of the Batyrevmirror satisfy the desired relationship. On the other hand, it is deeply rooted in toricgeometry. This has led skeptics to suggest that mirror symmetry is an intrinsically toricphenomenon, despite work [3, 40] extending Batyrev’s point of view some ways beyond thetoric setting.The construction proposed by Strominger-Yau-Zaslow in 1996 has quite a different flavor.It is directly inspired by a physical duality, the so-called T-duality between sigma-modelswhose targets are dual tori. Remarkably, although it is supposed to transform one projectivevariety into another, the construction is not algebraic, or even K¨ahler, in nature. Rather, itis symplectic: one must find a foliation of M by special Lagrangian tori, and replace eachtorus with its dual.This bold idea has already led to some interesting work on the existence of families ofLagrangiantoriin Calabi-Yau3-folds [22, 23,41], which isessentially aproblem in symplectictopology. But it is not yet sufficiently advanced that the mirror can be constructed in anyprecise sense, nor any of its invariants computed beyond the Euler characteristic. It is noteven known how to construct families of tori which are special Lagrangian (as opposed tojust Lagrangian). And the further questions of what complex structure to place on the dualfamily, and how to deal with singular fibers, remain mysterious.

Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation

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.

Pfaffian Calabi-Yau threefolds and mirror symmetry

Abstract: The aim of this article is to report on recent progress in un- derstanding mirror symmetry for some non-complete intersection Calabi- Yau threefolds. We first construct four new smooth non-complete in- tersection Calabi-Yau threefolds with h 1,1 = 1, whose existence was previously conjectured by C. van Enckevort and D. van Straten in (19). We then compute the period integrals of candidate mirror families of F. Tonoli's degree 13 Calabi-Yau threefold and three of the new Calabi- Yau threefolds. The Picard-Fuchs equations coincide with the expected Calabi-Yau equations listed in (18, 19). Some of the mirror families turn out to have two maximally unipotent monodromy points.