Mirror Manifolds Research Articles

Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory

We analyze the moduli spaces of Calabi-Yau three-folds and their associated conformally invariant nonlinear σ-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kähler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological Calabi-Yau space—the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kähler and complex structure moduli spaces of a Calabi-Yau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kähler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized by the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory.

Mirror manifolds in higher dimension

We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We isolate certain simplifying features present only in dimension three, and supply alternative methods that do not rely on these special characteristics and hence can be generalized to other dimensions. Although the moduli spaces for Calabi-Yaud-folds are not “special Kähler manifolds” whend>3, they still have a restricted geometry, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi-Yau manifolds. Our mirror-aided calculations agree with those Chern class calculations in the limited number of cases for which the latter can be carried out with current mathematical tools. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra.

Generalized Calabi-Yau manifolds and the mirror of a rigid manifold

The Z ̃ manifold is a Calabi-Yau manifold with b 21 = 0. At first sight it seems to provide a counter-example to the mirror hypothesis since its mirror would have b 11 = 0 and hence could not be Kähler. However, by identifying the Z ̃ manifold with the Gepner model 1 9 we are able to ascribe a geometrical interpretation to the mirror, Z ̃ , as a certain seven-dimensional manifold. The mirror manifold Z ̃ is a representative of a class of generalized Calabi-Yau manifolds, which we describe, that can be realized as manifolds of dimension five and seven. Despite their dimension these generalized Calabi-Yau manifolds correspond to superconformal theories with c = 9 and so are perfectly good for compactifying the heterotic string to the four dimensions of space-time. As a check of mirror symmetry we compute the structure of the space of complex structures of the mirror Z ̃ and check that this reproduces the known results for the Yukawa couplings and metric appropriate to the Kähler class parameters on the Z ̃ orbifold together with their instanton corrections. In addition to reproducing known results we can calculate the periods of the manifold to arbitrary order in the blowing up parameters. This provides a means of calculating the Yukawa couplings and metric as functions also to arbitrary order in the blowing up parameters, which is difficult to do by traditional methods.

Where are the mirror manifolds?

The recent classification of Landau-Ginzburg potentials and their abelian symmetries focuses attention on a number of models with large positive Euler number for which no mirror partner is known. All of these models are related to Calabi-Yau manifolds in weighted P 4, with a characteristic structure of the defining polynomials. A closer look at these potentials suggests a series of non-linear transformations, which relate the model to configurations for which a construction of the mirror is known, though only at certain points in moduli space. A special case of these transformations generalizes the Z 2 orbifold representation of the D invariant, implying a hidden symmetry in tensor products of minimal models.

Multiple mirror manifolds and topology change in string theory

We use mirror symmetry to establish the first concrete arena of spacetime topology change in string theory. In particular, we establish that the quantum theories based on certain nonlinear sigma models with topologically distinct target spaces can be smoothly connected even though classically a physical singularity would be encountered. We accomplish this by rephrasing the description of these nonlinear sigma models in terms of their mirror manifold partners - a description in which the full quantum theory can be described exactly using lowest order geometrical methods. We establish that, for the known class of mirror manifolds, the moduli space of the corresponding conformal field theory requires not just two but numerous topologically distinct Calabi-Yau manifolds for its geometric interpretation. A single family of continously connected conformal theories thereby probes a host of topologically distinct geometrical spaces giving rise to multiple mirror manifolds.

A generalized construction of mirror manifolds

We generalize the known method for explicit construction of mirror pairs of (2,2)-superconformal field theories, using the formalism of Landau-Ginzburg orbifolds. Geometrically, these theories ae realized as Calabi-Yau hypersurfaces in weighted projective spaces. This generalization makes it possible to construct the mirror partners of many manifolds for which the mirror was not previously known.

Considerations of one-modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps

We consider Calabi-Yau compactifications with one Kähler modulus. Following the method of Candelas et al. we use the mirror hypotheses to solve the quantum theory exactly in dependence of this modulus by performing the calculation for the corresponding complex structure deformation on the mirror manifold. Here the information is accessible by techniques of classical geometry. It is encoded in the Picard-Fuchs differential equation which has to be supplemented by requirements on the global properties of its solutions.

Instantons and mirror K3 surfaces

The instanton moduli space of a real 4-dimensional torus is an 8-dimensional Calabi-Yau manifold. Associated to this Calabi-Yau manifold are two (singular) K3 surfaces, one a quotient, the other a submanifold of the moduli space; both carry a natural Calabi-Yau metric. They are curiously related in much the same way as special examples of complex 3-dimensional mirror manifolds; however, in our case the “mirror” is present in the form of instanton moduli.

Moduli space of Calabi-Yau manifolds

We present an accessible account of the local geometry of the parameter space of Calabi-Yau manifolds. It is shown that the parameter space decomposes, at least locally, into a product with the space of parameters of the complex structure as one factor and a complex extension of the parameter space of the Kähler class as the other. It is also shown that each of these spaces is itself a Kähler manifold and is moreover a Kähler manifold of restricted type. There is a remarkable symmetry in the intrinsic structures of the two parameter spaces and the relevance of this to the conjectured existence of mirror manifolds is discussed. The two parameter spaces behave differently with respect to modular transformations and it is argued that the role of quantum corrections is to restore the symmetry between the two types of parameters so as to enforce modular invariance.

Geometry of mirror manifolds

We analyze the mirror manifold hypothesis in one and three dimensions using the simplest available representations of the N = 2 superconformal algebra. The symmetries of these tensor models can be divided out to give an explicit representation of the mirror, and we give a simple group theoretical algorithm for determining which symmetries should be used. We show that the mirror of a superconformal field theory does not always have a geometrical interpretation, but when it does, deformations of complex structure of one manifold are reflected in deformations of the Kähler form of the mirror manifold, and we show how the large radius limit of a manifold corresponds to a large complex structure limit in the mirror manifold. The mirror of the Tian-Yau three generation model is constructed both as a conformal field theory and as an algebraic variety with Euler number six. The Hodge numbers of this manifold are fixed, but the intersection numbers are highly ambiguous, presumably reflecting a rich structure of multicritical points in the moduli space of the field theory.

Calabi-Yau supermoduli space, field strength duality and mirror manifolds

Calabi-Yau moduli space have a super extension due to their relation to type II superstring compactifications. Supermoduli can be defined as N = 2 target space vector-supermultiplets containing both the moduli and their vector field bosonic superpartners. A change of basis in the cohomology vector space corresponds to (modular) duality transformations of target space vector field strengths. Target space two-forms will take values in the cohomology spaces of Calabi-Yau manifolds, and are therefore related to the Kähler class and complex structure deformations, which may be relevant for the description of mirror symmetries of (2, 2) superconformal field theories.

Construction and couplings of mirror manifolds

We present an analysis of the conjectured existence of Calabi-Yau “mirror manifolds” for the case where the starting manifold is Y 4,5. We construct mirror pairs with equal but opposite values for the Euler characteristic and the Hodge numbers h 2,1 and h 1,1 interchanged. In one particular example we show that the couplings of (1,1)-forms equal the couplings of (2,1)-forms in the mirror manifold, provided that a suitable limit is taken of the complex structure which corresponds to the large-radius limit appropriate for the mirror manifold. This leads to a determination, via deformation theory, of corrections to the topologically determined couplings of the (1,1)-forms.

From special Lagrangian to Hermitian–Yang–Mills via Fourier–Mukai transform

We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and Bcycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section. We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of supersymmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa’s mirror conjecture with bundles in this special case.