Mirror Transformation Research Articles

Holomorphic Line Bundles on Projective Toric Manifolds from Lagrangian Sections of their Mirrors by SYZ Transformations

The mirror of a projective toric manifold is given by a Landau–Ginzburg model (Y, W). We introduce a class of Lagrangian submanifolds in (Y, W) and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant Hermitian metrics on holomorphic line bundles over ⁠. Through this geometric correspondence, we also identify the mirrors of Hermitian–Einstein metrics, which are given by distinguished Lagrangian sections whose potentials satisfy certain Laplace-type equations.

Mirrorfolds with K3 fibrations

We study a class of non-geometric string vacua realized as completely soluble superconformal field theory (SCFT). These models are defined as `interpolating orbifolds' of K3 × S1 by the mirror transformation acting on the K3 fiber combined with the half-shift on the S1-base. They are variants of the T-folds, the interpolating orbifolds by T-duality transformations, and thus may be called `mirrorfolds'. Starting with arbitrary (compact or non-compact) Gepner models for the K3 fiber, we construct modular invariant partition functions of general mirrorfold models. In the case of compact K3 fiber the mirrorfolds only yield non-supersymmetric string vacua. They exhibit IR instability due to winding tachyon condensation which is similar to the Scherk-Schwarz type circle compactification. When the fiber SCFT is non-compact (say, the ALE space in the simplest case), on the other hand, both supersymmetric and non-supersymmetric vacua can be constructed. The non-compact non-supersymmetric mirrorfolds can get stabilised at the level of string perturbation theory. We also find that in the non-compact supersymmeric mirrorfolds D-branes are always non-BPS. These D-branes can get stabilized against both open- and closed-string marginal deformations.

Quantum [formula omitted]-modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D -modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D -module after a mirror transformation when the first Chern class c 1 ( M ) of the tangent bundle is nef. In this paper, even when c 1 ( M ) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D -module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].

Rapid Prototyping and Reverse Engineering Application for Orthopedic Surgery Planning

This paper describes rapid prototyping (RP) and reverse engineering (RE) application for orthopedic surgery planning to improve the efficiency and accuracy of the orthopedic surgery. Using the symmetrical characteristics of the human body, CAD data of undamaged bone of the injured area are generated from a mirror transformation of undamaged bone data for the uninjured area. The physical model before the injury is manufactured from Polyjet RP process. The surgical plan, including the selection of the proper implant, pre-forming of the implant and decision of fixation positions, etc., is determined by a physical simulation using the physical model. In order to examine the applicability and efficiency of the surgical planning technology, two case studies, such as a distal tibia comminuted fracture and an iliac wing fracture of pelvis, are carried out. From the results of the examination, it has been shown that the RP and RE can be applied to orthopedic surgical planning and can be an efficient surgical tool.

COORDINATE CHANGE OF GAUSS–MANIN SYSTEM AND GENERALIZED MIRROR TRANSFORMATION

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.

Quantum cohomology via [formula omitted]-modules

We propose a new point of view on quantum cohomology, motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is a D -module which “quantizes” a commutative algebra associated to the (uncompactified) space of rational curves. Under appropriate conditions, we show that the associated flat connection may be gauged to the flat connection underlying quantum cohomology. This method clarifies the role of the Birkhoff factorization in the “mirror transformation”, and it gives a new algorithm (requiring construction of a Groebner basis and solution of a system of o.d.e.) for computation of the quantum product.

Mirror apraxia affects the peripersonal mirror space. A combined lesion and cerebral activation study.

Mirror apraxia is a condition in which patients with lesions of the posterior parietal cortex have deficits in reaching to objects presented through a mirror. The aim of the present study was to investigate possible mechanisms underlying this disorder. First, we addressed the question of whether mirror apraxia is exhibited to the same extent in peripersonal and in body space. Four patients with lesions of the posterior parietal lobe on either side and with marked mirror apraxia were required to reach for objects that were presented to them through a mirror and located either in body space (i.e. on the body surface) or in peripersonal space (i.e. in the reaching distance). Whereas reaching for objects located in body space was flawless in all patients, the performance deteriorated when the same objects were transferred to the peripersonal space. Although the objects were located only a few centimetres above the body surface, the patients reached towards the virtual object in the mirror. Based on these results we suggest that mirror apraxia may originate from a dissociation between the representations of body schema and peripersonal space and that objects located on the body surface become integrated into the body schema. In the second part of the study, using positron emission tomography study (PET), we studied the cerebral activation pattern during reaching to objects presented through a mirror in the peripersonal space in healthy subjects. The results show that increased neural activity in the anterior part of the intraparietal sulcus and in the dorsal premotor cortex was bound to the transformation of the target position from the mirror space to the real space. In contrast, the activity related to object localization in the mirror occurred at the parieto-occipital junction. Both mirror and arm transformation involved the medial posterior part of the superior parietal lobule, putatively area V6a. The results demonstrate that acting through a mirror is processed in a number of cortical areas of the dorsal stream.

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let $ Y \subset X $ be a smooth very ample hypersurface such that $ -K_Y $ is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry.

Riemannian Geometry Over Different Normed Division Algebras

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number (in the appropriate cases) and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra $\mathbb{A}$. For example, special Riemannian $\mathbb{A}$-manifolds are oriented Riemannian, Calabi-Yau, hyperkähler and $G_2$-manifolds respectively. For vector bundles over such manifolds, we introduce (special) $\mathbb{A}$-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) $\frac{1}{2}\mathbb{A}$-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds. We also discuss geometric dualities from this viewpoint: Fourier transformations on $\mathbb{A}$-geometry for flat tori and a conjectural SYZ mirror transformation from (special) $\mathbb{A}$-geometry to (special) $\frac{1}{2} \mathbb{A}$-Lagrangian geometry on mirror special $\mathbb{A}$-manifolds.

Mirror transformations of Hamiltonian systems

We demonstrate, through various examples of Hamiltonian systems, that symplectic structures have been encoded into the Painlevé test. Each principal balance in the Painlevé test induces a mirror transformation that regularizes movable singularities. Moreover, for finite-dimensional Hamiltonian systems, the mirror transformations are canonical.

ON THE QUANTUM COHOMOLOGY RINGS OF GENERAL TYPE PROJECTIVE HYPERSURFACES AND GENERALIZED MIRROR TRANSFORMATION

In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with nonpositive first Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi–Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three-point Gromov–Witten invariants up to cubic rational curves. We also construct a projective space resolution of the moduli space of polynomial maps, which is in good correspondence with the terms that appear in the generalized mirror transformation.

VIRTUAL GROMOV–WITTEN INVARIANTS AND THE QUANTUM COHOMOLOGY RINGS OF GENERAL TYPE PROJECTIVE HYPERSURFACES

In this letter, we propose another characterization of the generalized mirror transformation on the quantum cohomology rings of general type projective hypersurfaces. This characteristic is useful for explicit determination of the form of the generalized mirror transformation. As an application, we rederive the generalized mirror transformation up to d=3 rational Gromov–Witten invariants obtained in our previous paper, and determine explicitly the generalized mirror transformation for the d=4,5 rational Gromov–Witten invariants in the case when the first Chern class of the hypersurface equals -H (i.e. k-N=1).

Mirror Maps in Chern–Simons Gauge Theory

We describe mirror symmetry inN=2 superconformal field theories in terms of a dynamical topology changing process of the principal fiber bundle associated with a topological membrane. We show that the topological symmetries of Calabi–Yau sigma-models can be obtained from discrete geometric transformations of compact Chern–Simons gauge theory coupled to charged matter fields. We demonstrate that the appearence of magnetic monopole-instantons, which interpolate between topologically inequivalent vacua of the gauge theory, implies that the discrete symmetry group of the worldsheet theory is realized kinematically in three dimensions as the magnetic flux symmetry group. From this we construct the mirror map and show that it corresponds to the interchange of topologically non-trivial matter field and gauge degrees of freedom. We also apply the mirror transformation to the mean field theory of the quantum Hall effect. We show that it maps the Jain hierarchy into a new hierarchy of states in which the lowest composite fermions have the same filling fractions.

A mirror pair of Calabi-Yau four-folds in Type II string theory

We give some simple examples of mirror Calabi-Yau four-folds in Type II string theory. These are realised as toroidal orbifolds. Motivated by the Strominger, Yau, Zaslow argument we explicitly give the mirror transformation which maps Type IIA/IIB on such a four-fold to Type IIA/IIB on the mirror. The mirrors are related by the inclusion/exclusion of discrete torsion. Implicit in the result is a confirmation of mirror symmetry to genus g in the string path integral. Finally, by considering the relationship between M-theory and Type IIA theory, we show how in M-theory on mirror Calabi-Yau four-folds, mirror symmetry exchanges the Coulomb branch with (one of the) Higgs branches of the theory. This result is relevant to duality in N = 2 supersymmetric gauge theories in three dimensions.

The Candelas-de la Ossa-Green-Parkes formula

In this note we discuss a recent proof of the formula for the worldsheet instanton prepotential predicted by Mirror Symmetry for the quintics in P 4. One of the key ingredients in the proof is the equivariant cohomology groups on the so-called linear sigma model moduli spaces. We introduce the notion of admissible data on the equivariant cohomology groups of the linear sigma model. An admissible data may be thought of as a sequence of equivariant classes satisfying certain algebraic conditions. It arises naturally from Kontsevich's stable map compactification of moduli spaces of maps from curves into projective manifolds. The structures of admissible data help reduce many counting problems to checking certain combinatorial structure of a compactification. The mirror transformation of Candelas et al turns out to be a transformation between two admissible data associated respectively to the linear and the non-linear sigma models. As an application, we prove the formula for the worldsheet instanton prepotential in terms of hypergeometric series. At the end we also interprete an infinite dimensional transformation group, called the mirror group, acting on admissible data, as a certain duality group of the linear sigma model.

The mirror transform of type I vacua in six dimensions

We study certain compactifications of the type I string on K3. The three topologically distinct choices of gauge bundles for the type I theory are shown to be equivalent to type IIB orientifolds with different choices of background anti-symmetric tensor field flux. Using a mirror transformation we relate these models to orientifolds with fixed seven planes, and without any anti-symmetric tensor field flux. This map allows us to relate these type I vacua to particular six-dimensional F-theory and heterotic string compactifications.

Mirror symmetry is T-duality

It is argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space Y. The mirror transformation is equivalent to T-duality on the 3-cycles. The geometry of moduli space is addressed in a general framework. Several examples are discussed.

Topological matter, mirror symmetry and non-critical (super)strings

We study the realization of the (super) conformal topological symmetry in two-dimensional field theories. The mirror automorphism of the topological algebra is represented as a reflection in the space of fields. As a consequence, a double BRST structure for topological matter theories is found. It is shown that the implementation of the topological symmetry in non-critical (super) string theories depends on the matter content of the two realizations connected by the mirror transformation.

A procedure for determining rigid body transformation parameters

For many biomechanical applications it is necessary to determine the parameters which describe the transformation of a rigid body from one reference frame to another. These parameters are a scaling factor, an attitude matrix, and a translation vector. The paper presents a new procedure for the determination of these parameters incorporating the work of Arun et al. [IEEE Trans. Pattern Anal. Machine Intell, 9, 698–700 (1987)] but expanding their analysis to allow for the determination of a scale factor, the scalar weighting of the least-squares problem, and the problem of obtaining the incorrect determinant when determining the attitude matrix. The procedure, which requires the coordinates of three or more noncollinear points, is based around the singular value decomposition, and provides a least-squares estimate of the rigid body transformation parameters. Examples are presented of the use of this procedure for determining the attitude of a rigid body, and for osteometric scaling. When used for osteometric scaling mirror transformations are possible, therefore a right-hand specimen can be scaled to the left-hand side of another specimen.

Axial-vector duality as a mirror symmetry

We study N = 2 supersymmetric SU (2) U (1) and SL(2, R) U (1) gauged Wess-Zumino-Witten models. It is shown that the vector gauged model is transformed to the axial gauged model by a mirror transformation. Therefore the vector gauged model and the axial gauged model are equivalent as N = (2,2) superconformal field theories. In the SL(2, R) U (1) model, it is known that axial-vector duality relates a background with a singularity to one without a singularity. Implications of the equivalence of these two models to space-time singularities are discussed.