Mirror symmetry

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

Symmetry breaking and Landau quantization in topological crystalline insulators

Abstract: In the recently discovered topological crystalline insulators SnTe and ${\mathrm{Pb}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}$(Te, Se), crystal symmetry and electronic topology intertwine to create topological surface states with many interesting features including Lifshitz transition, Van-Hove singularity, and fermion mass generation. These surface states are protected by mirror symmetry with respect to the (110) plane. In this work we present a comprehensive study of the effects of different mirror-symmetry-breaking perturbations on the (001) surface band structure. Pristine (001) surface states have four branches of Dirac fermions at low energy. We show that ferroelectric-type structural distortion generates a mass and gaps out some or all of these Dirac points, while strain shifts Dirac points in the Brillouin zone. An in-plane magnetic field leaves the surface state gapless, but introduces asymmetry between Dirac points. Finally, an out-of-plane magnetic field leads to discrete Landau levels. We show that the Landau level spectrum has an unusual pattern of degeneracy and interesting features due to the unique underlying band structure. This suggests that Landau level spectroscopy can detect and distinguish between different mechanisms of symmetry breaking in topological crystalline insulators.

Mirror symmetry via logarithmic degeneration data, II

Abstract: This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing.

N=1 Mirror Symmetry and Open/Closed String Duality

Abstract: We show that the exact N=1 superpotential of a class of 4d string compactifications is computed by the closed topological string compactified to two dimensions. A relation to the open topological string is used to define a special geometry for N=1 mirror symmetry. Flat coordinates, an N=1 mirror map for chiral multiplets and the exact instanton corrected superpotential are obtained from the periods of a system of differential equations. The result points to a new class of open/closed string dualities which map individual string world-sheets with boundary to ones without. It predicts an mathematically unexpected coincidence of the closed string Gromov-Witten invariants of one Calabi-Yau geometry with the open string invariants of the dual Calabi-Yau.

Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds

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.

Open string Gromov-Witten invariants: Calculations and a mirror 'theorem'

Abstract: We propose localization techniques for computing Gromov-WitteninvariantsofmapsfromRiemannsurfaceswith boundariesintoaCalabi-Yau, with the boundaries mapped to a Lagrangian submanifold. Thecomputations can be expressed in terms of Gromov-Witten invariantsof one-pointed maps. In genus zero, an equivariant version of the mir-ror theorem allows us to write down a hypergeometric series, whichtogether with a mirror map allows one to compute the invariants to allorders, similar to the closed string model or the physics approach viamirror symmetry. In the noncompact example where the Calabi-Yau isK P 2 ,our results agree with physics predictions at genus zero obtainedusing mirror symmetry for open strings. At higher genera, our resultssatisfy strong integrality checks conjectured from physics. 1 Introduction 1.1 The Physics Mirror symmetry is famous for being able to predict Gromov-Witten invari-ants of Calabi-Yau manifolds. The basic conjecture is that there is a dualitybetween string theories on mirror Calabi-Yau manifolds. As a consequence,the topological field theory defined from the A-twist of one Calabi-Yau man-ifold is equal to the topological B-twist of the mirror. Both twists can beperformed on Calabi-Yau target manifolds. From a practical point of view,in order to obtain enumerative predictions, one needs to know the theoryon the B-model (in this case, defined through classical period integrals) aswell as an identification of the parameter spaces for both theories – the“mirror map.” To extract integer-valued invariants, one needs an all-genus“multiple-cover” formula. The technology for finding mirror manifolds [3]1