Related papers: Some integrality properties in local mirror symmet…
We propose a monodromy invariant pairing $K_{hol}(X) \otimes H_3(X^\vee,\ZZ) \to \IQ$ for a mirror pair of Calabi-Yau manifolds, $(X,X^\vee)$. This pairing is utilized implicitly in the previous calculations of the prepotentials for…
Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$…
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…
Landau has characterized the integrality of certain ratios of factorials. Delaygue has characterized the integrality of the Taylor coefficients of certain mirror maps constructed from series involving those ratios. Using the…
We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold…
We propose an extended set of differential operators for local mirror symmetry. If $X$ is Calabi-Yau such that $\dim H_4(X,\Z)=0$, then we show that our operators fully describe mirror symmetry. In the process, a conjecture for intersection…
We show that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic p>0, exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds…
We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate…
Lecture notes from 1993 Park City lectures and 1994 Trento lectures. The focus of these lectures is on giving a mathematical description of the A-model and B-model correlation functions on a Calabi--Yau manifold, and a precise mathematical…
The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners.…
We discuss how the Gross-Siebert reconstruction theorem applies to the local mirror symmetry of Chiang, Klemm, Yau and Zaslow. The reconstruction theorem associates to certain combinatorial data a degeneration of (log) Calabi-Yau varieties.…
We apply mirror symmetry to the super Calabi-Yau manifold CP^{(n|n+1)} and show that the mirror can be recast in a form which depends only on the superdimension and which is reminiscent of a generalized conifold. We discuss its geometrical…
In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror…
We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit…
We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for affine-K\"{a}hler manifolds and for elliptic curves. Their geometric properties, especially the link with special lagrangian fibrations and the…
We give a class of examples of $A$-hypergeometric systems that display integrality of mirror maps. Specifically, these systems have solutions $F(\lambda_1,\dots,\lambda_N) = 1$ and $\log\lambda^l + G(\lambda_1,\dots,\lambda_N)$ (for certain…
Mirror symmetry, a phenomenon in superstring theory, has recently been used to give tentative calculations of several numbers in algebraic geometry. In this paper, the numbers of lines and conics on various hypersurfaces which satisfy…
We construct a wide class of non-geometric compactifications of type II superstring theories preserving N=1 space-time supersymmetry in four dimensions, starting from Calabi-Yau compactifications at Gepner points. Particular examples of…
We study open string mirror symmetry for one-parameter Calabi-Yau hypersurfaces in weighted projective space. We identify mirror pairs of D-brane configurations, derive the corresponding inhomogeneous Picard-Fuchs equations, and solve for…
We study an example of complete intersection Calabi-Yau threefold due to Libgober and Teitelbaum arXiv:alg-geom/9301001, and verify mirror symmetry at a cohomological level. Direct computations allow us to propose an analogue to the…