Related papers: Gamma conjecture and tropical geometry
We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $\mathbb{Q}$ in degree one when the manifold has at worst symple…
Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau--Ginzburg models dual to…
We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly…
We study the mod $2$ cohomology of real Calabi-Yau threefolds given by real structures which preserve the torus fibrations constructed by Gross. We extend the results of Casta\~no-Bernard-Matessi and Arguz-Prince to the case of real…
We propose and give strong evidence for a duality relating Type II theories on Calabi-Yau spaces and heterotic strings on $K3 \times T^2$, both of which have $N=2$ spacetime supersymmetry. Entries in the dictionary relating the dual…
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…
This work establishes a subtle connection between mirror symmetry for Calabi-Yau threefolds and that of curves of higher genus. The linking structure is what we call a perverse curve. We show how to obtain such from Calabi-Yau threefolds in…
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…
We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear sigma-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kahler sector of the…
We study the worldsheet CFTs of type II strings on compact $G_2$ orbifolds obtained as quotients of a product of a Calabi-Yau threefold and a circle. For such models, we argue that the Calabi-Yau mirror map implies a mirror map for the…
The four-form field strength in F-theory compactifications on Calabi-Yau fourfolds takes its value in the middle cohomology group $H^4$. The middle cohomology is decomposed into a vertical, a horizontal and a remaining component, all three…
It is shown that in string theory mirror duality is a gauge symmetry (a Weyl transformation) in the moduli space of $N=2$ backgrounds on group manifolds, and we conjecture on the possible generalization to other backgrounds, such as…
In this paper, we study the relation between the zeta function of a Calabi-Yau hypersurface and the zeta function of its mirror. Two types of arithmetic relations are discovered. This motivates us to formulate two general arithmetic mirror…
For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg…
In this paper, we compute the integral cohomology groups for all examples of Calabi-Yau 3-folds obtained from hypersurfaces in 4-dimensional Gorenstein toric Fano varieties. Among 473 800 776 families of Calabi-Yau 3-folds $X$ corresponding…
Given a Tyurin degeneration of a Calabi-Yau complete intersection in a toric variety, we prove gluing formulas relating the generalized functional invariants, periods, and $I$-functions of the mirror Calabi-Yau family and those of the two…
A Calabi-Yau threefold is called of type K if it admits an \'etale Galois covering by the product of a K3 surface and an elliptic curve. In our previous paper, based on Oguiso-Sakurai's fundamental work, we provide the full classification…
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…
The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries…
We construct \Lambda-adic de Rham and crystalline analogues of Hida's ordinary \Lambda-adic etale cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \Q_p, we prove appropriate…