Related papers: The Gamma conjecture for $G$-functions
Hodge-theoretic mirror symmetry for a Calabi-Yau mirror pair says that the variation of Hodge structure arising from quantum cohomology of a Calabi-Yau manifold and that arising from deformation of complex structures on the dual Calabi-Yau…
Let $\sum\_{n=0}^\infty a\_n z^n\in \overline{\mathbb Q}[[z]]$ be a $G$-function, and, for any $n\ge0$, let $\delta\_n\ge 1$ denote the least integer such that $\delta\_n a\_0, \delta\_n a\_1, ..., \delta\_n a\_n$ are all algebraic…
De Giorgi conjectured in 1979 that if a sequence of parabolic functionals Gamma converges to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. This paper studies the…
Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual…
This paper contains a preliminary study of the monodromy of certain fourth order differential equations, that were called of Calabi-Yau type in math.NT/0402386. Some of these equations can be interpreted as the Picard-Fuchs equations of a…
By applying mirror symmetry to D-branes in a Calabi-Yau geometry we shed light on a $G_2$ flop in M-theory relevant for large $N$ dualities in ${\cal N}=1$ supersymmetric gauge theories. Furthermore, we derive superpotential for M-theory on…
Given two Calabi--Yau threefolds which are believed to constitute a mirror pair, there are very precise predictions about the enumerative geometry of rational curves on one of the manifolds which can be made by performing calculations on…
We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric function field Gamma-function are explained…
The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors…
In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a…
Assuming the standard framework of mirror symmetry, a conjecture is formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y should act by families of Fourier-Mukai transforms over the complex moduli space of the mirror…
Localization methods have produced explicit expressions for the sphere partition functions of (2,2) superconformal field theories. The mirror symmetry conjecture predicts an IR duality between pairs of Abelian gauged linear sigma models, a…
We study mirror symmetric pairs of Calabi--Yau manifolds over finite fields. In particular we compute the number of rational points of the manifolds as a function of the complex structure parameters. The data of the number of rational…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
We relate a coherent sheaf supported on a holomorphic curve with its mirror Langrangian submanifold in local mirror symmetry through a tropical curve by interpreting their central charges using the combinatorial information of the tropical…
We analyse the geometric properties of the high derivatives of the distance function from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its covariant derivatives of…
De Giorgi conjectured in 1979 that if a sequence of functionals converges in the sense of Gamma-convergence to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. It is…
Let $X$ be a Fano variety, and $D\subset X$ be an snc anticanonical divisor. We study relative mirror symmetry for the log Calabi--Yau pair $(X,D)$. (1) We prove a relative mirror theorem for snc pairs without assuming the divisors are nef.…
Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.
This paper primarily deals with the study of G-derivations associated with Lie-Yamaguti algebras. Taking G as an automorphism group, the concept of G-derivations, which is a derivation under both the bilinear and trilinear operations is…