Related papers: Periods of mirrors and multiple zeta values
We introduce an algebraic formulation of cyclic sum formulas for multiple zeta values and for multiple zeta-star values. We also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta-star values by…
We interpret the $q$-refined theta function $\vartheta_1$ of a log Calabi-Yau surface $(\mathbb{P},E)$ as a natural $q$-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on the…
In this paper, we will study the connections between the mirror symmetry of Calabi-Yau threefolds and Deligne's conjecture on the special values of the $L$-functions of critical motives. Using the theory of mirror symmetry, we will develop…
In this paper, we introduce and study new classes of Ap\'ery-type series involving the multiple $t$-harmonic sums by combining the methods of iterated integral and Fourier--Legendre series expansions, where the multiple $t$-harmonic sums…
In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier in their recent paper in Ann. Sci. \'{E}cole Normale Sup. that $R_\zeta=c_\zeta^2$, where $R_\zeta$ is their map on $K_3$ defined using the cyclic quantum…
Hirose, Murahara, and Saito proved that some $t$-adic symmetric multiple zeta values, for indices in which $1$ and $3$ appear alternately in succession, can be expressed as polynomials in Riemann zeta values, and conjectured similar…
We introduce a symbolic representation of $r$-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these…
The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real…
We give a representation, in terms of iterated Mellin-Barnes integrals, of periods on multi-moduli Calabi-Yau manifolds arising in superstring theory. Using this representation and the theory of multidimensional residues, we present a…
The aim of this article is to report on recent progress in understanding mirror symmetry for some non-complete intersection Calabi-Yau threefolds. We first construct four new smooth non-complete intersection Calabi-Yau threefolds with…
We define polynomials of one variable t whose values at t=0 and 1 are the multiple zeta values and the multiple zeta-star values, respectively. We give an application to the two-one conjecture of Ohno-Zudilin, and also prove the cyclic sum…
For a del Pezzo surface of degree $\geq 3$, we compute the oscillatory integral for its mirror Landau-Ginzburg model in the sense of Gross-Hacking-Keel [Mark Gross, Paul Hacking, and Sean Keel, "Mirror symmetry for log Calabi-Yau surfaces…
Schneps [J. Lie Theory 16 (2006), 19--37] has found surprising links between Ihara brackets and even period polynomials. These results can be recovered and generalized by considering some identities relating Ihara brackets and classical Lie…
Classical multiple zeta values can be viewed as iterated integrals of the differentials $\frac{dt}{t}, \frac{dt}{1-t}$ from $0$ to $1$. In this paper, we reprove Brown's theorem: For $a_i, b_i, c_{ij}\in \mathbb{Z}$, the iterated integral…
The role in string theory of manifolds of complex dimension $D_{crit} + 2(Q-1)$ and positive first Chern class is described. In order to be useful for string theory, the first Chern class of these spaces has to satisfy a certain relation.…
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.…
We consider multi-polytopes to describe non-Fano toric varieties and their associated anticanonical Calabi-Yau hypersurfaces. From the periods of the mirror manifold the $\widehat{\Gamma}$-conjecture is shown to hold for examples of…
We prove that every multiple zeta value is a $\mathbb{Z}$-linear combination of $\zeta(k_1,\dots, k_r)$ where $k_i\geq 2$. Our proof also yields an explicit algorithm for such an expansion. The key ingredient is to introduce modified…
We study the depth filtration on multiple zeta values, the motivic Galois group of mixed Tate motives over $\mathbb{Z}$ and the Grothendieck-Teichm\"uller group, and its relation to modular forms. Using period polynomials for cusp forms for…
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…