Mirror maps, modular relations and hypergeometric series II

For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 / t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's T -function in terms of hypergeometric functions.

We show that one can express Frobenius transformation on middle-dimensional p-adic cohomology of Calabi–Yau threefold in terms of mirror map and instanton numbers. We express the mirror map in terms of Frobenius transformation on p-adic cohomology. We discuss a p-adic interpretation of the conjecture about integrality of Gopakumar–Vafa invariants.

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to , hypergeometric functions, and show that one just reduces to the previous cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

We study the modular symmetry in heterotic string theory on Calabi-Yau threefolds. In particular, we examine whether moduli-dependent holomorphic Yukawa couplings are described by modular forms in the context of heterotic string theory with standard embedding. We find that SL(2, ℤ) modular symmetry emerges in asymptotic regions of the Calabi-Yau moduli space. The instanton-corrected holomorphic Yukawa couplings are then given by modular forms under SL(2, ℤ) or its congruence subgroups such as Γ0(3) and Γ0(4). In addition to the modular symmetry, it turns out that another coupling selection rule controls the structure of holomorphic Yukawa couplings. Furthermore, the coexistence of both the positive and negative modular weights for matter fields leads to a hierarchical structure of matter field Kähler metric. Thus, these holomorphic modular forms and the matter field Kähler metric play an important role in realizing a hierarchical structure of physical Yukawa couplings.

Perturbation theory and zero modes in O(N) lattice σ models

We provide formulas for traces of p t h p^{th} Hecke operators in level 1 in terms of values of finite field 2 F 1 _2F_1 -hypergeometric functions, extending previous work of the author to all odd primes p p instead of only those p ≡ 1 ( mod 12 ) p \equiv 1 \pmod {12} . We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes p > 3 p > 3 .

The l=0 partial wave projected Coulomb off-shell T matrix Tc,l=0 in momentum representation is obtained in closed form. Problems existing in the literature concerning the half- and on-shell behavior of Tc and Tc,l are discussed and clarified by means of explicit formulas. The remaining derivations in this paper are based on Tc,l=0. We consider the class of N-term separable potentials where the form factors are rational functions of p2 (in momentum representation). We prove that the l=0 T matrix corresponding to the Coulomb potential plus any such so-called rational separable potential has a very simple form, namely, it can be written in terms of rational functions and the (simple) hypergeometric function with parameters (1, iγ; 1+iγ), where γ is the well-known Coulomb parameter. Explicit analytic formulas are derived for a number of simple members of the class, the Yamaguchi potential being one of them. In this particular case the expressions of Zachary and of Bajzer are reproduced who used a method based on the O4 symmetry.

Sharp Landen transformation inequalities for hypergeometric functions, with applications

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

There are connections, already seen by Gauss himself, between his analytic work and the theory of quadratic forms. Some fragments from the early 1800s show that Gauss was familiar with the rudiments of a theory which eventually found its completion in Klein’s and Fricke’s work on modular forms. Both F. Klein and R. Fricke were deeply involved in the publication of Gauss’s work and had the opportunity to study Gauss’s fragments closely.

I will illustrate Ogg’s lectures by giving a few computations of modular relations, in an honest nineteenth-century spirit. A modular relation may, I suppose, be practically any pretty relation between modular functions; but classically the greatest interest has always been in the relation between j = j(z) and JN = j(Nz) for natural numbers N, and that is the case I will keep to.

Mirror maps are power series which occur in Mirror Symmetry as the inverse for composition of \(q(z)=\exp (f(z)/g(z))\), called local q-coordinates, where f and g are particular solutions of the Picard–Fuchs differential equations associated with certain one-parameter families of Calabi–Yau varieties. In several cases, it has been observed that such power series have integral Taylor coefficients at the origin. In the case of hypergeometric equations, we discuss p-adic tools and techniques that enable one to prove a criterion for the integrality of the coefficients of mirror maps. This is a joint work with T. Rivoal and J. Roques. This note is an extended abstract of the talk given by the author in January 2017 at the conference “Hypergeometric motives and Calabi–Yau differential equations” in Creswick, Australia.

Mirror symmetry for two-parameter models — II

A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 1990, 264–273]. It also includes many new evaluations of q-integrals.

If R(ω,q) denotes Dyson’s partition rank generating function, due to work of Bringmann and Ono, it is known that for roots of unity ω≠1, R(ω,q) is the “holomorphic part” of a harmonic weak Maass form. Dating back to Ramanujan, it is also known that \(\widehat{R}(\omega,q):=R(\omega,q^{-1})\) is given by Eichler integrals and modular forms. In analogy to these results, more recently Monks and Ono have shown that modular forms arise in a natural way from G(ω,q), the generating function for ranks of partitions into distinct parts. Moreover, Monks and Ono pose the following problem: determine whether the function \(\widehat{G}(\omega,q):=G(\omega,q^{-1})\) appears naturally in the theory of modular forms. Here we answer this question of Monks and Ono, and show that \(\widehat{G}(\omega,q)\) , when combined with \(\widehat{G}(\omega^{-1},q)\) and a twisted third-order mock theta of Ramanujan, form a weight 1 modular form. We provide a more general result on the modularity of certain expressions involving basic hypergeometric series and then show that our result on \(\widehat {G}(\omega,q)\) may be deduced from this as a special case.