Terms Of Modular Forms Research Articles

Minimal left–right symmetric model with A4 modular symmetry

In this paper, we have realized the left–right symmetric model (LRSM) with modular symmetry. Most of the previous works on LRSM have been done considering discrete flavor symmetry, but this work has been carried out using [Formula: see text](3) modular group which is isomorphic to nonabelian discrete symmetry group [Formula: see text]. The advantage of using modular symmetry is the nonrequirement for the use of extra particles called “flavons”. In this model, the Yukawa couplings are expressed in terms of modular forms [Formula: see text]. In this work, we have studied minimal LRSM for both type-I and type-II dominances. Here, we have calculated the values for the Yukawa couplings and then plotted it against the sum of the neutrino masses. The results obtained are well within the experimental limits for the desired values of sum of neutrino masses. We have also briefly analyzed the effects of the implications of modular symmetry on neutrinoless double beta decay with the new physics contributions and the correlation of lepton flavor violation and lightest neutrino mass within the framework of modular symmetric LRSM.

Quark mass hierarchies and CP violation in A4 × A4 × A4 modular symmetric flavor models

We study A4× A4× A4 modular symmetric flavor models to realize quark mass hierarchies and mixing angles without fine-tuning. Mass matrices are written in terms of modular forms. At modular fixed points τ = i∞ and ω, A4 is broken to Z3 residual symmetry. When the modulus τ is deviated from the fixed points, modular forms show hierarchies depending on their residual charges. Thus, we obtain hierarchical structures in mass matrices. Since we begin with A4× A4× A4, the residual symmetry is Z3× Z3× Z3 which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix |VCKM| without fine-tuning. Furthermore, CP violation is studied. We present necessary conditions for CP violation caused by the value of τ. We also show possibilities to realize observed values of the Jarlskog invariant JCP, quark mass ratios and CKM matrix |VCKM| simultaneously, if (10) adjustments in coefficients of Yukawa couplings are allowed or moduli values are non-universal.

Sl(2)ˆ Decomposition of denominator formulae of some BKM Lie superalgebras - II

The square-root of Siegel modular forms of CHL Z_N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N=1,2,3,4. We study the decomposition of these Siegel modular forms in terms of characters of two sub-algebras: one is a $\widehat{sl(2)}$ and the second is a Borcherds extension of the $\widehat{sl(2)}$. This is a continuation of our previous work where we studied the case of Siegel modular forms appearing in the context of Umbral moonshine. This situation is more intricate and provides us with a new example (for N=5) that did not appear in that case. We restrict our analysis to the first N terms in the expansion as a first attempt at deconstructing the Siegel modular forms and unravelling the structure of potentially new Lie algebras that occur for N=5,6.

Quark hierarchical structures in modular symmetric flavor models at level 6

We study modular symmetric quark flavor models without fine-tuning. Mass matrices are written in terms of modular forms, and modular forms in the vicinity of the modular fixed points become hierarchical depending on their residual charges. Thus modular symmetric flavor models in the vicinity of the modular fixed points have a possibility to describe mass hierarchies without fine-tuning. Since describing quark hierarchies without fine-tuning requires ${Z}_{n}$ residual symmetry with $n\ensuremath{\ge}6$, we focus on ${\mathrm{\ensuremath{\Gamma}}}_{6}$ modular symmetry in the vicinity of the cusp $\ensuremath{\tau}=i\ensuremath{\infty}$ where ${Z}_{6}$ residual symmetry remains. We use only modular forms belonging to singlet representations of ${\mathrm{\ensuremath{\Gamma}}}_{6}$ to make our analysis simple. Consequently, viable quark flavor models are obtained without fine-tuning.

Double cover of modular S4 for flavour model building

We develop the formalism of the finite modular group Γ4′≡S4′, a double cover of the modular permutation group Γ4≃S4, for theories of flavour. The integer weight k>0 of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight (k=1) modular forms in terms of two Jacobi theta constants, denoted as ε(τ) and θ(τ), τ being the modulus. We show that these forms furnish a 3D representation of S4′ not present for S4. Having derived the S4′ multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to k=10. These are expressed as polynomials in ε and θ, bypassing the need to search for non-linear constraints. We further show that within S4′ there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

Supersymmetric phases of 4d mathcal{N} = 4 SYM at large N

We find a family of complex saddle-points at large N of the matrix model for the superconformal index of SU(N ) mathcal{N} = 4 super Yang-Mills theory on S3× S1 with one chemical potential τ . The saddle-point configurations are labelled by points (m, n) on the lattice Λτ = ℤτ + ℤ with gcd(m, n) = 1. The eigenvalues at a given saddle are uniformly distributed along a string winding (m, n) times along the (A, B) cycles of the torus ℂ/Λτ . The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and the related Bloch formula allows us to calculate the action at the saddle-points in terms of real-analytic Eisenstein series. The actions of (0, 1) and (1, 0) agree with that of pure AdS5 and the supersymmetric AdS5 black hole, respectively. The black hole saddle dominates the canonical ensemble when τ is close to the origin, and there are new saddles that dominate when τ approaches rational points. The extension of the action in terms of modular forms leads to a simple treatment of the Cardy-like limit τ → 0.

Elliptic Curves in Hyper-Kähler Varieties

Abstract We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic four-fold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3,780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kähler varieties with fixed $j$-invariant in terms of Gromov–Witten invariants. In $K3^{[2]}$-type this leads to explicit formulas of these counts in terms of modular forms.

Reflective modular forms and applications

The reflective modular forms of orthogonal type are fundamental automorphic objects generalizing the classical Dedekind eta-function. This article describes two methods for constructing such modular forms in terms of Jacobi forms: automorphic products and Jacobi lifting. In particular, it is proved that the first non-zero Fourier–Jacobi coefficient of the Borcherds modular form (the generating function of the so-called Fake Monster Lie Algebra) in any of the 23 one-dimensional cusps coincides with the Kac–Weyl denominator function of the affine algebra of the root system of the corresponding Niemeier lattice. This article gives a new simple construction of the automorphic discriminant of the moduli space of Enriques surfaces as a Jacobi lifting of the product of eight theta-functions and considers three towers of reflective modular forms. One of them, the tower of , gives a solution to a problem of Yoshikawa (2009) concerning the construction of Lorentzian Kac–Moody algebras from the automorphic discriminants connected with del Pezzo surfaces and analytic torsions of Calabi–Yau manifolds. The article also formulates some conditions on sublattices, making it possible to produce families of `daughter' reflective forms from a fixed modular form. As a result, nearly 100 such functions are constructed here. Bibliography: 77 titles.

Rankin–Cohen brackets and Serre derivatives as Poincaré series

We give expressions for the Serre derivatives of Eisenstein and Poincar\'e series as well as their Rankin-Cohen brackets with arbitrary modular forms in terms of the Poincar\'e averaging construction, and derive several identities for the Ramanujan tau function as applications.

Holomorphic anomaly and quantum mechanics

We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory.

On the number of representations of integers by quadratic forms with congruence conditions

The generating functions of the number of representations of integers by quadratic forms under congruence conditions on the variables will be characterized in terms of modular forms. According to this characterization we present several formulas for the representation numbers as examples.

Generalized Donaldson–Thomas invariants on the local projective plane

We show that the generating series of generalized Donaldson–Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type series for indefinite lattices. In particular, it absolutely converges to give a holomorphic function on the upper half plane.

Topological string amplitudes for the local $\frac{1}{2}$K3 surface

We study topological string amplitudes for the local half K3 surface. We develop a method of computing higher-genus amplitudes along the lines of the direct integration formalism, making full use of the Seiberg-Witten curve expressed in terms of modular forms and E_8-invariant Jacobi forms. The Seiberg-Witten curve was constructed previously for the low-energy effective theory of the non-critical E-string theory in R^4 x T^2. We clarify how the amplitudes are written as polynomials in a finite number of generators expressed in terms of the Seiberg-Witten curve. We determine the coefficients of the polynomials by solving the holomorphic anomaly equation and the gap condition, and construct the amplitudes explicitly up to genus three. The results encompass topological string amplitudes for all local del Pezzo surfaces.

THE KATZ–KLEMM–VAFA CONJECTURE FOR SURFACES

We prove the KKV conjecture expressing Gromov–Witten invariants of $K3$ surfaces in terms of modular forms. Our results apply in every genus and for every curve class. The proof uses the Gromov–Witten/Pairs correspondence for $K3$-fibered hypersurfaces of dimension 3 to reduce the KKV conjecture to statements about stable pairs on (thickenings of) $K3$ surfaces. Using degeneration arguments and new multiple cover results for stable pairs, we reduce the KKV conjecture further to the known primitive cases. Our results yield a new proof of the full Yau–Zaslow formula, establish new Gromov–Witten multiple cover formulas, and express the fiberwise Gromov–Witten partition functions of $K3$-fibered 3-folds in terms of explicit modular forms.

Modular forms, de Rham cohomology and congruences

In this paper we show that Atkin and Swinnerton- Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of ``differentials of the second kind''. As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve $X^N+Y^N=Z^N$. We show that the Galois representation associated to this space is given by a Gr\ossencharacter of the cyclotomic field $\Q(\zeta_N)$. Moreover, for $N=5$ the space does not admit a ``$p$-adic Hecke eigenbasis'' for (non-ordinary) primes $p\equiv 2, 3 \pmod{; ; 5}; ; $, which provides a counterexample to Atkin and Swinnerton-Dyer's original speculation.

Bounds on sup-norms of half-integral weight modular forms

Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup-norm of a half-integral weight cusp form is bounded in terms of the level: we prove that $\|y^{\kappa/2}\tilde{f}\|_\infty \ll_{\varep

A theory of theta functions to the quintic base

Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobiʼs quartic theta function identity and counterparts of Jacobiʼs Principles of Duplication, Dimidiation and Change of Sign Formulas. The resulting library of quintic transformation formulas is used to describe series multisections for modular forms in terms of simple matrix operations. These efforts culminate in a formal technique for deducing congruences modulo powers of five for a variety of combinatorial generating functions, including the partition function. Further analysis of the quintic theta functions is undertaken by exploring their modular properties and their connection to Eisenstein series. The resulting relations lead to a coupled system of differential equations for the quintic theta functions.

The Ω Deformed B-model for Rigid N = 2 Theories

We give an interpretation of the Ω deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four-dimensional rigid N = 2 theories explicitly in general Ω-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N = 2 supersymmetric theories. The rigid N = 2 field theories we focus on are the conformal rank one N = 2 Seiberg–Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N = 2 theories arising from compactifications on local Calabi–Yau manifolds, we consider the theory of local $${\mathbb{P}^2}$$ . We calculate motivic Donaldson–Thomas invariants for this geometry and make predictions for generalized Gromov–Witten invariants at the orbifold point.

Eisenstein Series and Modular Differential Equations

AbstractThe purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions, and equivariant forms.

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi–Yau equations

We give the exact expressions of the partial susceptibilities χ(3)d and χ(4)d for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi–Yau ODEs, and more specifically, 3F2([1/3, 2/3, 3/2], [1, 1]; z) and 4F3([1/2, 1/2, 1/2, 1/2], [1, 1, 1]; z) hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for χ(3)d and χ(4)d. We also give new results for χ(5)d. We see, in particular, the emergence of a remarkable order-6 operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the n-fold integrals of the Ising model are not only ‘derived from geometry’ (globally nilpotent), but actually correspond to ‘special geometry’ (homomorphic to their formal adjoint). This raises the question of seeing if these ‘special geometry’ Ising operators are ‘special’ ones, reducing, in fact systematically, to (selected, k-balanced, ...) q + 1Fq hypergeometric functions, or correspond to the more general solutions of Calabi–Yau equations.