Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings by Jan E. Gerken, Axel Kleinschmidt, Oliver Schlotterer

“…In the low-energy expansion of the heterotic-string amplitude, the integrals over torus punctures are systematically evaluated in terms of modular graph forms, certain non-holomorphic modular forms. For a specific torus integral, the modular graph forms in the low-energy expansion are related to the elliptic multiple zeta values from the analogous open-string integrations over cylinder boundaries. …”
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Rademacher expansions and the spectrum of 2d CFT by Alday, LF, Bae, J-B

“…A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and c > 1. …”

Modular binary octahedral symmetry for flavor structure of Standard Model by Gui-Jun Ding, Xiang-Gan Liu, Jun-Nan Lu, Ming-Hua Weng

“…The vector-valued modular forms in all irreducible representations of this group are constructed. …”
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Exact half-BPS black hole entropies in CHL models from Rademacher series by Richard Nally

“…We begin by developing a Rademacher-like expansion for the Fourier coefficients of the partition functions for these theories, which are modular forms for congruence subgroups. We then describe a possible macroscopic interpretation of these results, emphasizing the role of twisted boundary conditions.…”
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All-order differential equations for one-loop closed-string integrals and modular graph forms by Jan E. Gerken, Axel Kleinschmidt, Oliver Schlotterer

“…The low-energy expansion of such torus integrals introduces infinite families of non-holomorphic modular forms known as modular graph forms. Our results generate homogeneous first- and second-order differential equations for arbitrary such modular graph forms and can be viewed as a step towards all-order low-energy expansions of closed-string integrals.…”
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ε-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves by Xuhang Jiang, Xing Wang, Li Lin Yang, Jingbang Zhao

“…The letters are combinations of modular forms on the corresponding elliptic curves and algebraic functions arising from the sub-sectors. …”
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Curves on K3 surfaces in divisibility 2 by Younghan Bae, Tim-Henrik Buelles

“…We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. …”
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Permutations of massive vacua by Antoine Bourget, Jan Troost

“…Our examples give rise to interesting field extensions of spaces of modular forms.…”
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On Eisenstein series in M2k(Γ0(N)) and their applications by Aygin, Zafer Selcuk

“…At last we give essential results to derive similar results for modular forms in a more general setting.…”
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$$A_4$$ A 4 modular flavour model of quark mass hierarchies close to the fixed point $$\tau = \omega $$ τ = ω by S. T. Petcov, M. Tanimoto

“…The model involves modular forms of level 3 and weights 6, 4 and 2, and contains eight constants, only two of which, $$g_u$$ g u and $$g_d$$ g d , can be a source of CP violation in addition to the VEV of the modulus, $$\tau = \omega + \epsilon $$ τ = ω + ϵ , $$(\epsilon )^* \ne \epsilon $$ ( ϵ ) ∗ ≠ ϵ , $$|\epsilon |\ll 1$$ | ϵ | ≪ 1 . …”
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Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields by Lauder, A, Darmon, H, Rotger, V

“…This article examines the Fourier expansions of certain non-classical <em>p</em>-adic modular forms of weight one: the eponymous <em>generalised eigenforms</em> of the title, so called because they lie in a generalised eigenspace for the Hecke operators. …”

Modular symmetry by orbifolding magnetized T 2 × T 2: realization of double cover of Γ N by Shota Kikuchi, Tatsuo Kobayashi, Hajime Otsuka, Shintaro Takada, Hikaru Uchida

“…Each of the wavefunctions on T 1 2 × T 2 2 $$ {T}_1^2\times {T}_2^2 $$ and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup Γ(N), N being 2 times the least common multiple of M 1 and M 2. …”
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On Certain Generalizations of Rational and Irrational Equivariant Functions by Isra Al-Shbeil, Afis Saliu, Abbas Kareem Wanas, Adriana Cătaş

“…In this sense, we establish a criterion in order to determine the rationality of equivariant functions derived from ratios of modular functions of low weight. Modular forms play an important role in number theory and many areas of mathematics and physics.…”
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Positivity of discrete information for CHL black holes by Suresh Govindarajan, Sutapa Samanta, P. Shanmugapriya, Amitabh Virmani

“…This leads to a specific prediction for the signs of certain linear combinations of Fourier coefficients of Siegel modular forms. We explicitly test these predictions for low charges. …”
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Duality and modular symmetry in the quantum Hall effect from Lifshitz holography by Brian P. Dolan

“…The temperature dependence of the infra-red conductivities is then linked to modular forms via gradient flow and the resulting flow diagrams show remarkable agreement with existing experimental data on the temperature flow of both integral and fractional quantum Hall conductivities.…”
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Modular and duality properties of surface operators in N = 2 ⋆ $$ \mathcal{N}={2}^{\star } $$ gauge theories by S. K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, R. R. John, A. Lerda

“…Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. …”
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Quark-lepton mass relations from modular flavor symmetry by Mu-Chun Chen, Stephen F. King, Omar Medina, José W. F. Valle

“…These relations are determined by both the Clebsch-Gordan coefficients of the specific finite modular group and the expansion coefficients of its modular forms, thus offering potential probes of modular invariant models. …”
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Hecke relations in rational conformal field theory by Jeffrey A. Harvey, Yuxiao Wu

“…Abstract We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). …”
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Lepton flavor mixing and CP violation in the minimal type-(I+II) seesaw model with a modular A4 symmetry by Xin Wang

“…We consider the most economical case where the right-handed neutrino and the Higgs triplets in this model are assigned into the trivial one-dimensional irreducible representation of the modular group A4, and all the modular forms are with the lowest weights they can take. …”
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Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series by Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi

“…Our formalism is based on a special case of a coaction on large classes of periods that is applied in particular to elliptic polylogarithms and iterated integrals of modular forms. We illustrate how to use our formalism to derive relations among elliptic polylogarithms, in complete analogy with the non-elliptic case. …”
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