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KP hierarchy for Hurwitz-type cohomological field theories Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-05-04 Reinier Kramer
We generalise a result of Kazarian regarding Kadomtsev–Petviashvili integrability for single Hodge integrals to general cohomological field theories related to Hurwitz-type counting problems or hypergeometric tau‑functions. The proof uses recent results on the relations between hypergeometric tau-functions and topological recursion, as well as the DOSS correspondence between topological recursion and
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Equivariant derived equivalence and rational points on K3 surfaces Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-05-04 Brendan Hassett, Yuri Tschinkel
We study arithmetic properties of derived equivalent K3 surfaces over the field of Laurent power series, using the equivariant geometry of K3 surfaces with cyclic groups actions.
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On quasi-tame Looijenga pairs Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-05-04 Andrea Brini, Yannik Schüler
We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact $\log$ Gromov–Witten invariants of Looijenga pairs to other curve counting invariants of Gromov–Witten/Gopakumar–Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the $\log$
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Completing the $c_2$ completion conjecture for $p=2$ Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-05-04 Simone Hu, Karen Yeats
The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain
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Resurgent Stokes data for Painlevé equations and two-dimensional quantum (super) gravity Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-05-04 Salvatore Baldino, Ricardo Schiappa, Maximilian Schwick, Roberto Vega
Resurgent-transseries solutions to Painlevé equations may be recursively constructed out of these nonlinear differential-equations—but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear
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On arithmetic Dijkgraaf–Witten theory Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-02-23 Hikaru Hirano, Junhyeong Kim, Masanori Morishita
We present basic constructions and properties in arithmetic Chern–Simons theory with finite gauge group along the line of topological quantum field theory. For a finite set $S$ of finite primes of a number field $k$, we construct arithmetic analogues of the Chern–Simons $1$-cocycle, the prequantization bundle for a surface and the Chern–Simons functional for a $3$-manifold. We then construct arithmetic
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On generalized stuffle relations between cell-zeta values Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-02-23 Nikita Markarian
We introduce a family of linear relations between cell-zeta values that have a form similar to product map relations and jointly with them imply stuffle relations between multiple zeta values.
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On the fundamental group of open Richardson varieties Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-02-23 Changzheng Li, Frank Sottile, Chi Zhang
We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these $\log$ Calabi–Yau varieties underlie a Landau–Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is $\log$ Calabi–Yau as it isomorphic to the complement
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Graph complexes and Feynman rules Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-02-23 Marko Berghoff, Dirk Kreimer
We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on the interplay between graph homology, Hopf-algebraic structures on Feynman graphs and the analytic structure of their associated integrals. Furthermore, we discuss the appearance of cubical complexes where the differential is formed by reducing internal edges and by putting edge-propagators on the
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Fourier expansions of vector-valued automorphic functions with non-unitary twists Commun. Number Theory Phys. (IF 1.9) Pub Date : 2023-02-23 Ksenia Fedosova, Anke Pohl, Julie Rowlett
We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed
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Fibers over infinity of Landau–Ginzburg models Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-21 Ivan Cheltsov, Victor Przyjalkowski
We conjecture that the number of components of the fiber over infinity of Landau–Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for $\log$ Calabi–Yau compactifications of toric Landau–Ginzburg models for smooth Fano threefolds, complete intersections in projective spaces, and some toric varieties.
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Orthosymplectic Satake equivalence Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-21 Alexander Braverman, Michael Finkelberg, Roman Travkin
This is a companion paper of [BFGT]. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $\mathrm{SO}(N-1,\mathbb{C} [\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $\mathrm{SO}_N$. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh
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Elliptic threefolds with high Mordell–Weil rank Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-21 Antonella Grassi, Timo Weigand
We present the first examples of smooth elliptic Calabi–Yau threefolds with Mordell–Weil rank 10, the highest currently known value. They are given by the Schoen threefolds introduced by Namikawa; there are six isolated fibers of Kodaira Type IV. We explicitly compute the Shioda homomorphism and the induced height pairing. Compactification of F‑theory on these threefolds gives an effective theory in
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Flux vacua: a voluminous recount Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-21 Miranda C. N. Cheng, Gregory W. Moore, Natalie M. Paquette
In this note, we apply mathematical results for the volume of certain symmetric spaces to the problem of counting flux vacua in simple IIB Calabi–Yau compactifications. In particular, we obtain estimates for the number of flux vacua including the geometric factor related to the Calabi–Yau moduli space, in the large flux limit, for the FHSV model and some closely related models. We see that these geometric
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Unipotent extensions and differential equations (after Bloch–Vlasenko) Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-21 Matt Kerr
S. Bloch and M. Vlasenko recently introduced a theory of motivic Gamma functions, given by periods of the Mellin transform of a geometric variation of Hodge structure. They tie properties of these functions to the monodromy and asymptotic behavior of certain unipotent extensions of the variation. In this article, we further examine their Gamma functions and the related Apéry and Frobenius invariants
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Berezin–Toeplitz quantization in real polarizations with toric singularities Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-21 Nai-Chung Conan Leung, Yu-Tung Yau
On a compact Kähler manifold $X$, Toeplitz operators determine a deformation quantization $(\mathrm{C}^\infty (X,\mathbb{C}) [\![\hbar]\!], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1,0} X, T^{0,1} X$ as $\hbar \to 0^{+}$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]
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$T \overline{T}$-deformed modular forms Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-04 John Cardy
Certain objects of conformal field theory, for example partition functions on the rectangle and the torus, and one-point functions on the torus, are either invariant or transform simply under the modular group, properties which should be preserved under the $T \overline{T}$ deformation. The formulation and proof of this statement in fact extents to more general functions such as $T \overline{T}$ deformed
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On the mixed-twist construction and monodromy of associated Picard–Fuchs systems Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-04 Andreas Malmendier, Michael T. Schultz
We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank $\rho \geq 16$. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard–Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by twoelementary
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Graphical functions in even dimensions Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-04 Michael Borinsky, Oliver Schnetz
Graphical functions are special position space Feynman integrals, which can be used to calculate Feynman periods and one- or two-scale processes at high loop orders. With graphical functions, renormalization constants have been calculated to loop orders seven and eight in four-dimensional $\phi^4$ theory and to order five in six-dimensional $\phi^3$ theory. In this article we present the theory of
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Degeneracy and hidden symmetry for the asymmetric quantum Rabi model with integral bias Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-10-04 Cid Reyes-Bustos, Masato Wakayama
The hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias was uncovered in recent studies by the explicit construction of operators $J_\ell$ commuting with the Hamiltonian. The existence of such symmetry has been widely believed to cause the degeneration of the spectrum, that is, the crossings on the energy curves. In this paper we propose a conjectural relation between
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Mirror symmetry of Calabi-Yau manifolds fibered by $(1,8)$-polarized abelian surfaces Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-04-27 Shinobu Hosono, Hiromichi Takagi
We study mirror symmetry of a family of Calabi–Yau manifolds fibered by $(1,8)$-polarized abelian surfaces with Euler characteristic zero. By describing the parameter space globally, we find all expected boundary points (LCSLs), including those correspond to Fourier–Mukai partners. Applying mirror symmetry at each boundary point, we calculate Gromov–Witten invariants $(g \leq 2)$ and observe nice (quasi-)modular
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Kapranov’s $L_\infty$ structures, Fedosov’s star products, and one-loop exact BV quantizations on Kähler manifolds Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-04-27 Kwokwai Chan, Naichung Conan Leung, Qin Li
We study quantization schemes on a Kähler manifold and relate several interesting structures. We first construct Fedosov’s star products on a Kähler manifold $X$ as quantizations of Kapranov’s $L_\infty$-algebra structure. Then we investigate the Batalin–Vilkovisky (BV) quantizations associated to these star products. A remarkable feature is that they are all one-loop exact, meaning that the Feynman
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Modular parametrization as Polyakov path integral: cases with CM elliptic curves as target spaces Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-04-27 Satoshi Kondo, Taizan Watari
For an elliptic curve $E$ over an abelian extension $k/K$ with CM by $K$ of Shimura type, the L-functions of its $[k:K]$ Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to $E$ pulls back the $1$-forms on $E$ to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain
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Diophantine equations with sum of cubes and cube of sum Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-04-27 Bogdan A. Dobrescu, Patrick J. Fox
We solve Diophantine equations of the type $a(x^3+y^3+z^3)=(x+y+z)^3$, where $x$, $y$, $z$ are integer variables, and the coefficient $a \neq 0$ is rational. We show that there are infinite families of such equations, including those where $a$ is any cube or certain rational fractions, that have nontrivial solutions. There are also infinite families of equations that do not have any nontrivial solution
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Intermediate and small scale limiting theorems for random fields Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-02-01 Dmitry Beliaev, Riccardo W. Maffucci
In this paper we study the nodal lines of random eigenfunctions of the Laplacian on the torus, the so-called ‘arithmetic waves’. To be more precise, we study the number of intersections of the nodal line with a straight interval in a given direction. We are interested in how this number depends on the length and direction of the interval and the distribution of spectral measure of the random wave.
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Identities among higher genus modular graph tensors Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-02-01 Eric D’Hoker, Oliver Schlotterer
Higher genus modular graph tensors map Feynman graphs to functions on the Torelli space of genus‑$h$ compact Riemann surfaces which transform as tensors under the modular group $Sp(2h, \mathbb{Z})$, thereby generalizing a construction of Kawazumi. An infinite family of algebraic identities between one-loop and tree-level modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank
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Amplitude recursions with an extra marked point Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-02-01 Johannes Broedel, Andre Kaderli
The recursive calculation of Selberg integrals by Aomoto and Terasoma using the Knizhnik–Zamolodchikov equation and the Drinfeld associator makes use of an auxiliary point and facilitates the recursive evaluation of string amplitudes at genus zero: open-string $N$‑point amplitudes can be obtained from those at $N-1$ points. We establish a similar formalism at genus one, which allows the recursive calculation
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On a class of non-simply connected Calabi-Yau $3$-folds with positive Euler characteristic Commun. Number Theory Phys. (IF 1.9) Pub Date : 2022-02-01 Tolga Karayayla
In this work we obtain a class of non-simply connected Calabi–Yau $3$ folds with positive Euler characteristic as the quotient of projective small resolutions of singular Schoen $3$ folds under the free action of finite groups. A Schoen $3$ fold is a fiber product $X = B_1 \times {}_{\mathbb{P}^1} \: B_2$ of two relatively minimal rational elliptic surfaces with section $\beta_i : B_i \to \mathbb{P}^1
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Wrońskian algebra and Broadhurst–Roberts quadratic relations Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-10-06 Yajun Zhou
Through algebraic manipulations onWrońskian matrices whose entries are reducible to Bessel moments, we present a new analytic proof of the quadratic relations conjectured by Broadhurst and Roberts, along with some generalizations. In the Wrońskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove’s differential operators, and compute the Betti intersection
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Geometries in perturbative quantum field theory Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-10-06 Oliver Schnetz
In perturbative quantum field theory one encounters certain, very specific geometries over the integers. These perturbative quantum geometries determine the number contents of the amplitude considered. In the article ‘Modular forms in quantum field theory’ F. Brown and the author report on a first list of perturbative quantum geometries using the $c_2$-invariant in $\varphi^4$ theory. A main tool was
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Graph hypersurfaces with torus action and a conjecture of Aluffi Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-07-15 Graham Denham, Delphine Pol, Mathias Schulze, Uli Walther
Generalizing the $\star$‑graphs of Müller–Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a non-trivial torus action. For such graphs, we show that the Euler characteristic of the corresponding projective graph hypersurface complement is zero. In contrast, we also show that the Euler characteristic in question can take any integer value for a suitable
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Local energy optimality of periodic sets Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-07-15 Renaud Coulangeon, Achill Schürmann
We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r) = e^{-cr}$ with $c \gt 0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive
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Weyl invariant $E_8$ Jacobi forms Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-07-15 Haowu Wang
We investigate the Jacobi forms for the root system $E_8$ invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov–Witten theory and string theory. In 1992, Wirthmüller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper
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Massive deformations of Maass forms and Jacobi forms Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-07-15 Marcus Berg, Kathrin Bringmann, Terry Gannon
We define one-parameter “massive” deformations of Maass forms and Jacobi forms. This is inspired by descriptions of plane gravitational waves in string theory. Examples include massive Green’s functions (that we write in terms of Kronecker–Eisenstein series) and massive modular graph functions.
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Rational $2$-functions are abelian Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-07-15 L. Felipe Müller
We show that the coefficients of rational $2$-functions are contained in an abelian number field. More precisely, we show that the poles of such functions are poles of order one and given by roots of unity and rational residue.
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KP integrability of triple Hodge integrals, I. From Givental group to hierarchy symmetries Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-07-15 Alexander Alexandrov
In this paper, we investigate a relation between the Givental group of rank one and the Heisenberg–Virasoro symmetry group of the KP hierarchy. We prove, that only a two-parameter family of the Givental operators can be identified with elements of the Heisenberg–Virasoro symmetry group. This family describes triple Hodge integrals satisfying the Calabi–Yau condition. Using the identification of the
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Motivic Galois coaction and one-loop Feynman graphs Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-06-01 Matija Tapušković
Following the work of Brown, we can canonically associate a family of motivic periods — called the motivic Feynman amplitude — to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown’s “small graphs principle”
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Entropy modulo a prime Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-06-01 Tom Leinster
Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability distribution in which the ‘probabilities’ are integers modulo a prime $p$. The entropy, too, is an integer $\operatorname{mod} p$. Entropy $\operatorname{mod} p$ is shown to be uniquely characterized by a functional equation identical to the one that characterizes ordinary Shannon entropy. We also establish
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Green’s functions for Vladimirov derivatives and Tate’s thesis Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-06-01 An Huang, Bogdan Stoica, Shing-Tung Yau, Xiao Zhong
Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $\chi$. These theories appear in the study of $p$‑adic string theory and $p$‑adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the
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On functional equations for Nielsen polylogarithms Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-06-01 Steven Charlton, Herbert Gangl, Danylo Radchenko
We derive new functional equations for Nielsen polylogarithms. We show that, when viewed $\operatorname{modulo} \mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to $8$, and general families of identities in
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Vertex operator algebras of rank $2$: The Mathur–Mukhi–Sen theorem revisited Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-03-01 Geoffrey Mason, Kiyokazu Nagatomo, Yuichi Sakai
Let $V$ be a strongly regular vertex operator algebra and let $\mathfrak{ch}_V$ be the space spanned by the characters of the irreducible $V$-modules. It is known that $\mathfrak{ch}_V$ is the space of solutions of a so-called modular linear differential equation (MLDE). In this paper we obtain a classification of those $V$ for which the corresponding MLDE is irreducible and monic of order $2$. It
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Resurgent expansion of Lambert series and iterated Eisenstein integrals Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-01-01 Daniele Dorigoni, Axel Kleinschmidt
We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity properties of iterated Eisenstein integrals that have recently attracted attention in the context of certain period integrals and string theory scattering amplitudes
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Gamma functions, monodromy and Frobenius constants Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-01-01 Spencer Bloch, Masha Vlasenko
In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as Frobenius constants associated to an ordinary linear differential operator L with a reflection type singularity. These numbers describe the variation around the reflection point of Frobenius solutions to L defined near other singular points. Golyshev and Zagier show that in certain geometric
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Vafa–Witten invariants from modular anomaly Commun. Number Theory Phys. (IF 1.9) Pub Date : 2021-01-01 Sergei Alexandrov
Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses the completion through the holomorphic generating functions of lower ranks. Here we show that for $U(N)$ Vafa-Witten theory on Hirzebruch and del Pezzo surfaces this formula can be used to extract the
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K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry I Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-12-01 Shinobu Hosono, Bong H. Lian, Hiromichi Takagi, Shing-Tung Yau
From the viewpoint of mirror symmetry, we revisit the hypergeometric system $E(3, 6)$ for a family of K3 surfaces. We construct a good resolution of the Baily–Borel–Satake compactification of its parameter space, which admits special boundary points (LCSLs) given by normal crossing divisors. We find local isomorphisms between the $E(3, 6)$ systems and the associated GKZ systems defined locally on the
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Aspects of $(2,2)$ and $(0,2)$ hybrid models Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Marco Bertolini, Mauricio Romo
In this work we study the topological rings of two dimensional (2,2) and (0,2) hybrid models. In particular, we use localization to derive a formula for the correlators in both cases, focusing on the B- and B/2-twists. Although our methods apply to a vast range of hybrid CFTs, we focus on hybrid models suitable for compactifications of the heterotic string. In this case, our formula provides unnormalized
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Absence of irreducible multiple zeta-values in melon modular graph functions Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Eric D’Hoker, Michael B. Green
The expansion of a modular graph function on a torus of modulus $\tau$ near the cusp is given by a Laurent polynomial in $y= \pi \Im (\tau)$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term whose coefficient is rational and exponentially suppressed terms. We prove that the coefficients of the non-leading terms in the Laurent polynomial
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Genus-zero and genus-one string amplitudes and special multiple zeta values Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Don Zagier, Federico Zerbini
In this paper we show that in perturbative string theory the genus-one contribution to formal 2-point amplitudes can be related to the genus-zero contribution to 4-point amplitudes. This is achieved by studying special linear combinations of multiple zeta values that appear as coefficients of the amplitudes. We also exploit our results to relate closed strings to open strings at genus one using Brown's
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Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Edward Frenkel, Davide Gaiotto
We review and extend the vertex algebra framework linking gauge theory constructions and a quantum deformation of the Geometric Langlands Program. The relevant vertex algebras are associated to junctions of two boundary conditions in a 4d gauge theory and can be constructed from the basic ones by following certain standard procedures. Conformal blocks of modules over these vertex algebras give rise
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Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Imma Gálvez-Carrillo, Ralph M. Kaufmann, Andrew Tonks
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects
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Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Imma Gálvez-Carrillo, Ralph M. Kaufmann, Andrew Tonks
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects
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From Möbius inversion to renormalisation Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Joachim Kock
This paper traces a straight line from classical Mobius inversion to Hopf-algebraic perturbative renormalisation. This line, which is logical but not entirely historical, consists of just a few main abstraction steps, and some intermediate steps dwelled upon for mathematical pleasure. The paper is largely expository, but contains many new perspectives on well-known results. For example, the equivalence
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A Cartesian diagram of Rapoport–Zink towers over universal covers of $p$-divisible groups Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Hadi S. Mohammad Hedayatzadeh
In their paper Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial $\ell$-adic etale cohomology classes appearing in the the generic fiber of Lubin-Tate and Rapoprt-Zink towers. We also study the behavior of the vector bundle functor
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CHL Calabi–Yau threefolds: curve counting, Mathieu moonshine and Siegel modular forms Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Jim Bryan, Georg Oberdieck
A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which
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Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell–Yan scattering Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Marco Besier, Dino Festi, Michael Harrison, Bartosz Naskręcki
We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations
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Stringy Hirzebruch classes of Weierstrass fibrations Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 James Fullwood, Mark van Hoeij
A Weierstrass fibration is an elliptic fibration $Y\to B$ whose total space $Y$ may be given by a global Weierstrass equation in a $\mathbb{P}^2$-bundle over $B$. In this note, we compute stringy Hirzebruch classes of singular Weierstrass fibrations associated with constructing non-Abelian gauge theories in $F$-theory. For each Weierstrass fibration $Y\to B$ we then derive a generating function $\
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Bounds for smooth Fano weighted complete intersections Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Victor Przyjalkowski, Constantin Shramov
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed $n+1$. Based on this bound we classify all smooth Fano complete intersections of dimensions $4$ and $5$, and
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On $E_1$-degeneration for the special fiber of a semistable family Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Mao Sheng, Junchao Shentu
We study the $E_1$-degeneration of the logarithmic Hodge to de Rham spectral sequence of the special fiber of a semistable family over a discrete valuation ring. On the one hand, we prove that the $E_1$-degeneration property is invariant under admissible blow-ups. Assuming functorial resolution of singularities over $\mathbb{Z}$, this implies that the $E_1$-degeneration property depends only on the
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On derived equivalence of general Clifford double mirrors Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Zhan Li
We show that general Clifford double mirrors constructed in "On Clifford double mirrors of toric complete intersections" are derived equivalent.
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Properties of extremal CFTs with small central charge Commun. Number Theory Phys. (IF 1.9) Pub Date : 2020-01-01 Francesca Ferrari, Sarah M. Harrison
We analyze aspects of extant examples of 2d extremal chiral (super)conformal field theories with $c\leq 24$. These are theories whose only operators with dimension smaller or equal to $c/24$ are the vacuum and its (super)Virasoro descendents. The prototypical example is the monster CFT, whose famous genus zero property is intimately tied to the Rademacher summability of its twined partition functions