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The cosmic Galois group, the sunrise Feynman integral, and the relative completion of $\Gamma^1(6)$ Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240715
Matija TapuškovićIn the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts of Feynman integrals associated to related Feynman graphs. Only one of the conjugates of the motivic lift of the sunrise, other than itself, can be expressed in

Vector spaces of generalized Euler integrals Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240715
Daniele Agostini, Claudia Fevola, AnnaLaura Sattelberger, Simon TelenWe study vector spaces associated to a family of generalized Euler integrals. Their dimension is given by the Euler characteristic of a very affine variety. Motivated by Feynman integrals from particle physics, this has been investigated using tools from homological algebra and the theory of $D$modules. We present an overview and uncover new relations between these approaches. We also provide new

Witten–Reshetikhin–Turaev invariants and homological blocks for plumbed homology spheres Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240715
Yuya MurakamiIn this paper, we prove a conjecture by Gukov–Pei–Putrov–Vafa for a wide class of plumbed $3$manifolds. Their conjecture states that Witten–Reshetikhin–Turaev (WRT) invariants are radial limits of homological blocks, which are $q$series introduced by them for plumbed $3$manifolds with negative definite linking matrices. The most difficult point in our proof is to prove the vanishing of weighted

Quantum KdV hierarchy and quasimodular forms Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240715
JanWillem M. van Ittersum, Giulio RuzzaDubrovin $\href{https://doi.org/10.1007/s0002301504492}{[10]}$ has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg–de Vries (KdV) hierarchy is given by shifted symmetric functions; the latter are related by the Bloch–Okounkov Theorem $\href{https://doi.org/10.1007/JHEP07(2014)141}{[1]}$ to quasimodular forms on the full modular

Colored Bosonic models and matrix coefficients Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240715
Daniel Bump, Slava NaprienkoWe develop the theory of colored bosonic models (initiated by Borodin and Wheeler). We will show how a family of such models can be used to represent the values of Iwahori vectors in the “spherical model” of representations of $\mathrm{GL}_r (F)$, where $F$ is a nonarchimedean local field. Among our results are a monochrome factorization, which is the realization of the Boltzmann weights by fusion

Rankin–Cohen brackets for Calabi–Yau modular forms Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240607
Younes Nikdelan$\def\M{\mathscr{M}}\def\Rscr{\mathscr{R}}\def\Rsf{\mathsf{R}}\def\Tsf{\mathsf{T}}\def\tildeM{\widetilde{\M}}$For any positive integer $n$, we introduce a modular vector field $\Rsf$ on a moduli space $\Tsf$ of enhanced Calabi–Yau $n$folds arising from the Dwork family. By Calabi–Yau quasimodular forms associated to $\Rsf$ we mean the elements of the graded $\mathbb{C}$algebra $\tildeM$ generated

Quantum geometry, stability and modularity Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240607
Sergei Alexandrov, Soheyla Feyzbakhsh, Albrecht Klemm, Boris Pioline, Thorsten Schimannekrelated to GopakumarVafa (GV) invariants, and rank 0 DonaldsonThomas (DT) invariants countingD4D2D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4D2D0 indices for compact oneparameter CalabiYau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that

On motivic and arithmetic refinements of DonaldsonThomas invariants Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240607
Felipe Espreafico, Johannes WalcherIn recent years, a version of enumerative geometry over arbitrary fields has been developed and studied by KassWickelgren, Levine, and others, in which the counts obtained are not integers but quadratic forms. Aiming to understand the relation to other “refined invariants”, and especially their possible interpretation in quantum theory, we explain how to obtain a quadratic version of DonaldsonThomas

Spectral geometry of functional metrics on noncommutative tori Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240607
Asghar Ghorbanpour, Masoud KhalkhaliWe introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature

Infinite families of quantum modular 3manifold invariants Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240607
Louisa Liles, Eleanor McSpiritOne of the first key examples of a quantum modular form, which unifies the WittenReshetikhinTuraev (WRT) invariants of the Poincaré homology sphere, appears in work of Lawrence and Zagier. We show that the series they construct is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3‑manifolds whose radial limits toward roots of unity may be thought of as

Laplace transform of the $xy$ symplectic transformation formula in Topological Recursion Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240124
Alexander HockThe functional relation coming from the $xy$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order momentcumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau $3$folds Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240124
Miroslav Rapčák, Yan Soibelman, Yaping Yang, Gufang ZhaoWe study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi–Yau $3$fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via “raising operators”. Conjecturally the COHA action extends to an action of the Drinfeld double by

Numerical experiments on coefficients of instanton partition functions Commun. Number Theory Phys. (IF 1.2) Pub Date : 20240124
Aradhita Chattopadhyaya, Jan ManschotWe analyze the coefficients of partition functions of Vafa–Witten (VW) theory on a fourmanifold. These partition functions factorize into a product of a function enumerating pointlike instantons and a function enumerating smooth instantons. For gauge groups $SU(2)$ and $SU(3)$ and fourmanifold the complex projective plane $\mathbb{CP}^2$, we experimentally study the latter functions, which are examples

Weyl invariant $E_8$ Jacobi forms and $E$strings Commun. Number Theory Phys. (IF 1.2) Pub Date : 20231107
Kaiwen Sun, Haowu WangIn 1992 Wirthmüller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free

Whittaker Fourier type solutions to differential equations arising from string theory Commun. Number Theory Phys. (IF 1.2) Pub Date : 20231107
Ksenia Fedosova, Kim KlingerLoganIn this article, we find the full Fourier expansion for solutions of $(\Delta\lambda)f(z) = E_k (z) E_\ell (z)$ for $z = x + i y \in \mathfrak{H}$ for certain values of parameters $k$, $\ell$ and $\lambda$. When such an $f$ is fully automorphic these functions are referred to as generalized nonholomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with

Enumeration of hypermaps and Hirota equations for extended rationally constrained KP Commun. Number Theory Phys. (IF 1.2) Pub Date : 20231107
G. Carlet, J. van de Leur, H. Posthuma, S. ShadrinWe consider the Hurwitz Dubrovin–Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin–Frobenius manifold is a tau function of a rational reduction of the Kadomtsev–Petviashvili

Resurgence, Stokes constants, and arithmetic functions in topological string theory Commun. Number Theory Phys. (IF 1.2) Pub Date : 20231107
Claudia RellaThe quantization of the mirror curve to a toric Calabi–Yau threefold gives rise to quantummechanical operators, whose fermionic spectral traces produce factorially divergent power series in the Planck constant. These asymptotic expansions can be promoted to resurgent transseries. They show infinite towers of periodic singularities in their Borel plane and infinitely many rational Stokes constants

KP hierarchy for Hurwitztype cohomological field theories Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230504
Reinier KramerWe generalise a result of Kazarian regarding Kadomtsev–Petviashvili integrability for single Hodge integrals to general cohomological field theories related to Hurwitztype counting problems or hypergeometric tau‑functions. The proof uses recent results on the relations between hypergeometric taufunctions and topological recursion, as well as the DOSS correspondence between topological recursion and

Equivariant derived equivalence and rational points on K3 surfaces Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230504
Brendan Hassett, Yuri TschinkelWe 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.

On quasitame Looijenga pairs Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230504
Andrea Brini, Yannik SchülerWe prove a conjecture of Bousseau, van Garrel and the firstnamed 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 closedform $q$hypergeometric resummation of the quantum tropical vertex calculation of the $\log$

Completing the $c_2$ completion conjecture for $p=2$ Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230504
Simone Hu, Karen YeatsThe $c_2$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

Resurgent Stokes data for Painlevé equations and twodimensional quantum (super) gravity Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230504
Salvatore Baldino, Ricardo Schiappa, Maximilian Schwick, Roberto VegaResurgenttransseries solutions to Painlevé equations may be recursively constructed out of these nonlinear differentialequations—but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connectionformulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear

On arithmetic Dijkgraaf–Witten theory Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230223
Hikaru Hirano, Junhyeong Kim, Masanori MorishitaWe 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

On generalized stuffle relations between cellzeta values Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230223
Nikita MarkarianWe introduce a family of linear relations between cellzeta values that have a form similar to product map relations and jointly with them imply stuffle relations between multiple zeta values.

On the fundamental group of open Richardson varieties Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230223
Changzheng Li, Frank Sottile, Chi ZhangWe 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

Graph complexes and Feynman rules Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230223
Marko Berghoff, Dirk KreimerWe investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on the interplay between graph homology, Hopfalgebraic 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 edgepropagators on the

Fourier expansions of vectorvalued automorphic functions with nonunitary twists Commun. Number Theory Phys. (IF 1.2) Pub Date : 20230223
Ksenia Fedosova, Anke Pohl, Julie RowlettWe provide Fourier expansions of vectorvalued eigenfunctions of the hyperbolic Laplacian that are twistperiodic in a horocycle direction. The twist may be given by any endomorphism of a finitedimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vectorvalued twisted automorphic forms of Fuchsian groups. We further provide a detailed

Fibers over infinity of Landau–Ginzburg models Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221021
Ivan Cheltsov, Victor PrzyjalkowskiWe 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.

Orthosymplectic Satake equivalence Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221021
Alexander Braverman, Michael Finkelberg, Roman TravkinThis is a companion paper of [BFGT]. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $\mathrm{SO}(N1,\mathbb{C} [\![t]\!])$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 BenZvi, Sakellaridis and Venkatesh

Elliptic threefolds with high Mordell–Weil rank Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221021
Antonella Grassi, Timo WeigandWe 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

Flux vacua: a voluminous recount Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221021
Miranda C. N. Cheng, Gregory W. Moore, Natalie M. PaquetteIn 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

Unipotent extensions and differential equations (after Bloch–Vlasenko) Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221021
Matt KerrS. 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

Berezin–Toeplitz quantization in real polarizations with toric singularities Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221021
NaiChung Conan Leung, YuTung YauOn 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 nonsingular real polarizations [13]

$T \overline{T}$deformed modular forms Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221004
John CardyCertain objects of conformal field theory, for example partition functions on the rectangle and the torus, and onepoint 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

On the mixedtwist construction and monodromy of associated Picard–Fuchs systems Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221004
Andreas Malmendier, Michael T. SchultzWe use the mixedtwist construction of Doran and Malmendier to obtain a multiparameter 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

Graphical functions in even dimensions Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221004
Michael Borinsky, Oliver SchnetzGraphical functions are special position space Feynman integrals, which can be used to calculate Feynman periods and one or twoscale processes at high loop orders. With graphical functions, renormalization constants have been calculated to loop orders seven and eight in fourdimensional $\phi^4$ theory and to order five in sixdimensional $\phi^3$ theory. In this article we present the theory of

Degeneracy and hidden symmetry for the asymmetric quantum Rabi model with integral bias Commun. Number Theory Phys. (IF 1.2) Pub Date : 20221004
Cid ReyesBustos, Masato WakayamaThe hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a halfintegral 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

Mirror symmetry of CalabiYau manifolds fibered by $(1,8)$polarized abelian surfaces Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220427
Shinobu Hosono, Hiromichi TakagiWe 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

Kapranov’s $L_\infty$ structures, Fedosov’s star products, and oneloop exact BV quantizations on Kähler manifolds Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220427
Kwokwai Chan, Naichung Conan Leung, Qin LiWe 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 oneloop exact, meaning that the Feynman

Modular parametrization as Polyakov path integral: cases with CM elliptic curves as target spaces Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220427
Satoshi Kondo, Taizan WatariFor an elliptic curve $E$ over an abelian extension $k/K$ with CM by $K$ of Shimura type, the Lfunctions 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

Diophantine equations with sum of cubes and cube of sum Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220427
Bogdan A. Dobrescu, Patrick J. FoxWe 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

Intermediate and small scale limiting theorems for random fields Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220201
Dmitry Beliaev, Riccardo W. MaffucciIn this paper we study the nodal lines of random eigenfunctions of the Laplacian on the torus, the socalled ‘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.

Identities among higher genus modular graph tensors Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220201
Eric D’Hoker, Oliver SchlottererHigher 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 oneloop and treelevel modular graph tensors are proven for arbitrary genus and arbitrary tensorial rank

Amplitude recursions with an extra marked point Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220201
Johannes Broedel, Andre KaderliThe 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: openstring $N$‑point amplitudes can be obtained from those at $N1$ points. We establish a similar formalism at genus one, which allows the recursive calculation

On a class of nonsimply connected CalabiYau $3$folds with positive Euler characteristic Commun. Number Theory Phys. (IF 1.2) Pub Date : 20220201
Tolga KarayaylaIn this work we obtain a class of nonsimply 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

Wrońskian algebra and Broadhurst–Roberts quadratic relations Commun. Number Theory Phys. (IF 1.2) Pub Date : 20211006
Yajun ZhouThrough 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

Geometries in perturbative quantum field theory Commun. Number Theory Phys. (IF 1.2) Pub Date : 20211006
Oliver SchnetzIn 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

Graph hypersurfaces with torus action and a conjecture of Aluffi Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210715
Graham Denham, Delphine Pol, Mathias Schulze, Uli WaltherGeneralizing the $\star$‑graphs of Müller–Stach and Westrich, we describe a class of graphs whose associated graph hypersurface is equipped with a nontrivial 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

Local energy optimality of periodic sets Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210715
Renaud Coulangeon, Achill SchürmannWe 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

Weyl invariant $E_8$ Jacobi forms Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210715
Haowu WangWe 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

Massive deformations of Maass forms and Jacobi forms Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210715
Marcus Berg, Kathrin Bringmann, Terry GannonWe define oneparameter “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.

Rational $2$functions are abelian Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210715
L. Felipe MüllerWe 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.

KP integrability of triple Hodge integrals, I. From Givental group to hierarchy symmetries Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210715
Alexander AlexandrovIn 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 twoparameter 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

Motivic Galois coaction and oneloop Feynman graphs Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210601
Matija TapuškovićFollowing 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”

Entropy modulo a prime Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210601
Tom LeinsterBuilding 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

Green’s functions for Vladimirov derivatives and Tate’s thesis Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210601
An Huang, Bogdan Stoica, ShingTung Yau, Xiao ZhongGiven 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

On functional equations for Nielsen polylogarithms Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210601
Steven Charlton, Herbert Gangl, Danylo RadchenkoWe 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 fiveterm relation. We also give some functional equations and evaluations for Nielsen polylogarithms in weights up to $8$, and general families of identities in

Vertex operator algebras of rank $2$: The Mathur–Mukhi–Sen theorem revisited Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210301
Geoffrey Mason, Kiyokazu Nagatomo, Yuichi SakaiLet $V$ 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 socalled 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

Resurgent expansion of Lambert series and iterated Eisenstein integrals Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210101
Daniele Dorigoni, Axel KleinschmidtWe 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

Gamma functions, monodromy and Frobenius constants Commun. Number Theory Phys. (IF 1.2) Pub Date : 20210101
Spencer Bloch, Masha VlasenkoIn 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