Pub Date : 2019-08-26DOI: 10.4310/cntp.2020.v14.n4.a1
Mohammad Hadi 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 on the fundamental curve in $p$-adic Hodge theory, defined by Fargues-Fontaine, under multilinear morphisms.
{"title":"A Cartesian diagram of Rapoport–Zink towers over universal covers of $p$-divisible groups","authors":"Mohammad Hadi Hedayatzadeh","doi":"10.4310/cntp.2020.v14.n4.a1","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n4.a1","url":null,"abstract":"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 on the fundamental curve in $p$-adic Hodge theory, defined by Fargues-Fontaine, under multilinear morphisms.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45824234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-20DOI: 10.4310/CNTP.2021.v15.n1.a3
S. 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 cases Frobenius constants are periods, and they raise the question quite generally how to describe these numbers motivically. In this paper we give a relation between Frobenius constants and Taylor coefficients of generalized gamma functions, from which it follows that Frobenius constants of Picard--Fuchs differential operators are periods. We also study the relation between these constants and periods of limiting Hodge structures. This is a major revision of the previous version of the manuscript. The notion of Frobenius constants and our main result are extended to the general case of regular singularities with any sets of local exponents. In addition, the generating function of Frobenius constants is given explicitly for all hypergeometric connections.
{"title":"Gamma functions, monodromy and Frobenius constants","authors":"S. Bloch, Masha Vlasenko","doi":"10.4310/CNTP.2021.v15.n1.a3","DOIUrl":"https://doi.org/10.4310/CNTP.2021.v15.n1.a3","url":null,"abstract":"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 cases Frobenius constants are periods, and they raise the question quite generally how to describe these numbers motivically. In this paper we give a relation between Frobenius constants and Taylor coefficients of generalized gamma functions, from which it follows that Frobenius constants of Picard--Fuchs differential operators are periods. We also study the relation between these constants and periods of limiting Hodge structures. \u0000This is a major revision of the previous version of the manuscript. The notion of Frobenius constants and our main result are extended to the general case of regular singularities with any sets of local exponents. In addition, the generating function of Frobenius constants is given explicitly for all hypergeometric connections.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46470712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-13DOI: 10.4310/cntp.2021.v15.n2.a4
Steven Charlton, H. Gangl, D. Radchenko
We derive new functional equations for Nielsen polylogarithms. We show that, when viewed 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 higher weight.
{"title":"On functional equations for Nielsen polylogarithms","authors":"Steven Charlton, H. Gangl, D. Radchenko","doi":"10.4310/cntp.2021.v15.n2.a4","DOIUrl":"https://doi.org/10.4310/cntp.2021.v15.n2.a4","url":null,"abstract":"We derive new functional equations for Nielsen polylogarithms. We show that, when viewed 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 higher weight.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44204238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-02DOI: 10.4310/cntp.2020.v14.n4.a4
M. Besier, Dino Festi, Michael C. 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 of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.
{"title":"Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell–Yan scattering","authors":"M. Besier, Dino Festi, Michael C. Harrison, Bartosz Naskręcki","doi":"10.4310/cntp.2020.v14.n4.a4","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n4.a4","url":null,"abstract":"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 of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43570450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-28DOI: 10.4310/cntp.2020.v14.n2.a4
D. 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 single-valued projection, proving a conjecture of Broedel, Schlotterer and the second author.
{"title":"Genus-zero and genus-one string amplitudes and special multiple zeta values","authors":"D. Zagier, Federico Zerbini","doi":"10.4310/cntp.2020.v14.n2.a4","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n2.a4","url":null,"abstract":"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 single-valued projection, proving a conjecture of Broedel, Schlotterer and the second author.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44729248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-02DOI: 10.4310/cntp.2022.v16.n1.a1
D. 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. We analyse the second factorial moment in the short interval regime and the persistence probability in the long interval regime. We also study relations between the Cilleruelo and Cilleruelo-type fields. We give an explicit coupling between these fields which on mesoscopic scales preserves the structure of the nodal sets with probability close to one.
{"title":"Intermediate and small scale limiting theorems for random fields","authors":"D. Beliaev, Riccardo W. Maffucci","doi":"10.4310/cntp.2022.v16.n1.a1","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n1.a1","url":null,"abstract":"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. We analyse the second factorial moment in the short interval regime and the persistence probability in the long interval regime. We also study relations between the Cilleruelo and Cilleruelo-type fields. We give an explicit coupling between these fields which on mesoscopic scales preserves the structure of the nodal sets with probability close to one.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47735753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-17DOI: 10.4310/cntp.2021.v15.n4.a2
O. 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 $phi^4$ theory. A main tool was denominator reduction which allowed the authors to examine graphs up to loop order (first Betti number) 10. We introduce an improved quadratic denominator reduction which makes it possible to extend the previous results to loop order 11 (and partially orders 12 and 13). For comparison, also non-$phi^4$ graphs are investigated. Here, we extend the results from loop order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously 157) -- while being consistent with all major $c_2$-conjectures -- leads to a more refined picture of perturbative quantum geometries. In the appendix, Friedrich Knop proves a Chevalley-Warning-Ax theorem for double covers of affine space.
{"title":"Geometries in perturbative quantum field theory","authors":"O. Schnetz","doi":"10.4310/cntp.2021.v15.n4.a2","DOIUrl":"https://doi.org/10.4310/cntp.2021.v15.n4.a2","url":null,"abstract":"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 $phi^4$ theory. A main tool was denominator reduction which allowed the authors to examine graphs up to loop order (first Betti number) 10. We introduce an improved quadratic denominator reduction which makes it possible to extend the previous results to loop order 11 (and partially orders 12 and 13). For comparison, also non-$phi^4$ graphs are investigated. Here, we extend the results from loop order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously 157) -- while being consistent with all major $c_2$-conjectures -- leads to a more refined picture of perturbative quantum geometries. In the appendix, Friedrich Knop proves a Chevalley-Warning-Ax theorem for double covers of affine space.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47015397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-13DOI: 10.4310/cntp.2020.v14.n2.a2
E. D'hoker, M. 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 of the modular graph function $D_N(tau)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary $N geq 0$. The proof proceeds by expressing a generating function for $D_N(tau)$ in terms of an integral over the Virasoro-Shapiro closed-string tree amplitude.
{"title":"Absence of irreducible multiple zeta-values in melon modular graph functions","authors":"E. D'hoker, M. B. Green","doi":"10.4310/cntp.2020.v14.n2.a2","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n2.a2","url":null,"abstract":"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 of the modular graph function $D_N(tau)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary $N geq 0$. The proof proceeds by expressing a generating function for $D_N(tau)$ in terms of an integral over the Virasoro-Shapiro closed-string tree amplitude.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47920371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-24DOI: 10.4310/cntp.2020.v14.n3.a4
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 generic fiber. On the other hand, we show by explicit examples that the decomposability of the logarithmic de Rham complex is not invariant under admissible blow-ups, which answer negatively an open problem of L. Illusie (Problem 7.14 cite{Illusie2002}). We also give an algebraic proof of an $E_1$-degeneration result in characteristic zero due to Steenbrink and Kawamata-Namikawa.
{"title":"On $E_1$-degeneration for the special fiber of a semistable family","authors":"Mao Sheng, Junchao Shentu","doi":"10.4310/cntp.2020.v14.n3.a4","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n3.a4","url":null,"abstract":"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 generic fiber. On the other hand, we show by explicit examples that the decomposability of the logarithmic de Rham complex is not invariant under admissible blow-ups, which answer negatively an open problem of L. Illusie (Problem 7.14 cite{Illusie2002}). We also give an algebraic proof of an $E_1$-degeneration result in characteristic zero due to Steenbrink and Kawamata-Namikawa.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43070263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-21DOI: 10.4310/cntp.2019.v13.n3.a3
Daniele Dorigoni, A. Kleinschmidt
In this note we study $SL(2,mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we choose to represent them as Poincare series. In this way we can combine different methods for asymptotic expansions and obtain the perturbative and non-perturbative contributions to their zero Fourier modes. In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/anti-instantons.
{"title":"Modular graph functions and asymptotic expansions of Poincaré series","authors":"Daniele Dorigoni, A. Kleinschmidt","doi":"10.4310/cntp.2019.v13.n3.a3","DOIUrl":"https://doi.org/10.4310/cntp.2019.v13.n3.a3","url":null,"abstract":"In this note we study $SL(2,mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we choose to represent them as Poincare series. In this way we can combine different methods for asymptotic expansions and obtain the perturbative and non-perturbative contributions to their zero Fourier modes. In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/anti-instantons.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42571051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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