Pub Date : 2017-11-28DOI: 10.4310/cntp.2019.v13.n4.a2
Michał Kapustka, M. Rampazzo
We construct a gauged linear sigma model with two non-birational K"alher phases which we prove to be derived equivalent, $mathbb{L}$-equivalent, deformation equivalent and Hodge equivalent. This provides a new counterexample to the birational Torelli problem which admits a simple GLSM interpretation.
{"title":"Torelli problem for Calabi–Yau threefolds with GLSM description","authors":"Michał Kapustka, M. Rampazzo","doi":"10.4310/cntp.2019.v13.n4.a2","DOIUrl":"https://doi.org/10.4310/cntp.2019.v13.n4.a2","url":null,"abstract":"We construct a gauged linear sigma model with two non-birational K\"alher phases which we prove to be derived equivalent, $mathbb{L}$-equivalent, deformation equivalent and Hodge equivalent. This provides a new counterexample to the birational Torelli problem which admits a simple GLSM interpretation.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45091097","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 : 2017-11-06DOI: 10.4310/CNTP.2018.v12.n2.a5
Yajun Zhou
Drawing on Vanhove's contributions to mixed Hodge structures for Feynman integrals in two-di-men-sion-al quantum field theory, we compute two families of determinants whose entries are Bessel moments. Via explicit factorizations of certain Wronskian determinants, we verify two recent conjectures proposed by Broadhurst and Mellit, concerning determinants of arbitrary sizes. With some extensions to our methods, we also relate two more determinants of Broadhurst--Mellit to the logarithmic Mahler measures of certain polynomials.
{"title":"Wrońskian factorizations and Broadhurst–Mellit determinant formulae","authors":"Yajun Zhou","doi":"10.4310/CNTP.2018.v12.n2.a5","DOIUrl":"https://doi.org/10.4310/CNTP.2018.v12.n2.a5","url":null,"abstract":"Drawing on Vanhove's contributions to mixed Hodge structures for Feynman integrals in two-di-men-sion-al quantum field theory, we compute two families of determinants whose entries are Bessel moments. Via explicit factorizations of certain Wronskian determinants, we verify two recent conjectures proposed by Broadhurst and Mellit, concerning determinants of arbitrary sizes. With some extensions to our methods, we also relate two more determinants of Broadhurst--Mellit to the logarithmic Mahler measures of certain polynomials.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"12 1","pages":"355-407"},"PeriodicalIF":1.9,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47861249","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 : 2017-10-29DOI: 10.4310/cntp.2020.v14.n3.a6
Francesca Ferrari, Sarah M. Harrison
We analyze aspects of extant examples of 2d extremal chiral (super)conformal field theories with $cleq 24$. 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, a property which also distinguishes the functions of Mathieu and umbral moonshine. However, there are now several additional known examples of extremal CFTs, all of which have at least $mathcal N=1$ supersymmetry and global symmetry groups connected to sporadic simple groups. We investigate the extent to which such a property, which distinguishes the monster moonshine module from other $c=24$ chiral CFTs, holds for the other known extremal theories. We find that in most cases, the special Rademacher summability property present for monstrous and umbral moonshine does not hold for the other extremal CFTs, with the exception of the Conway module and two $c=12, ~mathcal N=4$ superconformal theories with $M_{11}$ and $M_{22}$ symmetry. This suggests that the connection between extremal CFT, sporadic groups, and mock modular forms transcends strict Rademacher summability criteria.
{"title":"Properties of extremal CFTs with small central charge","authors":"Francesca Ferrari, Sarah M. Harrison","doi":"10.4310/cntp.2020.v14.n3.a6","DOIUrl":"https://doi.org/10.4310/cntp.2020.v14.n3.a6","url":null,"abstract":"We analyze aspects of extant examples of 2d extremal chiral (super)conformal field theories with $cleq 24$. 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, a property which also distinguishes the functions of Mathieu and umbral moonshine. However, there are now several additional known examples of extremal CFTs, all of which have at least $mathcal N=1$ supersymmetry and global symmetry groups connected to sporadic simple groups. We investigate the extent to which such a property, which distinguishes the monster moonshine module from other $c=24$ chiral CFTs, holds for the other known extremal theories. We find that in most cases, the special Rademacher summability property present for monstrous and umbral moonshine does not hold for the other extremal CFTs, with the exception of the Conway module and two $c=12, ~mathcal N=4$ superconformal theories with $M_{11}$ and $M_{22}$ symmetry. This suggests that the connection between extremal CFT, sporadic groups, and mock modular forms transcends strict Rademacher summability criteria.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46211538","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 : 2017-09-27DOI: 10.4310/CNTP.2019.V13.N1.A1
S. Cynk, D. Straten
We report on $25$ families of projective Calabi-Yau threefolds that do not have a point of maximal unipotent monodromy in their moduli space. The construction is based on an analysis of certain pencils of octic arrangements that were found by C. Meyer. There are seven cases where the Picard-Fuchs operator is of order two and $18$ cases where it is of order four. The birational nature of the Picard-Fuchs operator can be used effectively to distinguish between families whose members have the same Hodge numbers.
{"title":"Picard–Fuchs operators for octic arrangements, I: The case of orphans","authors":"S. Cynk, D. Straten","doi":"10.4310/CNTP.2019.V13.N1.A1","DOIUrl":"https://doi.org/10.4310/CNTP.2019.V13.N1.A1","url":null,"abstract":"We report on $25$ families of projective Calabi-Yau threefolds that do not have a point of maximal unipotent monodromy in their moduli space. The construction is based on an analysis of certain pencils of octic arrangements that were found by C. Meyer. There are seven cases where the Picard-Fuchs operator is of order two and $18$ cases where it is of order four. The birational nature of the Picard-Fuchs operator can be used effectively to distinguish between families whose members have the same Hodge numbers.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48543149","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 : 2017-09-03DOI: 10.4310/CNTP.2018.V12.N4.A1
Jingyue Chen, An Huang, B. Lian, S. Yau
In this paper, we study the zero loci of local systems of the form $deltaPi$, where $Pi$ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space $X$, and $delta$ is a given differential operator on the space of sections $V^vee=Gamma(X,K_X^{-1})$. Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of $deltaPi$. As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.
{"title":"Differential zeros of period integrals and generalized hypergeometric functions","authors":"Jingyue Chen, An Huang, B. Lian, S. Yau","doi":"10.4310/CNTP.2018.V12.N4.A1","DOIUrl":"https://doi.org/10.4310/CNTP.2018.V12.N4.A1","url":null,"abstract":"In this paper, we study the zero loci of local systems of the form $deltaPi$, where $Pi$ is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space $X$, and $delta$ is a given differential operator on the space of sections $V^vee=Gamma(X,K_X^{-1})$. Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of $deltaPi$. As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"12 1","pages":"609-655"},"PeriodicalIF":1.9,"publicationDate":"2017-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45714889","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 : 2017-06-14DOI: 10.4310/CNTP.2019.v13.n1.a2
K. Sakai
We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm"uller.
{"title":"$E_n$ Jacobi forms and Seiberg–Witten curves","authors":"K. Sakai","doi":"10.4310/CNTP.2019.v13.n1.a2","DOIUrl":"https://doi.org/10.4310/CNTP.2019.v13.n1.a2","url":null,"abstract":"We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm\"uller.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46868785","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 : 2017-04-16DOI: 10.4310/CNTP.2019.V13.N2.A2
P. Ángel, C. Doran, J. Iyer, M. Kerr, James D. Lewis, S. Muller-Stach, D. Patel
A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve.
{"title":"Specialization of cycles and the $K$-theory elevator","authors":"P. Ángel, C. Doran, J. Iyer, M. Kerr, James D. Lewis, S. Muller-Stach, D. Patel","doi":"10.4310/CNTP.2019.V13.N2.A2","DOIUrl":"https://doi.org/10.4310/CNTP.2019.V13.N2.A2","url":null,"abstract":"A general specialization map is constructed for higher Chow groups and used to prove a \"going-up\" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48121050","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 : 2017-04-06DOI: 10.4310/cntp.2019.v13.n3.a1
L. Candelori, C. Franc
This article lays the foundations for the study of modular forms transforming with respect to representations of Fuchsian groups of genus zero. More precisely, we define geometrically weighted graded modules of such modular forms, where the graded structure comes from twisting with all isomorphism classes of line bundles on the corresponding compactified modular curve, and we study their structure by relating it to the structure of vector bundles over orbifold curves of genus zero. We prove that these modules are free whenever the Fuchsian group has at most two elliptic points. For three or more elliptic points, we give explicit constructions of indecomposable vector bundles of rank two over modular orbifold curves, which give rise to non-free modules of geometrically weighted modular forms.
{"title":"Vector bundles and modular forms for Fuchsian groups of genus zero","authors":"L. Candelori, C. Franc","doi":"10.4310/cntp.2019.v13.n3.a1","DOIUrl":"https://doi.org/10.4310/cntp.2019.v13.n3.a1","url":null,"abstract":"This article lays the foundations for the study of modular forms transforming with respect to representations of Fuchsian groups of genus zero. More precisely, we define geometrically weighted graded modules of such modular forms, where the graded structure comes from twisting with all isomorphism classes of line bundles on the corresponding compactified modular curve, and we study their structure by relating it to the structure of vector bundles over orbifold curves of genus zero. We prove that these modules are free whenever the Fuchsian group has at most two elliptic points. For three or more elliptic points, we give explicit constructions of indecomposable vector bundles of rank two over modular orbifold curves, which give rise to non-free modules of geometrically weighted modular forms.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49638499","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 : 2017-01-01DOI: 10.4310/CNTP.2017.V11.N3.A1
F. Brown
The first part of a set of notes based on lectures given at the IHES in May 2015 on Feynman amplitudes and motivic periods. 0.1. Some motivation for physicists. Scattering amplitudes are ubiquitous in high energy physics and have been intensively studied from at least three angles: (1) in phenomenology, where amplitudes in quantum field theory are obtained as a sum of Feynman integrals associated to graphs which represent interactions between fundamental particles. This presents a huge computational challenge with important applications to collider experiments. (2) in superstring perturbation theory, where amplitudes are expressed as integrals over moduli spaces of curves with marked points. (3) in various modern approaches, most notably in the planar limit of N = 4 SYM, which avoid the use of Feynman graphs altogether and seek to construct the amplitude directly, either via the bootstrap method, or via geometric approaches such as on-shell diagrams or the amplituhedron. The goal of these notes is to study a new kind of structure which is potentially satisfied by amplitudes in all three situations. To motivate it, consider first the case of the dilogarithm function, defined for |z| < 1 by the sum
{"title":"Feynman amplitudes, coaction principle, and cosmic Galois group","authors":"F. Brown","doi":"10.4310/CNTP.2017.V11.N3.A1","DOIUrl":"https://doi.org/10.4310/CNTP.2017.V11.N3.A1","url":null,"abstract":"The first part of a set of notes based on lectures given at the IHES in May 2015 on Feynman amplitudes and motivic periods. 0.1. Some motivation for physicists. Scattering amplitudes are ubiquitous in high energy physics and have been intensively studied from at least three angles: (1) in phenomenology, where amplitudes in quantum field theory are obtained as a sum of Feynman integrals associated to graphs which represent interactions between fundamental particles. This presents a huge computational challenge with important applications to collider experiments. (2) in superstring perturbation theory, where amplitudes are expressed as integrals over moduli spaces of curves with marked points. (3) in various modern approaches, most notably in the planar limit of N = 4 SYM, which avoid the use of Feynman graphs altogether and seek to construct the amplitude directly, either via the bootstrap method, or via geometric approaches such as on-shell diagrams or the amplituhedron. The goal of these notes is to study a new kind of structure which is potentially satisfied by amplitudes in all three situations. To motivate it, consider first the case of the dilogarithm function, defined for |z| < 1 by the sum","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"11 1","pages":"453-556"},"PeriodicalIF":1.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423325","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 : 2017-01-01DOI: 10.4310/CNTP.2017.V11.N2.A1
K. Hikami, Jeremy Lovejoy
Every closed orientable 3-manifold can be constructed by surgery on a link in S. In the case of surgery along a torus knot, one obtains a Seifert fibered manifold. In this paper we consider three families of such manifolds and study their unified WittenReshetikhin-Turaev (WRT) invariants. Thanks to recent computation of the coefficients in the cyclotomic expansion of the colored Jones polynomial for (2, 2t+ 1)-torus knots, these WRT invariants can be neatly expressed as q-hypergeometric series which converge inside the unit disk. Using the Rosso-Jones formula and some rather non-standard techniques for Bailey pairs, we find Hecke-type formulas for these invariants. We also comment on their mock and quantum modularity.
{"title":"Hecke-type formulas for families of unified Witten-Reshetikhin-Turaev invariants","authors":"K. Hikami, Jeremy Lovejoy","doi":"10.4310/CNTP.2017.V11.N2.A1","DOIUrl":"https://doi.org/10.4310/CNTP.2017.V11.N2.A1","url":null,"abstract":"Every closed orientable 3-manifold can be constructed by surgery on a link in S. In the case of surgery along a torus knot, one obtains a Seifert fibered manifold. In this paper we consider three families of such manifolds and study their unified WittenReshetikhin-Turaev (WRT) invariants. Thanks to recent computation of the coefficients in the cyclotomic expansion of the colored Jones polynomial for (2, 2t+ 1)-torus knots, these WRT invariants can be neatly expressed as q-hypergeometric series which converge inside the unit disk. Using the Rosso-Jones formula and some rather non-standard techniques for Bailey pairs, we find Hecke-type formulas for these invariants. We also comment on their mock and quantum modularity.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"11 1","pages":"249-272"},"PeriodicalIF":1.9,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423265","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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