Pub Date : 2021-06-05DOI: 10.4310/CNTP.2019.V13.N1.A3
Wei Luo, Shengmao Zhu
The Labastida-Marin˜o-Ooguri-Vafa (LMOV) invariants are the open string BPS invariants which are expected to be integers based on the string duality conjecture from M-theory. Several explicit formulae of LMOV invariants for framed unknot have been obtained in the literature. In this paper, we present a unified method to deal with the integrality of such explicit formulae. Furthermore, we also prove the integrality of certain LMOV invariants for framed unknot in higher genera.
{"title":"Integrality of the LMOV invariants for framed unknot","authors":"Wei Luo, Shengmao Zhu","doi":"10.4310/CNTP.2019.V13.N1.A3","DOIUrl":"https://doi.org/10.4310/CNTP.2019.V13.N1.A3","url":null,"abstract":"The Labastida-Marin˜o-Ooguri-Vafa (LMOV) invariants are the open string BPS invariants which are expected to be integers based on the string duality conjecture from M-theory. Several explicit formulae of LMOV invariants for framed unknot have been obtained in the literature. In this paper, we present a unified method to deal with the integrality of such explicit formulae. Furthermore, we also prove the integrality of certain LMOV invariants for framed unknot in higher genera.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47521152","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 : 2021-06-04DOI: 10.4310/cntp.2023.v17.n1.a1
Hikaru Hirano, Junhyeong Kim, M. 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 analogues for $k$ and $S$ of the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten partition function in (2+1)-dimensional Chern-Simons TQFT. We show some basic and functorial properties of those arithmetic analogues. Finally we show decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions.
{"title":"On arithmetic Dijkgraaf–Witten theory","authors":"Hikaru Hirano, Junhyeong Kim, M. Morishita","doi":"10.4310/cntp.2023.v17.n1.a1","DOIUrl":"https://doi.org/10.4310/cntp.2023.v17.n1.a1","url":null,"abstract":"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 analogues for $k$ and $S$ of the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten partition function in (2+1)-dimensional Chern-Simons TQFT. We show some basic and functorial properties of those arithmetic analogues. Finally we show decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48289179","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 : 2021-05-11DOI: 10.4310/CNTP.2022.v16.n3.a3
M. Borinsky, O. 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 φ 4 theory and to order five in six-dimensional φ 3 theory. In this article we present the theory of graphical functions in even dimensions ≥ 4 with detailed reviews of known properties and full proofs whenever possible.
{"title":"Graphical functions in even dimensions","authors":"M. Borinsky, O. Schnetz","doi":"10.4310/CNTP.2022.v16.n3.a3","DOIUrl":"https://doi.org/10.4310/CNTP.2022.v16.n3.a3","url":null,"abstract":". 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 φ 4 theory and to order five in six-dimensional φ 3 theory. In this article we present the theory of graphical functions in even dimensions ≥ 4 with detailed reviews of known properties and full proofs whenever possible.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46554856","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 : 2021-05-06DOI: 10.4310/cntp.2022.v16.n4.a3
A. Grassi, T. 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 for the generators of the Mordell-Weil group and their induced height pairing. Compactification of F-theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors. We compute the massless matter spectrum. In particular, we show that the charged singlet matter need not reside at enhancement loci of Type $I_2$, as previously believed. We relate the multiplicities of the massless spectrum to genus-zero Gopakumar-Vafa invariants and other geometric quantities of the Calabi-Yau. We show that the gravitational and abelian anomaly cancellation conditions are satisfied. We prove a Geometric Anomaly Cancellation equation and we deduce birational equivalence for the quantities in the spectrum. We explicitly describe a Weierstrass model over $mathbb P^2$ of the Calabi-Yau threefolds as a log canonical model and compare it to a construction by Elkies and classical results of Burkhardt.
{"title":"Elliptic threefolds with high Mordell–Weil rank","authors":"A. Grassi, T. Weigand","doi":"10.4310/cntp.2022.v16.n4.a3","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n4.a3","url":null,"abstract":"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 for the generators of the Mordell-Weil group and their induced height pairing. Compactification of F-theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors. We compute the massless matter spectrum. In particular, we show that the charged singlet matter need not reside at enhancement loci of Type $I_2$, as previously believed. We relate the multiplicities of the massless spectrum to genus-zero Gopakumar-Vafa invariants and other geometric quantities of the Calabi-Yau. We show that the gravitational and abelian anomaly cancellation conditions are satisfied. We prove a Geometric Anomaly Cancellation equation and we deduce birational equivalence for the quantities in the spectrum. We explicitly describe a Weierstrass model over $mathbb P^2$ of the Calabi-Yau threefolds as a log canonical model and compare it to a construction by Elkies and classical results of Burkhardt.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48654758","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 : 2021-05-06DOI: 10.4310/cntp.2022.v16.n4.a6
N. Leung, Y. Yau
On a compact K"ahler manifold $X$, Toeplitz operators determine a deformation quantization $(operatorname{C}^infty(X, mathbb{C})[[hbar]], star)$ 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]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $hbar to 0^+$.
在紧致Kähler流形$X$上，Toeplitz算子确定了一个变形量化$(operatorname{C}^infty(X, mathbb{C})[[hbar]], star)$，其中变量[10]相对于横向复极化$T^{1, 0}X, T^{0, 1}X$的分离为$hbar to 0^+$[15]。对于具有横向非奇异实极化[13]的紧辛流形证明了类似的命题。本文建立了紧环辛流形$X$上的横向奇异实极化的类似结果。由于环奇点的存在，我们的Toeplitz算子的半形式校正和局部化是必不可少的。通过范数估计，我们证明这些Toeplitz算子确定$X$上的明星产品为$hbar to 0^+$。
{"title":"Berezin–Toeplitz quantization in real polarizations with toric singularities","authors":"N. Leung, Y. Yau","doi":"10.4310/cntp.2022.v16.n4.a6","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n4.a6","url":null,"abstract":"On a compact K\"ahler manifold $X$, Toeplitz operators determine a deformation quantization $(operatorname{C}^infty(X, mathbb{C})[[hbar]], star)$ 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]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $hbar to 0^+$.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46945315","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 : 2021-01-01DOI: 10.4310/cntp.2021.v15.n1.a2
G. Mason, K. Nagatomo, Yuichi Sakai
{"title":"Vertex operator algebras of rank $2$: The Mathur–Mukhi–Sen theorem revisited","authors":"G. Mason, K. Nagatomo, Yuichi Sakai","doi":"10.4310/cntp.2021.v15.n1.a2","DOIUrl":"https://doi.org/10.4310/cntp.2021.v15.n1.a2","url":null,"abstract":"","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"15 1","pages":"59-90"},"PeriodicalIF":1.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70423039","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 : 2020-12-07DOI: 10.4310/CNTP.2021.v15.n4.a1
Yajun Zhou
Through algebraic manipulations on Wronskian 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 Wronskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove's differential operators, and compute the Betti intersection pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst--Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne's conjectures for critical values of motivic $L$-functions.
{"title":"Wrońskian algebra and Broadhurst–Roberts quadratic relations","authors":"Yajun Zhou","doi":"10.4310/CNTP.2021.v15.n4.a1","DOIUrl":"https://doi.org/10.4310/CNTP.2021.v15.n4.a1","url":null,"abstract":"Through algebraic manipulations on Wronskian 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 Wronskian framework, we reinterpret the de Rham intersection pairing through polynomial coefficients in Vanhove's differential operators, and compute the Betti intersection pairing via linear sum rules for on-shell and off-shell Feynman diagrams at threshold momenta. From the ideal generated by Broadhurst--Roberts quadratic relations, we derive new non-linear sum rules for on-shell Feynman diagrams, including an infinite family of determinant identities that are compatible with Deligne's conjectures for critical values of motivic $L$-functions.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2020-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45095162","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 : 2020-10-02DOI: 10.4310/cntp.2022.v16.n1.a2
E. D'hoker, O. 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. We also derive a family of identities that apply to modular graph tensors of higher loop order.
{"title":"Identities among higher genus modular graph tensors","authors":"E. D'hoker, O. Schlotterer","doi":"10.4310/cntp.2022.v16.n1.a2","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n1.a2","url":null,"abstract":"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. We also derive a family of identities that apply to modular graph tensors of higher loop order.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2020-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46657233","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 : 2020-08-21DOI: 10.4310/CNTP.2023.v17.n1.a4
Marko Berghoff, D. Kreimer
We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on graph homology and on the appearance of cubical complexes when either reducing internal edges or when removing them by putting them on the massshell.
{"title":"Graph complexes and Feynman rules","authors":"Marko Berghoff, D. Kreimer","doi":"10.4310/CNTP.2023.v17.n1.a4","DOIUrl":"https://doi.org/10.4310/CNTP.2023.v17.n1.a4","url":null,"abstract":"We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on graph homology and on the appearance of cubical complexes when either reducing internal edges or when removing them by putting them on the massshell.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41810219","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 : 2020-08-08DOI: 10.4310/cntp.2022.v16.n4.a5
M. Kerr
S. Bloch and M. Vlasenko recently introduced a theory of emph{motivic Gamma functions}, given by periods of the Mellin transform of a geometric variation of Hodge structure, which they tie to the monodromy and asymptotic behavior of certain unipotent extensions of the variation. Here we further examine these Gamma functions and the related emph{Apery and Frobenius invariants} of a VHS, and establish a relationship to motivic cohomology and solutions to inhomogeneous Picard-Fuchs equations.
{"title":"Unipotent extensions and differential equations (after Bloch–Vlasenko)","authors":"M. Kerr","doi":"10.4310/cntp.2022.v16.n4.a5","DOIUrl":"https://doi.org/10.4310/cntp.2022.v16.n4.a5","url":null,"abstract":"S. Bloch and M. Vlasenko recently introduced a theory of emph{motivic Gamma functions}, given by periods of the Mellin transform of a geometric variation of Hodge structure, which they tie to the monodromy and asymptotic behavior of certain unipotent extensions of the variation. Here we further examine these Gamma functions and the related emph{Apery and Frobenius invariants} of a VHS, and establish a relationship to motivic cohomology and solutions to inhomogeneous Picard-Fuchs equations.","PeriodicalId":55616,"journal":{"name":"Communications in Number Theory and Physics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2020-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44943491","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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