Pub Date : 2021-01-02DOI: 10.4310/atmp.2022.v26.n3.a5
Jean de Dieu Maniraguha, K. Marciniak, C'elestin Kurujyibwami
In this paper we discuss two canonical transformations that turn St"{a}ckel separable Hamiltonians of Benenti type into polynomial form: transformation to Vi`ete coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of St"{a}ckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Vi`ete and Newton coordinates.
{"title":"Transforming Stäckel Hamiltonians of Benenti type to polynomial form","authors":"Jean de Dieu Maniraguha, K. Marciniak, C'elestin Kurujyibwami","doi":"10.4310/atmp.2022.v26.n3.a5","DOIUrl":"https://doi.org/10.4310/atmp.2022.v26.n3.a5","url":null,"abstract":"In this paper we discuss two canonical transformations that turn St\"{a}ckel separable Hamiltonians of Benenti type into polynomial form: transformation to Vi`ete coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of St\"{a}ckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Vi`ete and Newton coordinates.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88381899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","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/atmp.2021.v25.n6.a3
S. Gates, Yangrui Hu, S.-N. Hazel Mak
{"title":"Advening to adynkrafields: Young tableaux to component fields of the 10D, $mathcal{N}=1$ scalar superfield","authors":"S. Gates, Yangrui Hu, S.-N. Hazel Mak","doi":"10.4310/atmp.2021.v25.n6.a3","DOIUrl":"https://doi.org/10.4310/atmp.2021.v25.n6.a3","url":null,"abstract":"","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81652124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","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/atmp.2021.v25.n3.a1
L. Alarie-Vézina, L. Lapointe, P. Mathieu
{"title":"The $mathcal{N}=2$ supersymmetric Calogero–Sutherland model and its eigenfunctions","authors":"L. Alarie-Vézina, L. Lapointe, P. Mathieu","doi":"10.4310/atmp.2021.v25.n3.a1","DOIUrl":"https://doi.org/10.4310/atmp.2021.v25.n3.a1","url":null,"abstract":"","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87018692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-21DOI: 10.4310/atmp.2022.v26.n3.a7
H. Schenck, M. Stillman, Beihui Yuan
A projectively normal Calabi-Yau threefold $X subseteq mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $mathrm{codim}(I)=4=mathrm{reg}(I)$, and that exactly 8 of these occur for smooth irreducible nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of $X$ with $h^{p,q}(X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties. A key tool in our approach is the use of inverse systems to identify possible betti tables for $X$.
{"title":"Calabi–Yau threefolds in $mathbb{P}^n$ and Gorenstein rings","authors":"H. Schenck, M. Stillman, Beihui Yuan","doi":"10.4310/atmp.2022.v26.n3.a7","DOIUrl":"https://doi.org/10.4310/atmp.2022.v26.n3.a7","url":null,"abstract":"A projectively normal Calabi-Yau threefold $X subseteq mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $mathrm{codim}(I)=4=mathrm{reg}(I)$, and that exactly 8 of these occur for smooth irreducible nondegenerate threefolds. We investigate the situation in codimension five or more, obtaining examples of $X$ with $h^{p,q}(X)$ not among those appearing for $I_X$ of lower codimension or as complete intersections in toric Fano varieties. A key tool in our approach is the use of inverse systems to identify possible betti tables for $X$.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91286345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-16DOI: 10.4310/atmp.2021.v25.n6.a5
J. Lenells, J. Roussillon
We study the recently introduced family of confluent Virasoro fusion kernels $mathcal{C}_k(b,boldsymbol{theta},sigma_s,nu)$. We study their eigenfunction properties and show that they can be viewed as non-polynomial generalizations of both the continuous dual $q$-Hahn and the big $q$-Jacobi polynomials. More precisely, we prove that: (i) $mathcal{C}_k$ is a joint eigenfunction of four different difference operators for any positive integer $k$, (ii) $mathcal{C}_k$ degenerates to the continuous dual $q$-Hahn polynomials when $nu$ is suitably discretized, and (iii) $mathcal{C}_k$ degenerates to the big $q$-Jacobi polynomials when $sigma_s$ is suitably discretized. These observations lead us to propose the existence of a non-polynomial generalization of the $q$-Askey scheme. The top member of this non-polynomial scheme is the Virasoro fusion kernel (or, equivalently, Ruijsenaars' hypergeometric function), and its first confluence is given by the $mathcal{C}_k$.
{"title":"The family of confluent Virasoro fusion kernels and a non-polynomial $q$-Askey scheme","authors":"J. Lenells, J. Roussillon","doi":"10.4310/atmp.2021.v25.n6.a5","DOIUrl":"https://doi.org/10.4310/atmp.2021.v25.n6.a5","url":null,"abstract":"We study the recently introduced family of confluent Virasoro fusion kernels $mathcal{C}_k(b,boldsymbol{theta},sigma_s,nu)$. We study their eigenfunction properties and show that they can be viewed as non-polynomial generalizations of both the continuous dual $q$-Hahn and the big $q$-Jacobi polynomials. More precisely, we prove that: (i) $mathcal{C}_k$ is a joint eigenfunction of four different difference operators for any positive integer $k$, (ii) $mathcal{C}_k$ degenerates to the continuous dual $q$-Hahn polynomials when $nu$ is suitably discretized, and (iii) $mathcal{C}_k$ degenerates to the big $q$-Jacobi polynomials when $sigma_s$ is suitably discretized. These observations lead us to propose the existence of a non-polynomial generalization of the $q$-Askey scheme. The top member of this non-polynomial scheme is the Virasoro fusion kernel (or, equivalently, Ruijsenaars' hypergeometric function), and its first confluence is given by the $mathcal{C}_k$.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86769731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-11DOI: 10.4310/ATMP.2022.v26.n5.a5
H. Iritani
Recently Bonisch-Fischbach-Klemm-Nega-Safari discovered, via numerical computation, that the leading asymptotics of the l-loop Banana Feynman amplitude at the large complex structure limit can be described by the Gamma class of a degree (1,...,1) Fano hypersurface F in (P^1)^{l+1}. We confirm this observation by using a Gamma-conjecture type result for F.
最近，Bonisch-Fischbach-Klemm-Nega-Safari通过数值计算发现，l-环Banana Feynman振幅在大复杂结构极限处的领先渐近性可以用(1，…，1)次Fano超曲面F in (P^1)^{1 +1}的Gamma类来描述。我们用F的伽玛猜想型结果证实了这一观察。
{"title":"Asymptotics of the Banana Feynman amplitudes at the large complex structure limit","authors":"H. Iritani","doi":"10.4310/ATMP.2022.v26.n5.a5","DOIUrl":"https://doi.org/10.4310/ATMP.2022.v26.n5.a5","url":null,"abstract":"Recently Bonisch-Fischbach-Klemm-Nega-Safari discovered, via numerical computation, that the leading asymptotics of the l-loop Banana Feynman amplitude at the large complex structure limit can be described by the Gamma class of a degree (1,...,1) Fano hypersurface F in (P^1)^{l+1}. We confirm this observation by using a Gamma-conjecture type result for F.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90500969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-21DOI: 10.4310/ATMP.2022.v26.n4.a1
P. Dunin-Barkowski, M. Kazarian, A. Popolitov, S. Shadrin, A. Sleptsov
We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).
{"title":"Topological recursion for the extended Ooguri–Vafa partition function of colored HOMFLY-PT polynomials of torus knots","authors":"P. Dunin-Barkowski, M. Kazarian, A. Popolitov, S. Shadrin, A. Sleptsov","doi":"10.4310/ATMP.2022.v26.n4.a1","DOIUrl":"https://doi.org/10.4310/ATMP.2022.v26.n4.a1","url":null,"abstract":"We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. \u0000We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type).","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74465737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-14DOI: 10.4310/atmp.2022.v26.n3.a1
L. Bieri
We investigate the Einstein vacuum equations as well as the Einstein-null fluid equations describing neutrino radiation. We find new structures in gravitational waves and memory for asymptotically-flat spacetimes of slow decay. It has been known that for stronger decay of the data, including data being stationary outside a compact set, gravitational wave memory is finite and of electric parity only. In this article, we investigate general spacetimes that are asymptotically flat in a rough sense. That is, the decay of the data to Minkowski space towards infinity is very slow. As a main new feature, we prove that there exists diverging magnetic memory sourced by the magnetic part of the curvature tensor (a) in the Einstein vacuum and (b) in the Einstein-null-fluid equations. The magnetic memory occurs naturally in the Einstein vacuum setting (a) of pure gravity. In case (b), in the ultimate class of solutions, the magnetic memory contains also a curl term from the energy-momentum tensor for neutrinos also diverging at the aforementioned rate. The electric memory diverges too, it is generated by the electric part of the curvature tensor and in the Einstein-null-fluid situation also by the corresponding energy-momentum component. In addition, we find a panorama of finer structures in these manifolds. Some of these manifest themselves as additional contributions to both electric and magnetic memory. Our theorems hold for any type of matter or energy coupled to the Einstein equations as long as the data decays slowly towards infinity and other conditions are satisfied. The new results have a multitude of applications ranging from mathematical general relativity to gravitational wave astrophysics, detecting dark matter and other topics in physics.
{"title":"New structures in gravitational radiation","authors":"L. Bieri","doi":"10.4310/atmp.2022.v26.n3.a1","DOIUrl":"https://doi.org/10.4310/atmp.2022.v26.n3.a1","url":null,"abstract":"We investigate the Einstein vacuum equations as well as the Einstein-null fluid equations describing neutrino radiation. We find new structures in gravitational waves and memory for asymptotically-flat spacetimes of slow decay. It has been known that for stronger decay of the data, including data being stationary outside a compact set, gravitational wave memory is finite and of electric parity only. In this article, we investigate general spacetimes that are asymptotically flat in a rough sense. That is, the decay of the data to Minkowski space towards infinity is very slow. As a main new feature, we prove that there exists diverging magnetic memory sourced by the magnetic part of the curvature tensor (a) in the Einstein vacuum and (b) in the Einstein-null-fluid equations. The magnetic memory occurs naturally in the Einstein vacuum setting (a) of pure gravity. In case (b), in the ultimate class of solutions, the magnetic memory contains also a curl term from the energy-momentum tensor for neutrinos also diverging at the aforementioned rate. The electric memory diverges too, it is generated by the electric part of the curvature tensor and in the Einstein-null-fluid situation also by the corresponding energy-momentum component. In addition, we find a panorama of finer structures in these manifolds. Some of these manifest themselves as additional contributions to both electric and magnetic memory. Our theorems hold for any type of matter or energy coupled to the Einstein equations as long as the data decays slowly towards infinity and other conditions are satisfied. The new results have a multitude of applications ranging from mathematical general relativity to gravitational wave astrophysics, detecting dark matter and other topics in physics.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86870009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-18DOI: 10.4310/ATMP.2021.v25.n8.a4
Christian Lange, A. Lytchak, Clemens Samann
We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $alpha$-Holder continuous Lorentzian metric admit a $mathcal{C}^{1,frac{alpha}{4}}$-parametrization.
{"title":"Lorentz Meets Lipschitz","authors":"Christian Lange, A. Lytchak, Clemens Samann","doi":"10.4310/ATMP.2021.v25.n8.a4","DOIUrl":"https://doi.org/10.4310/ATMP.2021.v25.n8.a4","url":null,"abstract":"We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $alpha$-Holder continuous Lorentzian metric admit a $mathcal{C}^{1,frac{alpha}{4}}$-parametrization.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75800809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-10DOI: 10.4310/ATMP.2022.v26.n6.a5
Yichen Huang
Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translationally invariant systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by $1/O(N)$, where $N$ is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is tight in systems satisfying the eigenstate thermalization hypothesis.
{"title":"Convergence of eigenstate expectation values with system size","authors":"Yichen Huang","doi":"10.4310/ATMP.2022.v26.n6.a5","DOIUrl":"https://doi.org/10.4310/ATMP.2022.v26.n6.a5","url":null,"abstract":"Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translationally invariant systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by $1/O(N)$, where $N$ is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is tight in systems satisfying the eigenstate thermalization hypothesis.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75940919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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