Pub Date : 2020-03-19DOI: 10.4310/ATMP.2022.v26.n5.a9
M. Rossi
The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called emph{framed} duality, so giving rise to a powerful method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.
{"title":"An extension of polar duality of toric varieties and its consequences in Mirror Symmetry","authors":"M. Rossi","doi":"10.4310/ATMP.2022.v26.n5.a9","DOIUrl":"https://doi.org/10.4310/ATMP.2022.v26.n5.a9","url":null,"abstract":"The present paper is dedicated to illustrating an extension of polar duality between Fano toric varieties to a more general duality, called emph{framed} duality, so giving rise to a powerful method of producing mirror partners of hypersurfaces and complete intersections in toric varieties, of any Kodaira dimension. In particular, the class of projective hypersurfaces and their mirror partners are studied in detail. Moreover, many connections with known Landau-Ginzburg mirror models, Homological Mirror Symmetry and Intrinsic Mirror Symmetry, are discussed.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"48 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73833962","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 : 2020-03-07DOI: 10.4310/atmp.2023.v27.n1.a3
T. Banks, W. Fischler
We revisit the construction of models of quantum gravity in d dimensional Minkowski space in terms of random tensor models, and correct some mistakes in our previous treatment of the subject. We find a large class of models in which the large impact parameter scattering scales with energy and impact parameter like Newton`s law. The scattering amplitudes in these models describe scattering of jets of particles, and also include amplitudes for the production of highly meta-stable states with all the parametric properties of black holes. These models have emergent energy, momentum and angular conservation laws, despite being based on time dependent Hamiltonians. The scattering amplitudes in which no intermediate black holes are produced have a time-ordered Feynman diagram space-time structure: local interaction vertices connected by propagation of free particles (really Sterman-Weinberg jets of particles). However, there are also amplitudes where jets collide to form large meta-stable objects, with all the scaling properties of black holes: energy, entropy and temperature, as well as the characteristic time scale for the decay of perturbations. We generalize the conjecture of Sekino and Susskind, to claim that all of these models are fast scramblers. The rationale for this claim is that the interactions are invariant under fuzzy subgroups of the group of volume preserving diffeomorphisms, so that they are highly non-local on the holographic screen. We review how this formalism resolves the Firewall Paradox.
{"title":"Holographic space-time, Newton’s law, and the dynamics of horizons","authors":"T. Banks, W. Fischler","doi":"10.4310/atmp.2023.v27.n1.a3","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n1.a3","url":null,"abstract":"We revisit the construction of models of quantum gravity in d dimensional Minkowski space in terms of random tensor models, and correct some mistakes in our previous treatment of the subject. We find a large class of models in which the large impact parameter scattering scales with energy and impact parameter like Newton`s law. The scattering amplitudes in these models describe scattering of jets of particles, and also include amplitudes for the production of highly meta-stable states with all the parametric properties of black holes. These models have emergent energy, momentum and angular conservation laws, despite being based on time dependent Hamiltonians. The scattering amplitudes in which no intermediate black holes are produced have a time-ordered Feynman diagram space-time structure: local interaction vertices connected by propagation of free particles (really Sterman-Weinberg jets of particles). However, there are also amplitudes where jets collide to form large meta-stable objects, with all the scaling properties of black holes: energy, entropy and temperature, as well as the characteristic time scale for the decay of perturbations. We generalize the conjecture of Sekino and Susskind, to claim that all of these models are fast scramblers. The rationale for this claim is that the interactions are invariant under fuzzy subgroups of the group of volume preserving diffeomorphisms, so that they are highly non-local on the holographic screen. We review how this formalism resolves the Firewall Paradox.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"37 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74824865","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 : 2020-03-04DOI: 10.4310/ATMP.2022.v26.n5.a6
Y. Kimura
We discuss a method for classifying the singularity types of 1/2 Calabi-Yau 3-folds, a family of rational elliptic 3-folds introduced in a previous study in relation to various U(1) factors in 6D F-theory models. A projective dual pair of del Pezzo manifolds recently studied by Mukai is used to analyze the singularity types. In particular, we studied the maximal rank seven singularity types of 1/2 Calabi-Yau 3-folds. The structures of the singular fibers are analyzed using blow-ups. Double covers of the 1/2 Calabi-Yau 3-folds yield elliptic Calabi-Yau 3-folds and applications to six-dimensional $N = 1$ F-theory on the Calabi-Yau 3-folds are also discussed. The deduced singular fibers have applications in studying the gauge groups formed in 6D F-theory compactifications. The blow-up methods used to analyze the singular fibers and sections utilized in this research might have applications in studying the U(1) factors and hypermultiplets charged under U(1) in 6D F-theory.
{"title":"Extremal $1/2$ Calabi–Yau $3$-folds and six-dimensional F-theory applications","authors":"Y. Kimura","doi":"10.4310/ATMP.2022.v26.n5.a6","DOIUrl":"https://doi.org/10.4310/ATMP.2022.v26.n5.a6","url":null,"abstract":"We discuss a method for classifying the singularity types of 1/2 Calabi-Yau 3-folds, a family of rational elliptic 3-folds introduced in a previous study in relation to various U(1) factors in 6D F-theory models. A projective dual pair of del Pezzo manifolds recently studied by Mukai is used to analyze the singularity types. In particular, we studied the maximal rank seven singularity types of 1/2 Calabi-Yau 3-folds. The structures of the singular fibers are analyzed using blow-ups. Double covers of the 1/2 Calabi-Yau 3-folds yield elliptic Calabi-Yau 3-folds and applications to six-dimensional $N = 1$ F-theory on the Calabi-Yau 3-folds are also discussed. The deduced singular fibers have applications in studying the gauge groups formed in 6D F-theory compactifications. The blow-up methods used to analyze the singular fibers and sections utilized in this research might have applications in studying the U(1) factors and hypermultiplets charged under U(1) in 6D F-theory.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":" 9","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72382604","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 : 2020-01-29DOI: 10.4310/ATMP.2022.v26.n1.a1
Kwokwai Chan, N. Leung, Qin Li
For a K"ahler manifold $X$ equipped with a prequantum line bundle $L$, we give a geometric construction of a family of representations of the Berezin-Toeplitz deformation quantization algebra $(C^infty(X)[[hbar]],star_{BT})$ parametrized by points $z_0 in X$. The key idea is to use peak sections to suitably localize the Hilbert spaces $H^{0}left( X,L^{otimes m}right) $ around $z_{0}$.
对于具有前量子线束$L$的Kähler流形$X$，我们给出了由点$z_0 in X$参数化的Berezin-Toeplitz变形量化代数$(C^infty(X)[[hbar]],star_{BT})$的一族表示的几何构造。关键思想是使用峰值截面来适当地定位$z_{0}$周围的希尔伯特空间$H^{0}left( X,L^{otimes m}right) $。
{"title":"A geometric construction of representations of the Berezin–Toeplitz quantization","authors":"Kwokwai Chan, N. Leung, Qin Li","doi":"10.4310/ATMP.2022.v26.n1.a1","DOIUrl":"https://doi.org/10.4310/ATMP.2022.v26.n1.a1","url":null,"abstract":"For a K\"ahler manifold $X$ equipped with a prequantum line bundle $L$, we give a geometric construction of a family of representations of the Berezin-Toeplitz deformation quantization algebra $(C^infty(X)[[hbar]],star_{BT})$ parametrized by points $z_0 in X$. The key idea is to use peak sections to suitably localize the Hilbert spaces $H^{0}left( X,L^{otimes m}right) $ around $z_{0}$.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"284 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73312469","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 : 2020-01-23DOI: 10.4310/ATMP.2022.v26.n3.a4
Matthias Ludewig, Guo Chuan Thiang
We prove that a spectral gap-filling phenomenon occurs whenever a Hamiltonian operator encounters a coarse index obstruction upon compression to a domain with boundary. Furthermore, the gap-filling spectra contribute to quantised current channels, which follow and are localised at the possibly complicated boundary. This index obstruction is shown to be insensitive to deformations of the domain boundary, so the phenomenon is generic for magnetic Laplacians modelling quantum Hall systems and Chern topological insulators. A key construction is a quasi-equivariant version of Roe's algebra of locally compact finite propagation operators.
{"title":"Cobordism invariance of topological edge-following states","authors":"Matthias Ludewig, Guo Chuan Thiang","doi":"10.4310/ATMP.2022.v26.n3.a4","DOIUrl":"https://doi.org/10.4310/ATMP.2022.v26.n3.a4","url":null,"abstract":"We prove that a spectral gap-filling phenomenon occurs whenever a Hamiltonian operator encounters a coarse index obstruction upon compression to a domain with boundary. Furthermore, the gap-filling spectra contribute to quantised current channels, which follow and are localised at the possibly complicated boundary. This index obstruction is shown to be insensitive to deformations of the domain boundary, so the phenomenon is generic for magnetic Laplacians modelling quantum Hall systems and Chern topological insulators. A key construction is a quasi-equivariant version of Roe's algebra of locally compact finite propagation operators.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"26 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81466835","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 : 2020-01-02DOI: 10.4310/ATMP.2021.v25.n5.a3
F. Han, V. Mathai
In this paper, we extend the T-duality Hori maps in [arXiv:hep-th/0306062], inducing isomorphisms of twisted cohomologies on T-dual circle bundles, to graded Hori maps and show that they induce isomorphisms of two-variable series of twisted cohomologies on the T-dual circle bundles, preserving Jacobi form properties. The composition of the graded Hori map with its dual equals the Euler operator. We also construct Witten gerbe modules arising from gerbe modules and show that their graded twisted Chern characters are Jacobi forms under an anomaly vanishing condition on gerbe modules, thereby giving interesting examples.
{"title":"$T$-duality, Jacobi forms and Witten–Gerbe modules","authors":"F. Han, V. Mathai","doi":"10.4310/ATMP.2021.v25.n5.a3","DOIUrl":"https://doi.org/10.4310/ATMP.2021.v25.n5.a3","url":null,"abstract":"In this paper, we extend the T-duality Hori maps in [arXiv:hep-th/0306062], inducing isomorphisms of twisted cohomologies on T-dual circle bundles, to graded Hori maps and show that they induce isomorphisms of two-variable series of twisted cohomologies on the T-dual circle bundles, preserving Jacobi form properties. The composition of the graded Hori map with its dual equals the Euler operator. We also construct Witten gerbe modules arising from gerbe modules and show that their graded twisted Chern characters are Jacobi forms under an anomaly vanishing condition on gerbe modules, thereby giving interesting examples.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"91 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79483528","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 : 2020-01-01DOI: 10.4310/atmp.2020.v24.n3.a4
Darrin D. Frey, Tom Rudelius
{"title":"6D SCFTs and the classification of homomorphisms $Gamma_{ADE} to E_8$","authors":"Darrin D. Frey, Tom Rudelius","doi":"10.4310/atmp.2020.v24.n3.a4","DOIUrl":"https://doi.org/10.4310/atmp.2020.v24.n3.a4","url":null,"abstract":"","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"9 1","pages":"709-756"},"PeriodicalIF":1.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72772881","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 : 2020-01-01DOI: 10.4310/atmp.2020.v24.n2.a3
Frédéric Hélein
For any positive integer n and any Lie group G, given a definite symmetric bilinear form on R n and an Ad-invariant scalar product on the Lie algebra of G, we construct a variational problem on fields defined on an arbitrary (n + dimG)-dimensional manifold Y. We show that, if G is compact and simply connected, any global solution of the Euler-Lagrange equations leads to identify Y with the total space of a principal bundle over an n-dimensional manifold X. Moreover X is automatically endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein-Yang-Mills system equation with a cosmological constant.
{"title":"A variational principle for Kaluza–Klein types theories","authors":"Frédéric Hélein","doi":"10.4310/atmp.2020.v24.n2.a3","DOIUrl":"https://doi.org/10.4310/atmp.2020.v24.n2.a3","url":null,"abstract":"For any positive integer n and any Lie group G, given a definite symmetric bilinear form on R n and an Ad-invariant scalar product on the Lie algebra of G, we construct a variational problem on fields defined on an arbitrary (n + dimG)-dimensional manifold Y. We show that, if G is compact and simply connected, any global solution of the Euler-Lagrange equations leads to identify Y with the total space of a principal bundle over an n-dimensional manifold X. Moreover X is automatically endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein-Yang-Mills system equation with a cosmological constant.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"27 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72774896","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 : 2019-12-31DOI: 10.4310/atmp.2022.v26.n1.a2
Cyril Closset, M. Zotto
We study the spectrum of BPS particles on the Coulomb branch of five-dimensional superconformal field theories (5d SCFTs) compactified on a circle. By engineering these theories in M-theory on ${mathbf X} times S^1 $, for ${mathbf X}$ an isolated Calabi-Yau threefold singularity, we naturally identify the BPS category of the 5d theory on a circle with the derived category of coherent sheaves on a resolution of ${mathbf X}$. It follows that the BPS spectrum can be studied in terms of 5d BPS quivers, which are the fractional-brane quivers for the singularity ${mathbf X}$. 5d BPS quivers generalize the well-studied 4d BPS quivers for 4d $mathcal{N}{=}2$ gauge theories that can be obtained from ${mathbf X}$ in so-called geometric engineering limits. We study the interplay between 4d and 5d BPS quivers in detail. We particularly focus on examples when ${mathbf X}$ is a toric singularity, in which case the 5d BPS quiver is given in terms of a brane tiling. For instance, the well-studied $Y^{p,q}$ brane tiling gives a 5d BPS quiver for the $SU(p)_q$ 5d gauge theory. We present a conjecture about the structure of the BPS spectra of a wide class of models, which we test in the simple case of the 5d $SU(2)_0$ theory (more precisely, the $E_1$ SCFT). We also argue that 5d UV dualities can be realized in terms of mutation sequences on the BPS quivers, which are in turn interpreted as autoequivalences of the BPS category.
{"title":"On 5D SCFTs and their BPS quivers. Part I: B-branes and brane tilings","authors":"Cyril Closset, M. Zotto","doi":"10.4310/atmp.2022.v26.n1.a2","DOIUrl":"https://doi.org/10.4310/atmp.2022.v26.n1.a2","url":null,"abstract":"We study the spectrum of BPS particles on the Coulomb branch of five-dimensional superconformal field theories (5d SCFTs) compactified on a circle. By engineering these theories in M-theory on ${mathbf X} times S^1 $, for ${mathbf X}$ an isolated Calabi-Yau threefold singularity, we naturally identify the BPS category of the 5d theory on a circle with the derived category of coherent sheaves on a resolution of ${mathbf X}$. It follows that the BPS spectrum can be studied in terms of 5d BPS quivers, which are the fractional-brane quivers for the singularity ${mathbf X}$. 5d BPS quivers generalize the well-studied 4d BPS quivers for 4d $mathcal{N}{=}2$ gauge theories that can be obtained from ${mathbf X}$ in so-called geometric engineering limits. We study the interplay between 4d and 5d BPS quivers in detail. We particularly focus on examples when ${mathbf X}$ is a toric singularity, in which case the 5d BPS quiver is given in terms of a brane tiling. For instance, the well-studied $Y^{p,q}$ brane tiling gives a 5d BPS quiver for the $SU(p)_q$ 5d gauge theory. We present a conjecture about the structure of the BPS spectra of a wide class of models, which we test in the simple case of the 5d $SU(2)_0$ theory (more precisely, the $E_1$ SCFT). We also argue that 5d UV dualities can be realized in terms of mutation sequences on the BPS quivers, which are in turn interpreted as autoequivalences of the BPS category.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"3 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75388607","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 : 2019-12-30DOI: 10.4310/ATMP.2021.v25.n7.a3
F. Finster, André Platzer
Asymptotically flat static causal fermion systems are introduced. Their total mass is defined as a limit of surface layer integrals which compare the measures describing the asymptotically flat spacetime and a vacuum spacetime near spatial infinity. Our definition does not involve any regularity assumptions; it even applies to singular or generalized"quantum"spacetimes. A positive mass theorem is proven. Our methods and results explain why and how the causal action principle incorporates the nonlinear effects of gravity for static systems.
{"title":"A positive mass theorem for static causal fermion systems","authors":"F. Finster, André Platzer","doi":"10.4310/ATMP.2021.v25.n7.a3","DOIUrl":"https://doi.org/10.4310/ATMP.2021.v25.n7.a3","url":null,"abstract":"Asymptotically flat static causal fermion systems are introduced. Their total mass is defined as a limit of surface layer integrals which compare the measures describing the asymptotically flat spacetime and a vacuum spacetime near spatial infinity. Our definition does not involve any regularity assumptions; it even applies to singular or generalized\"quantum\"spacetimes. A positive mass theorem is proven. Our methods and results explain why and how the causal action principle incorporates the nonlinear effects of gravity for static systems.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"32 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80303735","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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