Pub Date : 2024-07-16DOI: 10.4310/atmp.2023.v27.n6.a4
Danilo Polo Ojito
We study the propagation of currents along the interface of two $2$-$d$ magnetic systems, where one of them occupies the first quadrant of the plane. By considering the tight-binding approximation model and K-theory, we prove that, for an integer number that is given by the difference of two bulk topological invariants of each system, such interface currents are quantized. We further state the necessary conditions to produce corner states for these kinds of underlying systems, and we show that they have topologically protected asymptotic invariants.
我们研究了电流沿两个 $2$-$d$ 磁系界面的传播，其中一个磁系占据平面的第一象限。通过考虑紧约束近似模型和 K 理论，我们证明，对于由每个系统的两个体拓扑不变量之差给出的整数，这种界面电流是量子化的。我们进一步阐述了产生这类底层系统角态的必要条件，并证明它们具有拓扑保护渐近不变性。
{"title":"Interface currents and corner states in magnetic quarter-plane systems","authors":"Danilo Polo Ojito","doi":"10.4310/atmp.2023.v27.n6.a4","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n6.a4","url":null,"abstract":"We study the propagation of currents along the interface of two $2$-$d$ magnetic systems, where one of them occupies the first quadrant of the plane. By considering the tight-binding approximation model and K-theory, we prove that, for an integer number that is given by the difference of two bulk topological invariants of each system, such interface currents are quantized. We further state the necessary conditions to produce corner states for these kinds of underlying systems, and we show that they have topologically protected asymptotic invariants.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721381","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 : 2024-07-16DOI: 10.4310/atmp.2023.v27.n6.a1
Bernardo Araneda
Using a combination of techniques from conformal and complex geometry, we show the potentialization of 4‑dimensional closed Einstein–Weyl structures which are half-algebraically special and admit a “half-integrable” almost-complex structure. That is, we reduce the Einstein–Weyl equations to a single, conformally invariant, non-linear scalar equation, that we call the “conformal HH equation”, and we reconstruct the conformal structure (curvature and metric) from a solution to this equation. We show that the conformal metric is composed of: a conformally flat part, a conformally half-flat part related to certain “constants” of integration, and a potential part that encodes the full non-linear curvature, and that coincides in form with the Hertz potential from perturbation theory. We also study the potentialization of the Dirac–Weyl, Maxwell (with and without sources), and Yang–Mills systems. We show how to deal with the ordinary Einstein equations by using a simple trick. Our results give a conformally invariant, coordinatefree, generalization of the hyper-heavenly construction of Plebański and collaborators.
{"title":"Conformal geometry and half-integrable spacetimes","authors":"Bernardo Araneda","doi":"10.4310/atmp.2023.v27.n6.a1","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n6.a1","url":null,"abstract":"Using a combination of techniques from conformal and complex geometry, we show the potentialization of 4‑dimensional closed Einstein–Weyl structures which are half-algebraically special and admit a “half-integrable” almost-complex structure. That is, we reduce the Einstein–Weyl equations to a single, conformally invariant, non-linear scalar equation, that we call the “conformal HH equation”, and we reconstruct the conformal structure (curvature and metric) from a solution to this equation. We show that the conformal metric is composed of: a conformally flat part, a conformally half-flat part related to certain “constants” of integration, and a potential part that encodes the full non-linear curvature, and that coincides in form with the Hertz potential from perturbation theory. We also study the potentialization of the Dirac–Weyl, Maxwell (with and without sources), and Yang–Mills systems. We show how to deal with the ordinary Einstein equations by using a simple trick. Our results give a conformally invariant, coordinatefree, generalization of the hyper-heavenly construction of Plebański and collaborators.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722439","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 : 2024-07-16DOI: 10.4310/atmp.2023.v27.n6.a5
Jin Chen, Zhuo Chen, Wei Cui, Babak Haghighat
$defd{mathrm{d}}$ In this work, we study compactifications of $6d$ $(1, 0)$ SCFTs, in particular those of conformal matter type, on Kähler 4-manifolds. We show how this can be realized via wrapping M5 branes on $4$-cycles of non-compact Calabi–Yau fourfolds with ADE singularity in the fiber. Such compactifications lead to domain walls in $3d$ $mathcal{N} = 2$ theories which flow to $2d N = (0, 2)$ SCFTs. We compute the central charges of such $2d$ CFTs via $6d$ anomaly polynomials by employing a particular topological twist along the $4$-manifold. Moreover, we study compactifications on non-compact $4$-manifolds leading to coupled $3d$-$2d$ systems. We show how these can be glued together consistently to reproduce the central charge and anomaly polynomial obtained in the compact case. Lastly, we study concrete CFT proposals for some special cases.
{"title":"MSW-type compactifications of 6d $(1,0)$ SCFTs on 4-manifolds","authors":"Jin Chen, Zhuo Chen, Wei Cui, Babak Haghighat","doi":"10.4310/atmp.2023.v27.n6.a5","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n6.a5","url":null,"abstract":"$defd{mathrm{d}}$ In this work, we study compactifications of $6d$ $(1, 0)$ SCFTs, in particular those of conformal matter type, on Kähler 4-manifolds. We show how this can be realized via wrapping M5 branes on $4$-cycles of non-compact Calabi–Yau fourfolds with ADE singularity in the fiber. Such compactifications lead to domain walls in $3d$ $mathcal{N} = 2$ theories which flow to $2d N = (0, 2)$ SCFTs. We compute the central charges of such $2d$ CFTs via $6d$ anomaly polynomials by employing a particular topological twist along the $4$-manifold. Moreover, we study compactifications on non-compact $4$-manifolds leading to coupled $3d$-$2d$ systems. We show how these can be glued together consistently to reproduce the central charge and anomaly polynomial obtained in the compact case. Lastly, we study concrete CFT proposals for some special cases.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721380","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 : 2024-06-06DOI: 10.4310/atmp.2023.v27.n4.a3
Per Berglund, Giorgi Butbaia, Tristan Hüubsch, Vishnu Jejjala, Damián Mayorga Peña, Challenger Mishra, Justin Tan
$defSingX{mathrm{Sing}X}$Finding Ricci-flat (Calabi–Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi–Yau metric within a given Kähler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi–Yau threefolds. Using these Ricci-flat metric approximations for the Cefalú family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely spectral networks, which correctly approximate the topological characteristic of a Calabi–Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov–Yau type inequality $3c_2 geq c^2_1$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $chi (X setminus SingX) + 2 {lvert SingX rvert} = 24$ also holds for our numerical approximations.
{"title":"Machine-learned Calabi–Yau metrics and curvature","authors":"Per Berglund, Giorgi Butbaia, Tristan Hüubsch, Vishnu Jejjala, Damián Mayorga Peña, Challenger Mishra, Justin Tan","doi":"10.4310/atmp.2023.v27.n4.a3","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n4.a3","url":null,"abstract":"$defSingX{mathrm{Sing}X}$Finding Ricci-flat (Calabi–Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi–Yau metric within a given Kähler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi–Yau threefolds. Using these Ricci-flat metric approximations for the Cefalú family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely spectral networks, which correctly approximate the topological characteristic of a Calabi–Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov–Yau type inequality $3c_2 geq c^2_1$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $chi (X setminus SingX) + 2 {lvert SingX rvert} = 24$ also holds for our numerical approximations.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550607","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 : 2024-06-06DOI: 10.4310/atmp.2023.v27.n4.a2
Xiaoyue Sun, Junya Yagi
We define three families of quivers in which the braid relations of the symmetric group $S_n$ are realized by mutations and automorphisms. A sequence of eight braid moves on a reduced word for the longest element of $S_4$ yields three trivial cluster transformations with 8, 32 and 32 mutations. For each of these cluster transformations, a unitary operator representing a single braid move in a quantum mechanical system solves the tetrahedron equation. The solutions thus obtained are constructed from the noncompact quantum dilogarithm and can be identified with the partition functions of three-dimensional $mathcal{N} = 2$ supersymmetric gauge theories on a squashed three-sphere.
{"title":"Cluster transformations, the tetrahedron equation, and three-dimensional gauge theories","authors":"Xiaoyue Sun, Junya Yagi","doi":"10.4310/atmp.2023.v27.n4.a2","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n4.a2","url":null,"abstract":"We define three families of quivers in which the braid relations of the symmetric group $S_n$ are realized by mutations and automorphisms. A sequence of eight braid moves on a reduced word for the longest element of $S_4$ yields three trivial cluster transformations with 8, 32 and 32 mutations. For each of these cluster transformations, a unitary operator representing a single braid move in a quantum mechanical system solves the tetrahedron equation. The solutions thus obtained are constructed from the noncompact quantum dilogarithm and can be identified with the partition functions of three-dimensional $mathcal{N} = 2$ supersymmetric gauge theories on a squashed three-sphere.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552940","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 : 2024-06-06DOI: 10.4310/atmp.2023.v27.n4.a4
Leonardo García-Heveling
Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde and Sorkin proposed a way to avoid this dilemma by considering topology changing spacetimes constructed from Morse functions, where the metric is allowed to vanish at isolated points. They conjectured that these Morse spacetimes are causally continuous (hence quite well behaved), as long as the index of the Morse points is different from $1$ and $n-1$. In this paper, we prove a special case of this conjecture. We also argue, heuristically, that the original conjecture is actually false, and formulate a refined version of it.
{"title":"Topology change with Morse functions: progress on the Borde–Sorkin conjecture","authors":"Leonardo García-Heveling","doi":"10.4310/atmp.2023.v27.n4.a4","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n4.a4","url":null,"abstract":"Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde and Sorkin proposed a way to avoid this dilemma by considering topology changing spacetimes constructed from Morse functions, where the metric is allowed to vanish at isolated points. They conjectured that these Morse spacetimes are causally continuous (hence quite well behaved), as long as the index of the Morse points is different from $1$ and $n-1$. In this paper, we prove a special case of this conjecture. We also argue, heuristically, that the original conjecture is actually false, and formulate a refined version of it.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550608","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 : 2024-06-06DOI: 10.4310/atmp.2023.v27.n4.a5
Damien Gobin, Niky Kamran
$defD{mathcal{D}}$We compute a Green’s function giving rise to the solution of the Cauchy problem for the source-free Maxwell’s equations on a causal domain $D$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 times mathbb{S}^2 times mathbb{S}^3$, where $AdS^5$ denotes the simply connected $5$-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on $D$ and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on $mathbb{S}^3$. This gives rise to a sequence of inhomogeneous Cauchy problems governing the form-valued Fourier coefficients corresponding to the Fourier modes and involving operators related to the Hodge Laplacian on $AdS^5 times mathbb{S}^2$, which we solve explicitly by using Riesz distributions and the method of spherical means for differential forms. Finally we put together into the Fourier expansion on $mathbb{S}^3$ the modes obtained by this procedure, producing a $2$-form on $D subset AdS^5 times mathbb{S}^2 times mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell’s equations.
$defD{mathcal{D}}$我们计算了一个格林函数，它给出了无源麦克斯韦方程组在因果域$D$上的考奇问题的解，该因果域包含在洛伦兹流形$AdS^5 times mathbb{S}^2 times mathbb{S}^3$的大地法域中，其中$AdS^5$表示简单连接的5$维反德-西特时空。我们的方法是将原始的考奇问题表述为$D$上霍奇拉普拉奇的等效考奇问题，并根据$mathbb{S}^3$上霍奇拉普拉奇的特征形式寻求傅里叶展开形式的解。这就产生了一系列非均质考奇问题，它们支配着与傅里叶模式相对应的形式值傅里叶系数，并涉及与 $AdS^5 times mathbb{S}^2$ 上霍奇拉普拉斯相关的算子，我们利用里兹分布和微分形式的球面手段方法明确地解决了这些问题。最后，我们把通过这个过程得到的模合并到 $mathbb{S}^3$ 上的傅里叶展开中，在 $D subset AdS^5 times mathbb{S}^2 times mathbb{S}^3$ 上产生了一个 2$ 形式，我们证明它是麦克斯韦方程的原始考奇问题的解。
{"title":"A Green’s function for the source-free Maxwell Equations on $AdS^5 times S^2 times S^3$","authors":"Damien Gobin, Niky Kamran","doi":"10.4310/atmp.2023.v27.n4.a5","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n4.a5","url":null,"abstract":"$defD{mathcal{D}}$We compute a Green’s function giving rise to the solution of the Cauchy problem for the source-free Maxwell’s equations on a causal domain $D$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 times mathbb{S}^2 times mathbb{S}^3$, where $AdS^5$ denotes the simply connected $5$-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on $D$ and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on $mathbb{S}^3$. This gives rise to a sequence of inhomogeneous Cauchy problems governing the form-valued Fourier coefficients corresponding to the Fourier modes and involving operators related to the Hodge Laplacian on $AdS^5 times mathbb{S}^2$, which we solve explicitly by using Riesz distributions and the method of spherical means for differential forms. Finally we put together into the Fourier expansion on $mathbb{S}^3$ the modes obtained by this procedure, producing a $2$-form on $D subset AdS^5 times mathbb{S}^2 times mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell’s equations.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552956","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 : 2024-06-06DOI: 10.4310/atmp.2023.v27.n4.a6
Joakim Arnlind, Andreas Sykora
We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out that the structure of these modules depends in a crucial way on the properties of the function spaces. Moreover, for a class of subalgebras related to compact manifolds, we provide a construction procedure for the corresponding fuzzy spaces, i.e. sequences of finite dimensional modules of increasing dimension as the deformation parameter tends to zero, as well as infinite dimensional modules related to fuzzy non-compact spaces.
{"title":"On the construction of fuzzy spaces and modules over shift algebras","authors":"Joakim Arnlind, Andreas Sykora","doi":"10.4310/atmp.2023.v27.n4.a6","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n4.a6","url":null,"abstract":"We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out that the structure of these modules depends in a crucial way on the properties of the function spaces. Moreover, for a class of subalgebras related to compact manifolds, we provide a construction procedure for the corresponding fuzzy spaces, i.e. sequences of finite dimensional modules of increasing dimension as the deformation parameter tends to zero, as well as infinite dimensional modules related to fuzzy non-compact spaces.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552779","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 : 2024-06-06DOI: 10.4310/atmp.2023.v27.n4.a1
Luca Cassia, Nicolò Piazzalunga, Maxim Zabzine
We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on $S^1$, $D^2$ and $D^2 times S^1$, respectively. We define these objects and study their dependence on equivariant parameters for non-compact toric Kähler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the non-equivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov–Witten invariants of the target.
我们考虑了分别作为 1d、2d 和 3d 超对称 GLSM 在 $S^1$、$D^2$ 和 $D^2 times S^1$ 上的分割函数而得到的非等边 GIT 商的等变体积的广义。我们定义了这些对象，并研究了它们对非紧凑环凯勒商的等变参数的依赖性。我们把等变体积服从的有限差分方程（移位方程）推广到这些分区函数。分区函数由微分/差分算子湮灭，这些算子代表了目标的等变量子同调/K 理论关系，而这些关系中紧凑除数的出现在非等变极限的分析中起着至关重要的作用。我们证明了等变参数的扩展包含了目标的零属格罗莫夫-维滕不变式的信息。
{"title":"From equivariant volumes to equivariant periods","authors":"Luca Cassia, Nicolò Piazzalunga, Maxim Zabzine","doi":"10.4310/atmp.2023.v27.n4.a1","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n4.a1","url":null,"abstract":"We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on $S^1$, $D^2$ and $D^2 times S^1$, respectively. We define these objects and study their dependence on equivariant parameters for non-compact toric Kähler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the non-equivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov–Witten invariants of the target.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550606","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 : 2023-06-30DOI: 10.4310/atmp.2022.v26.n6.a7
Siu-Cheong Lau, Tsung-Ju Lee, Yu-Shen Lin
Given any smooth cubic curve $E subseteq mathbb{P}^2$, we show that the complex affine structure of the special Lagrangian fibration of $mathbb{P}^2 : backslash : E$ constructed by Collins–Jacob–Lin [12] coincides with the affine structure used in Carl–Pomperla–Siebert [15] for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl–Pomperla–Siebert.
{"title":"On the complex affine structures of SYZ-fibration of del Pezzo surfaces","authors":"Siu-Cheong Lau, Tsung-Ju Lee, Yu-Shen Lin","doi":"10.4310/atmp.2022.v26.n6.a7","DOIUrl":"https://doi.org/10.4310/atmp.2022.v26.n6.a7","url":null,"abstract":"Given any smooth cubic curve $E subseteq mathbb{P}^2$, we show that the complex affine structure of the special Lagrangian fibration of $mathbb{P}^2 : backslash : E$ constructed by Collins–Jacob–Lin [<b>12</b>] coincides with the affine structure used in Carl–Pomperla–Siebert [<b>15</b>] for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl–Pomperla–Siebert.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138527187","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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