The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $mathbf {k}$-vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.
当 M 是有限维实向量空间时，柏原-沙皮拉（Kashiwara-Schapira）最近引入了 M 上 $mathbf {k}$- 向量空间的舍维之间的卷积距离。在本文中，我们描述了实有限维向量空间上可构造函数群的距离，这些距离可以通过剪切-函数对应关系由卷积距离控制。我们的主要结果断言，这种距离几乎是微不足道的：只要两个可构造函数具有相同的欧拉积分，它们就会消失。我们提出了我们的结果对拓扑数据分析的影响：不可能存在对交织距离而言是连续的持久性模块的非难加不变式。
{"title":"Persistence and the Sheaf-Function Correspondence","authors":"Nicolas Berkouk","doi":"10.1017/fms.2023.115","DOIUrl":"https://doi.org/10.1017/fms.2023.115","url":null,"abstract":"<p>The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold <span>M</span> with the Grothendieck group of constructible sheaves on <span>M</span>. When <span>M</span> is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215022810799-0847:S2050509423001159:S2050509423001159_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {k}$</span></span></img></span></span>-vector spaces on <span>M</span>. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"242 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138717222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any branched double covering of compact Riemann surfaces, we consider the associated character varieties that are unitary in the global sense, which we call $operatorname {mathrm {GL}}_nrtimes !<!sigma {>}$-character varieties. We restrict the monodromies around the branch points to generic semi-simple conjugacy classes contained in $operatorname {mathrm {GL}}_nsigma $ and compute the E-polynomials of these character varieties using the character table of $operatorname {mathrm {GL}}_n(q)rtimes !<!sigma !>!$. The result is expressed as the inner product of certain symmetric functions associated to the wreath product $(mathbb {Z}/2mathbb {Z})^Nrtimes mathfrak {S}_N$. We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.
{"title":"E-Polynomials of Generic -Character Varieties: Branched Case","authors":"Cheng Shu","doi":"10.1017/fms.2023.119","DOIUrl":"https://doi.org/10.1017/fms.2023.119","url":null,"abstract":"<p>For any branched double covering of compact Riemann surfaces, we consider the associated character varieties that are unitary in the global sense, which we call <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_nrtimes !<!sigma {>}$</span></span></img></span></span>-character varieties. We restrict the monodromies around the branch points to generic semi-simple conjugacy classes contained in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_nsigma $</span></span></img></span></span> and compute the E-polynomials of these character varieties using the character table of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {GL}}_n(q)rtimes !<!sigma !>!$</span></span></img></span></span>. The result is expressed as the inner product of certain symmetric functions associated to the wreath product <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231215154237435-0861:S2050509423001196:S2050509423001196_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$(mathbb {Z}/2mathbb {Z})^Nrtimes mathfrak {S}_N$</span></span></img></span></span>. We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"118 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138717524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Sigma $ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in $Sigma $ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $Sigma $ . In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.
让 $Sigma $ 是一个封闭的双曲面。对于固定的 g，我们研究在 $Sigma $ 中最长为 L 的周期性大地线的数量的渐近性，这些大地线可以写成 g 换向器的乘积。其基本思想是将这些结果简化为能够计算 $Sigma $ 中三价图的临界实现。在附录中，我们用同样的策略给出了胡贝尔几何素数定理的证明。
{"title":"Counting geodesics of given commutator length","authors":"Viveka Erlandsson, Juan Souto","doi":"10.1017/fms.2023.114","DOIUrl":"https://doi.org/10.1017/fms.2023.114","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001147_inline1.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a closed hyperbolic surface. We study, for fixed <jats:italic>g</jats:italic>, the asymptotics of the number of those periodic geodesics in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001147_inline2.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> having at most length <jats:italic>L</jats:italic> and which can be written as the product of <jats:italic>g</jats:italic> commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509423001147_inline3.png\" /> <jats:tex-math> $Sigma $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"145 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138685932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rune Haugseng, Fabian Hebestreit, Sil Linskens, Joost Nuiten
We prove a universal property for $infty $ -categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose. As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an $(infty ,2)$ -category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the $infty $ -category of $infty $ -categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen).
我们证明了在巴维克适当三元组的一般性中，$infty $ -跨类的一个普遍性质，明确描述了与跨函子相对应的卡方傅立叶，并证明了后者限制在正交适当三元组类上的自等价性，我们为此引入了正交适当三元组类。作为我们开发的机制的应用，我们给出了巴维克展开定理的快速证明，证明了当且仅当一个正交因式分解系统形成一个适当的三元组时，它才会从一个卡方振动中产生（概括了拉纳里的工作），将巴维克、格拉斯曼和纳丁对二（共）卡方振动的描述扩展到二变量振动，明确描述了参数化的邻接（扩展了鸟居的工作）、在阿贝兰-加西亚（Abellán García）和斯特恩（Stern）的工作的基础上，确定了对$(infty ,2)$ -类的映射类别函子进行分类的正振动（orthofibration）；正式确定了$infty $ -类的$infty $ -类上的身份函子的非直化（unstraightenings）与点的（op）lax下类（under-categories of a point）；并推导了米田嵌入（Yoneda embedding）的某个自然属性（回答了克劳森（Clausen）的一个问题）。
{"title":"Two-variable fibrations, factorisation systems and -categories of spans","authors":"Rune Haugseng, Fabian Hebestreit, Sil Linskens, Joost Nuiten","doi":"10.1017/fms.2023.107","DOIUrl":"https://doi.org/10.1017/fms.2023.107","url":null,"abstract":"We prove a universal property for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline2.png\" /> <jats:tex-math> $infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose. As applications of the machinery we develop, we give a quick proof of Barwick’s unfurling theorem, show that an orthogonal factorisation system arises from a cartesian fibration if and only if it forms an adequate triple (generalising work of Lanari), extend the description of dual (co)cartesian fibrations by Barwick, Glasman and Nardin to two-variable fibrations, explicitly describe parametrised adjoints (extending work of Torii), identify the orthofibration classifying the mapping category functor of an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline3.png\" /> <jats:tex-math> $(infty ,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-category (building on work of Abellán García and Stern), formally identify the unstraightenings of the identity functor on the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline4.png\" /> <jats:tex-math> $infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-category of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300107X_inline5.png\" /> <jats:tex-math> $infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-categories with the (op)lax under-categories of a point, and deduce a certain naturality property of the Yoneda embedding (answering a question of Clausen).","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"37 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite $2$ -groups with Kantor familes, by showing that finite groups K with a Kantor family $(mathcal {F},mathcal {F}^*)$ having distinct members $A, B in mathcal {F}$ such that $A^* cap B^*$ is a central subgroup of K and the quotient $K/(A^* cap B^*)$ is abelian cannot exist if the center of K has exponent $4$ and the members of $mathcal {F}$ are elementary abelian. Then we give a short geometrical proof of a recent result of Ott which says that finite skew translation quadrangles of even order $(t,t)$ (where t is not a square) are always translation generalized quadrangles. This is a consequence of a complete classification of finite cyclic skew translation quadrangles of order $(t,t)$ that we carry out in the present paper.
我们解决了Frohardt在1988年的论文[6]中提出的关于具有Kantor族的有限$2$群的一个基本问题，通过证明具有Kantor族的有限群K $(mathcal {F}，mathcal {F}^*)$在mathcal {F}$中具有不同的成员$ a， $ B，使得$ a ^* cap B^*$是K的中心子群，而商$K/(a ^* cap B^*)$是阿贝尔，如果K的中心有指数$4$且$mathcal {F}$的成员是初等阿贝尔，则不存在。然后给出了一个简短的几何证明，证明了奥特最近的一个结果，即偶阶的有限斜平移四边形(t,t)$(其中t不是正方形)总是平移广义四边形。这是我们在本文中给出的阶$(t,t)$的有限循环斜平移四边形完全分类的结果。
{"title":"A question of Frohardt on -groups, skew translation quadrangles of even order and cyclic STGQs","authors":"Koen Thas","doi":"10.1017/fms.2023.105","DOIUrl":"https://doi.org/10.1017/fms.2023.105","url":null,"abstract":"Abstract We solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite \u0000$2$\u0000 -groups with Kantor familes, by showing that finite groups K with a Kantor family \u0000$(mathcal {F},mathcal {F}^*)$\u0000 having distinct members \u0000$A, B in mathcal {F}$\u0000 such that \u0000$A^* cap B^*$\u0000 is a central subgroup of K and the quotient \u0000$K/(A^* cap B^*)$\u0000 is abelian cannot exist if the center of K has exponent \u0000$4$\u0000 and the members of \u0000$mathcal {F}$\u0000 are elementary abelian. Then we give a short geometrical proof of a recent result of Ott which says that finite skew translation quadrangles of even order \u0000$(t,t)$\u0000 (where t is not a square) are always translation generalized quadrangles. This is a consequence of a complete classification of finite cyclic skew translation quadrangles of order \u0000$(t,t)$\u0000 that we carry out in the present paper.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A semiclassical analysis based on spin-coherent states is used to establish a classification and novel simple formulae for the spectral gap of mean-field spin Hamiltonians. For gapped systems, we provide a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian, hence justifying fluctuation theory at zero temperature for this case. We also point out a shift caused by the spherical geometry in these second-order approximations.
{"title":"The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems","authors":"Chokri Manai, Simone Warzel","doi":"10.1017/fms.2023.111","DOIUrl":"https://doi.org/10.1017/fms.2023.111","url":null,"abstract":"A semiclassical analysis based on spin-coherent states is used to establish a classification and novel simple formulae for the spectral gap of mean-field spin Hamiltonians. For gapped systems, we provide a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian, hence justifying fluctuation theory at zero temperature for this case. We also point out a shift caused by the spherical geometry in these second-order approximations.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"238 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is approximable if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that $mathsf {D}_{mathsf {qc}}( X )$ is approximable when X is a quasi compact, separated scheme led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article, we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which, in conjunction with the groundwork laid in [16], has interesting applications in existing and forthcoming work by the authors.
给定环上模的上有界协链复合体，用射影分辨率替换它是标准的，并且这样做是非常有用的。最近，除了环的派生范畴外，在三角化范畴中引入了一个改进的版本。如果这个修改的程序是可能的，一个三角分类是近似的。不出所料，这已被证明是一个强大的工具。例如:$mathsf {D}_{mathsf {qc}}(X)$在X是拟紧的分离格式时是近似的，这一事实导致了Bondal, Van den Bergh和Rouquier对旧定理的重大改进。在本文中，我们证明了在弱假设下，两个可逼近三角化范畴的回积是可逼近的。特别地，这表明在非交换代数几何中出现的许多三角化范畴是近似的。此外，本文中开发的引理和技术形成了一个强大的工具箱，与[16]中奠定的基础相结合，在作者现有和即将开展的工作中具有有趣的应用。
{"title":"Gluing approximable triangulated categories","authors":"Jesse Burke, Amnon Neeman, Bregje Pauwels","doi":"10.1017/fms.2023.97","DOIUrl":"https://doi.org/10.1017/fms.2023.97","url":null,"abstract":"Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is <jats:italic>approximable</jats:italic> if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S205050942300097X_inline1.png\" /> <jats:tex-math> $mathsf {D}_{mathsf {qc}}( X )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is approximable when <jats:italic>X</jats:italic> is a quasi compact, separated scheme led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article, we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which, in conjunction with the groundwork laid in [16], has interesting applications in existing and forthcoming work by the authors.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"16 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory. We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight. We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.
{"title":"Categorical and K-theoretic Donaldson–Thomas theory of (part II)","authors":"Tudor Pădurariu, Yukinobu Toda","doi":"10.1017/fms.2023.103","DOIUrl":"https://doi.org/10.1017/fms.2023.103","url":null,"abstract":"Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory. We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight. We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"81 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Angelo Lucia, David Pérez-García, Antonio Pérez-Hernández
We show that every ergodic Davies generator associated to any 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.
{"title":"Thermalization in Kitaev’s quantum double models via tensor network techniques","authors":"Angelo Lucia, David Pérez-García, Antonio Pérez-Hernández","doi":"10.1017/fms.2023.98","DOIUrl":"https://doi.org/10.1017/fms.2023.98","url":null,"abstract":"We show that every ergodic Davies generator associated to <jats:italic>any</jats:italic> 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"25 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove some results on the nilpotent orbit theorem for complex variations of Hodge structure.
我们证明了Hodge结构复变的幂零轨道定理的一些结果。
{"title":"On the nilpotent orbit theorem of complex variations of Hodge structure","authors":"Ya Deng","doi":"10.1017/fms.2023.109","DOIUrl":"https://doi.org/10.1017/fms.2023.109","url":null,"abstract":"<p>We prove some results on the nilpotent orbit theorem for complex variations of Hodge structure.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"8 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138529173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$mathbf {k}$-vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.
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