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Fully extended $r$-spin TQFTs 完全扩展的$r$-spin tqft
2区 数学
Quantum Topology Pub Date : 2023-10-15 DOI: 10.4171/qt/193
Nils Carqueville, Lóránt Szegedy
{"title":"Fully extended $r$-spin TQFTs","authors":"Nils Carqueville, Lóránt Szegedy","doi":"10.4171/qt/193","DOIUrl":"https://doi.org/10.4171/qt/193","url":null,"abstract":"We prove the $r$-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer $r$: the 2-groupoid of 2-dimensional fully extended $r$-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced $mathrm{Spin}_2^r$-action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the $r$th power of their Serre automorphisms. For $r=1$, we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to $r=2$. To construct examples, we explicitly describe $mathrm{Spin}_2^r$-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau–Ginzburg models gives rise to fully extended spin TQFTs and that half of these do not factor through the oriented bordism 2-category.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"238 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136185733","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}
引用次数: 3
The adjoint Reidemeister torsion for the connected sum of knots 结的连通和的伴随Reidemeister扭转
2区 数学
Quantum Topology Pub Date : 2023-09-15 DOI: 10.4171/qt/180
Joan Porti, Seokbeom Yoon
{"title":"The adjoint Reidemeister torsion for the connected sum of knots","authors":"Joan Porti, Seokbeom Yoon","doi":"10.4171/qt/180","DOIUrl":"https://doi.org/10.4171/qt/180","url":null,"abstract":"Let $K$ be the connected sum of knots $K_1,ldots,K_n$. It is known that the $mathrm{SL}_2(mathbb{C})$-character variety of the knot exterior of $K$ has a component of dimension $geq 2$ as the connected sum admits a so-called bending. We show that there is a natural way to define the adjoint Reidemeister torsion for such a high-dimensional component and prove that it is locally constant on a subset of the character variety where the trace of a meridian is constant. We also prove that the adjoint Reidemeister torsion of $K$ satisfies the vanishing identity if each $K_i$ does so.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"176 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353864","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}
引用次数: 2
Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots 捻结的几何三角剖分和teichm<e:1> ller TQFT体积猜想
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-09-01 DOI: 10.4171/qt/178
Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa
{"title":"Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots","authors":"Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa","doi":"10.4171/qt/178","DOIUrl":"https://doi.org/10.4171/qt/178","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"110 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76007100","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}
引用次数: 0
Triangular decomposition of $mathrm{SL}_3$ skein algebras $ mathm {SL}_3$ skein代数的三角分解
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-06-29 DOI: 10.4171/qt/177
Vijay Higgins
{"title":"Triangular decomposition of $mathrm{SL}_3$ skein algebras","authors":"Vijay Higgins","doi":"10.4171/qt/177","DOIUrl":"https://doi.org/10.4171/qt/177","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"214 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75577123","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}
引用次数: 0
$A_infty$-category of Lagrangian cobordisms in the symplectization of $Ptimes {mathbb R}$ $A_infty$的化简中的拉格朗日协数范畴 $Ptimes {mathbb R}$
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-06-29 DOI: 10.4171/qt/179
N. Legout
{"title":"$A_infty$-category of Lagrangian cobordisms in the symplectization of $Ptimes {mathbb R}$","authors":"N. Legout","doi":"10.4171/qt/179","DOIUrl":"https://doi.org/10.4171/qt/179","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"60 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87396258","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}
引用次数: 0
The next-to-top term in knot Floer homology 结花同源性的下一个最上面的术语
IF 1.1 2区 数学
Quantum Topology Pub Date : 2021-04-29 DOI: 10.4171/qt/174
Yi Ni
{"title":"The next-to-top term in knot Floer homology","authors":"Yi Ni","doi":"10.4171/qt/174","DOIUrl":"https://doi.org/10.4171/qt/174","url":null,"abstract":"Let $K$ be a null-homologous knot in a generalized L-space $Z$ with $b_1(Z)le1$. Let $F$ be a Seifert surface of $K$ with genus $g$. We show that if $widehat{HFK}(Z,K,[F],g)$ is supported in a single $mathbb Z/2mathbb Z$--grading, then [mathrm{rank}widehat{HFK}(Z,K,[F],g-1)gemathrm{rank}widehat{HFK}(Z,K,[F],g).]","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"19 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79137376","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}
引用次数: 2
Non-formality in PIN(2)-monopole Floer homology PIN(2)中的非正式性-单极花同源性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2021-03-26 DOI: 10.4171/QT/151
Francesco Lin
{"title":"Non-formality in PIN(2)-monopole Floer homology","authors":"Francesco Lin","doi":"10.4171/QT/151","DOIUrl":"https://doi.org/10.4171/QT/151","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80331455","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}
引用次数: 1
Instanton Floer homology, sutures, and Euler characteristics 瞬花同源性、缝合线和欧拉特性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2021-01-13 DOI: 10.4171/qt/182
Zhenkun Li, Fan Ye
{"title":"Instanton Floer homology, sutures, and Euler characteristics","authors":"Zhenkun Li, Fan Ye","doi":"10.4171/qt/182","DOIUrl":"https://doi.org/10.4171/qt/182","url":null,"abstract":"This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $chi_{rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $chi_{rm gr}(SHI(M,gamma))=chi_{rm gr}(SFH(M,gamma))$ for any balanced sutured manifold $(M,gamma)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $chi_{rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $underline{rm KHI}^-(Y,K)$ introduced by the first author.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86984989","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}
引用次数: 9
Quantum invariants of three-manifolds obtained by surgeries along torus knots 三流形沿环面结整形的量子不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-11-11 DOI: 10.4171/qt/175
H. Murakami, Anh T. Tran
{"title":"Quantum invariants of three-manifolds obtained by surgeries along torus knots","authors":"H. Murakami, Anh T. Tran","doi":"10.4171/qt/175","DOIUrl":"https://doi.org/10.4171/qt/175","url":null,"abstract":"We study the asymptotic behavior of the Witten-Reshetikhin-Turaev invariant associated with the square of the $n$-th root of unity with odd $n$ for a Seifert fibered space obtained by an integral Dehn surgery along a torus knot. We show that it can be described as a sum of the Chern-Simons invariants and the twisted Reidemeister torsions both associated with representations of the fundamental group to the two-dimensional complex special linear group.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"21 6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90409169","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}
引用次数: 0
Web calculus and tilting modules in type $C_2$ Web演算和倾斜模块类型$C_2$
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-09-29 DOI: 10.4171/qt/166
Elijah Bodish
{"title":"Web calculus and tilting modules in type $C_2$","authors":"Elijah Bodish","doi":"10.4171/qt/166","DOIUrl":"https://doi.org/10.4171/qt/166","url":null,"abstract":"Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we describe a \"light leaves\" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for $mathfrak{so}_5cong mathfrak{sp}_4$ (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when $[2]_qne 0$, the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules for the divided powers quantum group $mathcal{U}_q^{mathbb{Z}}(mathfrak{sp}_4)$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"64 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74067365","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}
引用次数: 8
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