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Homological mirror symmetry for invertible polynomials in two variables 二元可逆多项式的同调镜像对称
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-03-02 DOI: 10.4171/QT/163
Matthew Habermann
{"title":"Homological mirror symmetry for invertible polynomials in two variables","authors":"Matthew Habermann","doi":"10.4171/QT/163","DOIUrl":"https://doi.org/10.4171/QT/163","url":null,"abstract":"In this paper we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and as an application, we prove derived equivalences between certain stacky nodal curves, some of whose connected components have non-trivial generic stabiliser.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"195 1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78328374","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
The FKB invariant is the 3d index FKB不变量是3d索引
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-29 DOI: 10.4171/qt/171
S. Garoufalidis, R. Veen
{"title":"The FKB invariant is the 3d index","authors":"S. Garoufalidis, R. Veen","doi":"10.4171/qt/171","DOIUrl":"https://doi.org/10.4171/qt/171","url":null,"abstract":"We identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte-Gaiotto-Gukov. This implies the topological invariance of the $q$-series of Frohman and Kania-Bartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte-Gaiotto-Gukov as a limit of quantum 6j-symbols.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"32 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72899437","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
Factorization homology and 4D TQFT 分解同调与4D TQFT
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-20 DOI: 10.4171/QT/159
A. Kirillov, Ying Hong Tham
{"title":"Factorization homology and 4D TQFT","authors":"A. Kirillov, Ying Hong Tham","doi":"10.4171/QT/159","DOIUrl":"https://doi.org/10.4171/QT/159","url":null,"abstract":"In [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category $mathcal{A}$ extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) is described using the space of \"stringnets with boundary conditions\" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory in [CY1993].","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"60 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84451492","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: 10
Mapping class group actions from Hopf monoids and ribbon graphs 从Hopf monoids和带状图映射类群动作
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-10 DOI: 10.4171/QT/158
C. Meusburger, T. Voss
{"title":"Mapping class group actions from Hopf monoids and ribbon graphs","authors":"C. Meusburger, T. Voss","doi":"10.4171/QT/158","DOIUrl":"https://doi.org/10.4171/QT/158","url":null,"abstract":"We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g geq 1$ with $n geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"7 Suppl 5 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76610518","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
Constructing modular categories from orbifold data 从轨道数据构造模块类别
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-03 DOI: 10.4171/qt/170
Vincentas Mulevičius, I. Runkel
{"title":"Constructing modular categories from orbifold data","authors":"Vincentas Mulevičius, I. Runkel","doi":"10.4171/qt/170","DOIUrl":"https://doi.org/10.4171/qt/170","url":null,"abstract":"In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $mathbb{A}$ in a modular fusion category $mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $mathbb{A}$ in $mathcal{C}$, we introduce a ribbon category $mathcal{C}_{mathbb{A}}$ and show that it is again a modular fusion category. The definition of $mathcal{C}_{mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $mathbb{A}$ is given by a simple commutative $Delta$-separable Frobenius algebra $A$ in $mathcal{C}$; (ii) when $mathbb{A}$ is an orbifold datum in $mathcal{C} = operatorname{Vect}$, built from a spherical fusion category $mathcal{S}$. We show that in case (i), $mathcal{C}_{mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $mathcal{S}$. The category $mathcal{C}_{mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88325039","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: 10
Non-loose negative torus knots 非松散的负环面结
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-01-21 DOI: 10.4171/qt/169
Irena Matkovič
{"title":"Non-loose negative torus knots","authors":"Irena Matkovič","doi":"10.4171/qt/169","DOIUrl":"https://doi.org/10.4171/qt/169","url":null,"abstract":"We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. \u0000Additionally, we study the Legendrian invariants of these knots in the minus version of the knot Floer homology, obtaining that $Ucdotmathfrak L(L)$ vanishes for any Legendrian negative torus knot $L$ in any overtwisted structure, and that the strongly non-loose transverse realizations $T$ are characterized by having non-zero invariant $mathfrak T(T)$. \u0000Along the way, we relate our Legendrian realizations to the tight contact structures on the Legendrian surgeries along them. Specifically, we realize all tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in overtwisted structures to the non-fillable tight structures on the large negative surgeries along the underlying knots.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"96 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75931620","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: 4
Exotic Lagrangian tori in Grassmannians 格拉斯曼语中的奇异拉格朗日环面
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-10-24 DOI: 10.4171/qt/173
Marco Castronovo
{"title":"Exotic Lagrangian tori in Grassmannians","authors":"Marco Castronovo","doi":"10.4171/qt/173","DOIUrl":"https://doi.org/10.4171/qt/173","url":null,"abstract":"For each plabic graph of type (k,n) in the sense of Postnikov satisfying a smallness condition, we construct a nondisplaceable monotone Lagrangian torus in the complex Grassmannian Gr(k,n). Among these we find examples that bound the same number of families of Maslov 2 pseudoholomorphic disks, whose Hamiltonian isotopy classes are distinguished by the number of critical points in different algebraic torus charts of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. The tori are fibers of local regular Lagrangian fibrations over Okounkov bodies for the frozen anticanonical divisor, which is singled out by the cluster structure of the Grassmannian and has been studied by Rietsch-Williams. Lagrangian tori of plabic graphs related by a combinatorial square move have disk potentials connected by a 3-term Plucker relation, while their Newton polytopes undergo width 2 mutation in the sense of Akhtar-Coates-Galkin-Kasprzyk.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"46 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78181876","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: 4
Flags and tangles 旗帜和缠结
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-10-09 DOI: 10.4171/QT/157
F. Haiden
{"title":"Flags and tangles","authors":"F. Haiden","doi":"10.4171/QT/157","DOIUrl":"https://doi.org/10.4171/QT/157","url":null,"abstract":"We show that two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori--Hecke algebras of type $A_n$ as endomorphism algebras of certain objects.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"15 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88346948","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
Seifert hypersurfaces of 2-knots and Chern–Simons functional 2节的Seifert超曲面和chen - simons泛函
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-10-05 DOI: 10.4171/qt/165
Masaki Taniguchi
{"title":"Seifert hypersurfaces of 2-knots and Chern–Simons functional","authors":"Masaki Taniguchi","doi":"10.4171/qt/165","DOIUrl":"https://doi.org/10.4171/qt/165","url":null,"abstract":"We introduce a real-valued functional on the $SU(2)$-representation space of the knot group for any oriented $2$-knot. We calculate the functionals for ribbon $2$-knots and the twisted spun $2$-knots of torus knots, $2$-bridge knots and Montesinos knots. We show several properties of the images of the functionals including a connected sum formula and relationship to the Chern-Simons functionals of Seifert hypersurfaces of $K$. As a corollary, we show that every oriented $2$-knot having a homology $3$-sphere of a certain class as its Seifert hypersurface admits an $SU(2)$-irreducible representation of a knot group. Moreover, we also relate the existence of embeddings from a homology $3$-sphere into a negative definite $4$-manifold to $SU(2)$-representations of their fundamental groups. For example, we prove that every closed definite $4$-manifold containing $Sigma(2,3,5,7)$ as a submanifold has an uncountable family of $SU(2)$-representations of its fundamental group. This implies that every $2$-knot having $Sigma(2,3,5,7)$ as a Seifert hypersurface has an uncountable family of $SU(2)$-representations of its knot group. The proofs of these results use several techniques from instanton Floer theory.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"86 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85876495","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: 4
Link homology theories and ribbon concordances 链接同源理论和条带一致性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-09-16 DOI: 10.4171/qt/162
Sungkyung Kang
{"title":"Link homology theories and ribbon concordances","authors":"Sungkyung Kang","doi":"10.4171/qt/162","DOIUrl":"https://doi.org/10.4171/qt/162","url":null,"abstract":"It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"158 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72472912","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
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