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Prime decomposition of modular tensor categories of local modules of type D D型局部模模张量范畴的素数分解
IF 1.1 2区 数学
Quantum Topology Pub Date : 2018-10-22 DOI: 10.4171/QT/140
Andrew Schopieray
{"title":"Prime decomposition of modular tensor categories of local modules of type D","authors":"Andrew Schopieray","doi":"10.4171/QT/140","DOIUrl":"https://doi.org/10.4171/QT/140","url":null,"abstract":"Let $mathcal{C}(mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $mathcal{C}(mathfrak{g},k)_R^0$ where $R$ is the regular algebra of Tannakian $text{Rep}(H)subsetmathcal{C}(mathfrak{g},k)_text{pt}$. For $mathfrak{g}=mathfrak{so}_5$ we describe the decomposition of $mathcal{C}(mathfrak{g},k)_R^0$ into prime factors explicitly and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of $mathcal{C}(mathfrak{so}_5,k)$ and $mathcal{C}(mathfrak{g}_2,k)$ for $kinmathbb{Z}_{geq1}$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"40 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82659243","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
Heegaard Floer invariants of contact structures on links of surface singularities 表面奇点连杆上接触结构的保花不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2018-09-28 DOI: 10.4171/QT/153
J'ozsef Bodn'ar, O. Plamenevskaya
{"title":"Heegaard Floer invariants of contact structures on links of surface singularities","authors":"J'ozsef Bodn'ar, O. Plamenevskaya","doi":"10.4171/QT/153","DOIUrl":"https://doi.org/10.4171/QT/153","url":null,"abstract":"Let a contact 3-manifold $(Y, xi_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $xi_0$. We prove a special property of such contact 3-manifolds of \"algebraic\" origin: the Heegaard Floer invariant $c^+(xi_0)in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. It follows that Karakurt's \"height of $U$-tower\" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of $U$-tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and N'emethi's lattice cohomology.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85375981","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
A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra 拓扑Hopf超代数中3-流形的亨宁斯型不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2018-06-21 DOI: 10.4171/qt/142
N. Ha
{"title":"A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra","authors":"N. Ha","doi":"10.4171/qt/142","DOIUrl":"https://doi.org/10.4171/qt/142","url":null,"abstract":"We prove the unrolled superalgebra $mathcal{U}_{xi}^{H}mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"100 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90067225","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
Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles 无迹SU(2)特征缠结的完整摄动与正则性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2018-06-03 DOI: 10.4171/QT/110
Christopher Herald, P. Kirk
{"title":"Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles","authors":"Christopher Herald, P. Kirk","doi":"10.4171/QT/110","DOIUrl":"https://doi.org/10.4171/QT/110","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"86 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77016786","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
A categorification of cyclotomic rings 切眼环的分类
IF 1.1 2区 数学
Quantum Topology Pub Date : 2018-04-04 DOI: 10.4171/QT/172
Robert Laugwitz, You Qi
{"title":"A categorification of cyclotomic rings","authors":"Robert Laugwitz, You Qi","doi":"10.4171/QT/172","DOIUrl":"https://doi.org/10.4171/QT/172","url":null,"abstract":"For any natural number $n geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers $mathbb{O}_n$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"32 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82198341","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
Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial 卫星统治多项式，DGA表示，和彩色HOMFLY-PT多项式
IF 1.1 2区 数学
Quantum Topology Pub Date : 2018-02-28 DOI: 10.4171/qt/133
C. Leverson, Dan Rutherford
{"title":"Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial","authors":"C. Leverson, Dan Rutherford","doi":"10.4171/qt/133","DOIUrl":"https://doi.org/10.4171/qt/133","url":null,"abstract":"We establish relationships between two classes of invariants of Legendrian knots in $mathbb{R}^3$: Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, $beta subset J^1S^1$, we give a precise formula in terms of representation numbers for the $m$-graded ruling polynomial $R^m_{S(K,beta)}(z)$ of the satellite of $K$ with $beta$ specialized at $z=q^{1/2}-q^{-1/2}$ with $q$ a prime power, and we use this formula to prove that arbitrary $m$-graded satellite ruling polynomials, $R^m_{S(K,L)}$, are determined by the Chekanov-Eliashberg DGA of $K$. Conversely, for $mneq 1$, we introduce an $n$-colored $m$-graded ruling polynomial, $R^m_{n,K}(q)$, in strict analogy with the $n$-colored HOMFLY-PT polynomial, and show that the total $n$-dimensional $m$-graded representation number of $K$ to $mathbb{F}_q^n$, $mbox{Rep}_m(K,mathbb{F}_q^n)$, is exactly equal to $R^m_{n,K}(q)$. In the case of $2$-graded representations, we show that $R^2_{n,K}=mbox{Rep}_2(K, mathbb{F}_q^n)$ arises as a specialization of the $n$-colored HOMFLY-PT polynomial.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"120 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2018-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87939914","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 $mathsf {Lie}$ Lie algebra $mathsf {Lie}$李代数
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-12-06 DOI: 10.4171/qt/99
James F. Conant
{"title":"The $mathsf {Lie}$ Lie algebra","authors":"James F. Conant","doi":"10.4171/qt/99","DOIUrl":"https://doi.org/10.4171/qt/99","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"25 1","pages":"667-714"},"PeriodicalIF":1.1,"publicationDate":"2017-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80517259","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
Line and surface defects in Reshetikhin–Turaev TQFT Reshetikhin-Turaev TQFT的线和表面缺陷
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-10-27 DOI: 10.4171/QT/121
Nils Carqueville, I. Runkel, Gregor Schaumann
{"title":"Line and surface defects in Reshetikhin–Turaev TQFT","authors":"Nils Carqueville, I. Runkel, Gregor Schaumann","doi":"10.4171/QT/121","DOIUrl":"https://doi.org/10.4171/QT/121","url":null,"abstract":"A modular tensor category $mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by $Delta$-separable symmetric Frobenius algebras and the line defects by \"multi-modules\" which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where it distinguishes non-isotopic embeddings of 2-tori.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89680397","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: 28
Symplectic instanton homology: naturality, and maps from cobordisms 辛瞬子同调:自然性，和从协点映射
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-10-11 DOI: 10.4171/qt/129
Guillem Cazassus
{"title":"Symplectic instanton homology: naturality, and maps from cobordisms","authors":"Guillem Cazassus","doi":"10.4171/qt/129","DOIUrl":"https://doi.org/10.4171/qt/129","url":null,"abstract":"We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. \u0000This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometric interpretation of the maps appearing in the long exact sequence for symplectic instanton homology, together with vanishing criterions.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"39 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90944371","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
A note on coherent orientations for exact Lagrangian cobordisms 关于精确拉格朗日协数的相干取向的注释
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-07-13 DOI: 10.4171/qt/132
Cecilia Karlsson
{"title":"A note on coherent orientations for exact Lagrangian cobordisms","authors":"Cecilia Karlsson","doi":"10.4171/qt/132","DOIUrl":"https://doi.org/10.4171/qt/132","url":null,"abstract":"Let $L subset mathbb R times J^1(M)$ be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold $M$. Assume that $L$ has cylindrical Legendrian ends $Lambda_pm subset J^1(M)$. It is well known that the Legendrian contact homology of $Lambda_pm$ can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of $M$. It is also known that this count can be lifted to a mod 2 count of pseudo-holomorphic disks in the symplectization $mathbb R times J^1(M)$, and that $L$ induces a morphism between the $mathbb Z_2$-valued DGA:s of the ends $Lambda_pm$ in a functorial way. We prove that this hold with integer coefficients as well. The proofs are built on the technique of orienting the moduli spaces of pseudo-holomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA:s change if we change the capping operators.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"112 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73881641","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: 20
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