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The symplectic properties of the PGL($n,mathbb C$)-gluing equations PGL($n，mathbb C$)-胶合方程的辛性质
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-01-01 DOI: 10.4171/QT/80
S. Garoufalidis, C. Zickert
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引用次数: 2
Homological algebra related to surfaces with boundary 与有边界曲面相关的同调代数
IF 1.1 2区 数学
Quantum Topology Pub Date : 2015-08-11 DOI: 10.4171/qt/144
K. Cieliebak, K. Fukaya, J. Latschev
{"title":"Homological algebra related to surfaces with boundary","authors":"K. Cieliebak, K. Fukaya, J. Latschev","doi":"10.4171/qt/144","DOIUrl":"https://doi.org/10.4171/qt/144","url":null,"abstract":"In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, which we call IBL$_infty$-algebras.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"2016 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2015-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86437072","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: 36
Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality Kauffman托架串代数的表示III:闭面和自然性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2015-05-06 DOI: 10.4171/QT/125
F. Bonahon, H. Wong
{"title":"Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality","authors":"F. Bonahon, H. Wong","doi":"10.4171/QT/125","DOIUrl":"https://doi.org/10.4171/QT/125","url":null,"abstract":"This is the third article in the series begun with [BonWon3, BonWon4], devoted to finite-dimensional representations of the Kauffman bracket skein algebra of an oriented surface $S$. In [BonWon3] we associated a classical shadow to an irreducible representation $rho$ of the skein algebra, which is a character $r_rho in mathcal R_{mathrm{SL}_2(mathbb C)}(S)$ represented by a group homomorphism $pi_1(S) to mathrm{SL}_2(mathbb C)$. The main result of the current article is that, when the surface $S$ is closed, every character $rin mathcal R_{mathrm{SL}_2(mathbb C)}(S)$ occurs as the classical shadow of an irreducible representation of the Kauffman bracket skein algebra. We also prove that the construction used in our proof is natural, and associates to each group homomorphism $rcolon pi_1(S) to mathrm{SL}_2(mathbb C)$ a representation of the skein algebra $mathcal S^A(S)$ that is uniquely determined up to isomorphism.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"30 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2015-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81681110","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
A graph TQFT for hat Heegaard Floer homology 一个图TQFT为该heegard花同调
IF 1.1 2区 数学
Quantum Topology Pub Date : 2015-03-19 DOI: 10.4171/qt/154
Ian Zemke
{"title":"A graph TQFT for hat Heegaard Floer homology","authors":"Ian Zemke","doi":"10.4171/qt/154","DOIUrl":"https://doi.org/10.4171/qt/154","url":null,"abstract":"In this paper we introduce an extension of the hat Heegaard Floer TQFT which allows cobordisms with disconnected ends. Our construction goes by way of sutured Floer homology, and uses some elementary results from contact geometry. We provide some model computations, which allow us to realize the $H_1(Y;mathbb{Z})/text{Tors}$ action and the first order term, $partial_1$, of the differential of $CF^infty$ as cobordism maps. As an application we prove a conjectured formula for the action of $pi_1(Y,p)$ on $hat{HF}(Y,p)$. We provide enough model computations to completely determine the new cobordism maps without the use of any contact geometric constructions.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"20 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2015-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80287401","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: 5
Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants Reshetikhin-Turaev和Turaev-Viro不变量的体积猜想
IF 1.1 2区 数学
Quantum Topology Pub Date : 2015-03-09 DOI: 10.4171/QT/111
Qingtao Chen, Tian Yang
{"title":"Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants","authors":"Qingtao Chen, Tian Yang","doi":"10.4171/QT/111","DOIUrl":"https://doi.org/10.4171/QT/111","url":null,"abstract":"We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic $3$-manifold, evaluated at the root of unity $exp({2pisqrt{-1}}/{r})$ instead of the standard $exp({pisqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten's Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin-Turaev invariants may have a geometric interpretation other than the original one via $SU(2)$ Chern-Simons gauge theory.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"103 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2015-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89281660","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: 55
Drinfeld centers of fusion categories arising from generalized Haagerup subfactors 广义Haagerup子因子引起的融合范畴的Drinfeld中心
IF 1.1 2区 数学
Quantum Topology Pub Date : 2015-01-30 DOI: 10.4171/qt/167
Pinhas Grossman, Masaki Izumi
{"title":"Drinfeld centers of fusion categories arising from generalized Haagerup subfactors","authors":"Pinhas Grossman, Masaki Izumi","doi":"10.4171/qt/167","DOIUrl":"https://doi.org/10.4171/qt/167","url":null,"abstract":"We consider generalized Haagerup categories such that $1 oplus X$ admits a $Q$-system for every non-invertible simple object $X$. We show that in such a category, the group of order two invertible objects has size at most four. We describe the simple objects of the Drinfeld center and give partial formulas for the modular data. We compute the remaining corner of the modular data for several examples and make conjectures about the general case. We also consider several types of equivariantizations and de-equivariantizations of generalized Haagerup categories and describe their Drinfeld centers. \u0000In particular, we compute the modular data for the Drinfeld centers of a number of examples of fusion categories arising in the classification of small-index subfactors: the Asaeda-Haagerup subfactor; the $3^{Z_4} $ and $3^{Z_2 times Z_2} $ subfactors; the $2D2$ subfactor; and the $4442$ subfactor. \u0000The results suggest the possibility of several new infinite families of quadratic categories. A description and generalization of the modular data associated to these families in terms of pairs of metric groups is taken up in the accompanying paper cite{GI19_2}.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"94 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2015-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80723564","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
$L$-space surgeries on links $L$-链接上的空间手术
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-08-30 DOI: 10.4171/QT/96
Yajing Liu
{"title":"$L$-space surgeries on links","authors":"Yajing Liu","doi":"10.4171/QT/96","DOIUrl":"https://doi.org/10.4171/QT/96","url":null,"abstract":"An $L$-space link is a link in $S^3$ on which all large surgeries are $L$-spaces. In this paper, we initiate a general study of the definitions, properties, and examples of $L$-space links. In particular, we find many hyperbolic $L$-space links, including some chain links and two-bridge links; from them, we obtain many hyperbolic $L$-spaces by integral surgeries, including the Weeks manifold. We give bounds on the ranks of the link Floer homology of $L$-space links and on the coefficients in the multi-variable Alexander polynomials. We also describe the Floer homology of surgeries on any $L$-space link using the link surgery formula of Ozsv'{a}th and Manolescu. As applications, we compute the graded Heegaard Floer homology of surgeries on 2-component $L$-space links in terms of only the Alexander polynomial and the surgery framing, and give a fast algorithm to classify $L$-space surgeries among them.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"91 1","pages":"505-570"},"PeriodicalIF":1.1,"publicationDate":"2014-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90266242","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: 22
A basis theorem for the affine oriented Brauer category and its cyclotomic quotients 仿射取向Brauer范畴及其分环商的一个基定理
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-04-25 DOI: 10.4171/QT/87
Jonathan Brundan, J. Comes, David Nash, Andrew Reynolds
{"title":"A basis theorem for the affine oriented Brauer category and its cyclotomic quotients","authors":"Jonathan Brundan, J. Comes, David Nash, Andrew Reynolds","doi":"10.4171/QT/87","DOIUrl":"https://doi.org/10.4171/QT/87","url":null,"abstract":"The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a basis theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"38 1","pages":"75-112"},"PeriodicalIF":1.1,"publicationDate":"2014-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82423024","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: 43
Fourier transform for quantum D-modules via the punctured torus mapping class group 通过穿孔环面映射类群的量子d模的傅里叶变换
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-03-07 DOI: 10.4171/QT/92
Adrien Brochier, D. Jordan
{"title":"Fourier transform for quantum D-modules via the punctured torus mapping class group","authors":"Adrien Brochier, D. Jordan","doi":"10.4171/QT/92","DOIUrl":"https://doi.org/10.4171/QT/92","url":null,"abstract":"We construct a certain cross product of two copies of the braided dual $tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to $H$. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if $H$ is factorizable. \u0000The universal property of $E_H$ endows it with an action by algebra automorphisms of the mapping class group $widetilde{SL_2(mathbb{Z})}$ of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when $H=U_q(mathfrak{g})$, the quantum Fourier transform degenerates to the classical Fourier transform on $D(mathfrak{g})$ as $qto 1$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"6 1","pages":"361-379"},"PeriodicalIF":1.1,"publicationDate":"2014-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80406597","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: 13
A categorification of quantum $mathfrak{sl}_3$ projectors and the $mathfrak{sl}_3$ Reshetikhin–Turaev invariant of tangles 量子$mathfrak{sl}_3$投影的分类和缠结$mathfrak{sl}_3$ Reshetikhin-Turaev不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2014-03-03 DOI: 10.4171/QT/46
David E. V. Rose
{"title":"A categorification of quantum $mathfrak{sl}_3$ projectors and the $mathfrak{sl}_3$ Reshetikhin–Turaev invariant of tangles","authors":"David E. V. Rose","doi":"10.4171/QT/46","DOIUrl":"https://doi.org/10.4171/QT/46","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":"1-59"},"PeriodicalIF":1.1,"publicationDate":"2014-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75582578","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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