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.
{"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":null,"pages":null},"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}
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. This 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.
{"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":null,"pages":null},"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}
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.
设L 子集mathbb R 乘以J^1(M)$是光滑流形$M$的1-射流空间的化中的自旋精确拉格朗日协。假设$L$具有圆柱形的勒让端$Lambda_pm 子集J^1(M)$。众所周知，$Lambda_pm$的Legendrian接触同调可以用整数系数来定义，通过$M$的余切束中的伪全纯盘的带符号计数。我们还知道，这个计数可以在$mathbb R 乘以J^1(M)$的化过程中提升到一个模2的伪全纯磁盘计数，并且$L$在$Lambda_pm$的末端$mathbb Z_2$值的DGA:s之间以函子方式诱导出一个态射。我们也用整数系数证明了这一点。这些证明是建立在利用Reeb弦上的封顶算子定向伪全纯盘的模空间的技术之上的。我们给出了当封顶操作符改变时DGA:s如何变化的表达式。
{"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":null,"pages":null},"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}
This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.
{"title":"Bordered knot algebras with matchings","authors":"P. Ozsváth, Z. Szabó","doi":"10.4171/qt/127","DOIUrl":"https://doi.org/10.4171/qt/127","url":null,"abstract":"This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72944201","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}
This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics (2000). We deduce a link between combinatorics and representations in finite group theory.
本文证明了具有分配双投影格的不可约子因子平面代数允许一个极小的2盒投影生成恒等双投影。它是Oystein Ore关于有限群的分布区间定理(1938)的推广(2013年推测)，也是代数组合学中Kenneth S. Brown猜想(2000)的自然子因子扩展的必然结果。我们在有限群论中推导了组合学与表示之间的联系。
{"title":"Ore's theorem on subfactor planar algebras","authors":"S. Palcoux","doi":"10.4171/QT/141","DOIUrl":"https://doi.org/10.4171/QT/141","url":null,"abstract":"This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics (2000). We deduce a link between combinatorics and representations in finite group theory.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86730633","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 an arbitrary graph $Gamma$ and non-negative integers $g_v$ for each vertex $v$ of $Gamma$, let $X_Gamma$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*Sigma_v$ according to this graph, where $Sigma_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_Gamma$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that $g_v=0$ for all $v$ and $Gamma$ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw arXiv:math/0404186. Along the way, we find a smaller model for the internal DG-algebra of Ekholm-Ng arXiv:1307.8436 associated to $1$-handles in the Legendrian surgery presentation of Weinstein $4$-manifolds which might be of independent interest.
{"title":"Fukaya categories of plumbings and multiplicative preprojective algebras","authors":"Tolga Etgu, Yankı Lekili","doi":"10.4171/QT/131","DOIUrl":"https://doi.org/10.4171/QT/131","url":null,"abstract":"Given an arbitrary graph $Gamma$ and non-negative integers $g_v$ for each vertex $v$ of $Gamma$, let $X_Gamma$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*Sigma_v$ according to this graph, where $Sigma_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_Gamma$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that $g_v=0$ for all $v$ and $Gamma$ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw arXiv:math/0404186. Along the way, we find a smaller model for the internal DG-algebra of Ekholm-Ng arXiv:1307.8436 associated to $1$-handles in the Legendrian surgery presentation of Weinstein $4$-manifolds which might be of independent interest.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81502094","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 construct a triangulated monoidal Karoubi closed category with the Grothendieck ring, naturally isomorphic to the ring of integers localized at two.
{"title":"How to categorify the ring of integers localized at two","authors":"M. Khovanov, Yin Tian","doi":"10.4171/qt/130","DOIUrl":"https://doi.org/10.4171/qt/130","url":null,"abstract":"We construct a triangulated monoidal Karoubi closed category with the Grothendieck ring, naturally isomorphic to the ring of integers localized at two.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77331757","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}
Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z:mathcal{B}to widehat{mathbb{A}}$, where $mathcal{B}$ is the category of bottom tangles in handlebodies and $widehat{mathbb{A}}$ is the degree-completion of the category $mathbb{A}$ of Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, $mathbb{A}$ is the linear PROP governing "Casimir Hopf algebras", which are cocommutative Hopf algebras equipped with a primitive invariant symmetric 2-tensor. The functor $Z$ induces a canonical isomorphism $hbox{gr}mathcal{B} cong mathbb{A}$, where $hbox{gr}mathcal{B}$ is the associated graded of the Vassiliev-Goussarov filtration on $mathcal{B}$. To each Drinfeld associator $varphi$ we associate a ribbon quasi-Hopf algebra $H_varphi$ in $hbox{gr}mathcal{B}$, and we prove that the braided Hopf algebra resulting from $H_varphi$ by "transmutation" is precisely the image by $Z$ of a canonical Hopf algebra in the braided category $mathcal{B}$. Finally, we explain how $Z$ refines the LMO functor, which is a TQFT-like functor extending the Le-Murakami-Ohtsuki invariant
{"title":"The Kontsevich integral for bottom tangles in handlebodies","authors":"K. Habiro, G. Massuyeau","doi":"10.4171/qt/155","DOIUrl":"https://doi.org/10.4171/qt/155","url":null,"abstract":"Using an extension of the Kontsevich integral to tangles in handlebodies \u0000similar to a construction given by Andersen, Mattes and Reshetikhin, we \u0000construct a functor $Z:mathcal{B}to widehat{mathbb{A}}$, where \u0000$mathcal{B}$ is the category of bottom tangles in handlebodies and \u0000$widehat{mathbb{A}}$ is the degree-completion of the category $mathbb{A}$ of \u0000Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, \u0000$mathbb{A}$ is the linear PROP governing \"Casimir Hopf algebras\", which are \u0000cocommutative Hopf algebras equipped with a primitive invariant symmetric \u00002-tensor. The functor $Z$ induces a canonical isomorphism $hbox{gr}mathcal{B} \u0000cong mathbb{A}$, where $hbox{gr}mathcal{B}$ is the associated graded of the \u0000Vassiliev-Goussarov filtration on $mathcal{B}$. To each Drinfeld associator \u0000$varphi$ we associate a ribbon quasi-Hopf algebra $H_varphi$ in \u0000$hbox{gr}mathcal{B}$, and we prove that the braided Hopf algebra resulting \u0000from $H_varphi$ by \"transmutation\" is precisely the image by $Z$ of a \u0000canonical Hopf algebra in the braided category $mathcal{B}$. Finally, we \u0000explain how $Z$ refines the LMO functor, which is a TQFT-like functor extending \u0000the Le-Murakami-Ohtsuki invariant","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89362676","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 that the Witten--Reshetikhin--Turaev $mathrm{SU}(2)$ quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by one. We thus generalize a well--known result due to J. Roberts.
{"title":"Irreducibility of quantum representations of mapping class groups with boundary","authors":"T. Koberda, Ramanujan Santharoubane","doi":"10.4171/QT/116","DOIUrl":"https://doi.org/10.4171/QT/116","url":null,"abstract":"We prove that the Witten--Reshetikhin--Turaev $mathrm{SU}(2)$ quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by one. We thus generalize a well--known result due to J. Roberts.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2017-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77567615","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}
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