{"title":"Invariants of $mathbb{Z}/p$-homology 3-spheres from the abelianization of the level-$p$ mapping class group","authors":"Wolfgang Pitsch, Ricard Riba","doi":"10.4171/qt/196","DOIUrl":"https://doi.org/10.4171/qt/196","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139264357","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}
A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of a G-crossed braided spherical fusion category.
{"title":"Evaluating TQFT invariants from $G$-crossed braided spherical fusion categories via Kirby diagrams with 3-handles","authors":"Manuel Bärenz","doi":"10.4171/qt/183","DOIUrl":"https://doi.org/10.4171/qt/183","url":null,"abstract":"A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of a G-crossed braided spherical fusion category. ","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136229844","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}
{"title":"Actions of $sltwo$ on algebras appearing in categorification","authors":"Ben Elias, You Qi","doi":"10.4171/qt/181","DOIUrl":"https://doi.org/10.4171/qt/181","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136282070","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}
{"title":"On $mathfrak{sl}(N)$ link homology with mod $N$ coefficients","authors":"Joshua Wang","doi":"10.4171/qt/194","DOIUrl":"https://doi.org/10.4171/qt/194","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818384","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 propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construction works under the assumption that the algebra is braided commutative. The resulting knot invariant is a module with a coadjoint action. Taking the coinvariants yields a new quantum character variety that may be thought of as an alternative to the skein module. We give concrete examples for a few of the simplest knots and links.
{"title":"Quantized representations of knot groups","authors":"Jun Murakami, Roland van der Veen","doi":"10.4171/qt/191","DOIUrl":"https://doi.org/10.4171/qt/191","url":null,"abstract":"We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construction works under the assumption that the algebra is braided commutative. The resulting knot invariant is a module with a coadjoint action. Taking the coinvariants yields a new quantum character variety that may be thought of as an alternative to the skein module. We give concrete examples for a few of the simplest knots and links.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818389","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}
In this paper we construct a topological model for the Witten-Reshetikhin-Turaev invariants for $3$-manifolds coming from the quantum group $U_q(sl(2))$, as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level $cN in N$ we show that the level $cN$ WRT invariant for a $3-$manifold is a state sum of Lagrangian intersections in a covering of a {bf fixed} configuration space in the punctured disk. This model brings a new perspective on the structure of the level $cN$ Witten-Reshetikhin-Turaev invariant, showing that it is completely encoded by the intersection points between certain Lagrangian submanifolds in a fixed configuration space, with additional gradings which come from a particular choice of a local system. This formula provides a new framework for investigating the open question about categorifications of the WRT invariants.
本文构造了来自量子群$U_q(sl(2))$的$3$流形的Witten-Reshetikhin-Turaev不变量的拓扑模型，作为组态空间中同调类的渐变交对。更准确地说，对于一个固定的水平$cN in N$，我们证明了一个$3-$流形的水平$cN$ WRT不变量是穿孔盘中一个{bf固定}位形空间覆盖上的拉格朗日交点的状态和。该模型对水平$cN$ Witten-Reshetikhin-Turaev不变量的结构带来了新的视角，表明它完全由固定位形空间中某些拉格朗日子流形之间的交点编码，并带有来自局部系统的特定选择的附加等级。该公式为研究WRT不变量的分类问题提供了一个新的框架。
{"title":"Witten–Reshetikhin–Turaev invariants for 3-manifolds from Lagrangian intersections in configuration spaces","authors":"Cristina Ana-Maria Anghel","doi":"10.4171/qt/190","DOIUrl":"https://doi.org/10.4171/qt/190","url":null,"abstract":"In this paper we construct a topological model for the Witten-Reshetikhin-Turaev invariants for $3$-manifolds coming from the quantum group $U_q(sl(2))$, as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level $cN in N$ we show that the level $cN$ WRT invariant for a $3-$manifold is a state sum of Lagrangian intersections in a covering of a {bf fixed} configuration space in the punctured disk. This model brings a new perspective on the structure of the level $cN$ Witten-Reshetikhin-Turaev invariant, showing that it is completely encoded by the intersection points between certain Lagrangian submanifolds in a fixed configuration space, with additional gradings which come from a particular choice of a local system. This formula provides a new framework for investigating the open question about categorifications of the WRT invariants.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103393","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 consider Legendrian links and tangles in $J^1S^1$ and $J^1[0,1]$ equipped with Morse complex families over a field $mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $mathbb{F}_2$, this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod $2$ spin number. We apply the classification to construct augmented Legendrian surfaces in $J^1M$ with $mathrm{dim} M = 2$ realizing any prescribed monodromy representation, $Phi:pi_1(M,x_0) to mathrm{GL}(mathbf{n}, mathbb{F})$.
我们考虑了在场$mathbb{F}$上具有莫尔斯复族的$J^1S^1$和$J^1[0,1]$中的勒让连链和缠结，并将它们分类为勒让连协。当系数域为$mathbb{F}_2$时，这为配备了Legendrian接触同调dg -代数的增广的Legendrian提供了一种协配分类。由光纤上同调、梯度单矩阵和模$2$自旋数提供了一组可以得到任意值的不变量。我们应用分类构造了$J^1M$中的增广Legendrian曲面，其中$mathrm{dim} M = 2$实现了任意规定的单形表示，$Phi:pi_1(M,x_0) to mathrm{GL}(mathbf{n}, mathbb{F})$。
{"title":"Augmented Legendrian cobordism in $J^1S^1$","authors":"Yu Pan, Dan Rutherford","doi":"10.4171/qt/195","DOIUrl":"https://doi.org/10.4171/qt/195","url":null,"abstract":"We consider Legendrian links and tangles in $J^1S^1$ and $J^1[0,1]$ equipped with Morse complex families over a field $mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $mathbb{F}_2$, this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod $2$ spin number. We apply the classification to construct augmented Legendrian surfaces in $J^1M$ with $mathrm{dim} M = 2$ realizing any prescribed monodromy representation, $Phi:pi_1(M,x_0) to mathrm{GL}(mathbf{n}, mathbb{F})$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134973344","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}
A modular fusion category $mathcal{C}$ allows one to define projective representations of the mapping class groups of closed surfaces of any genus. We show that if all these representations are irreducible, then $mathcal{C}$ has a unique Morita class of simple non-degenerate algebras, namely, that of the tensor unit. This improves on a result by Andersen and Fjelstad, albeit under stronger assumptions. One motivation to look at this problem comes from questions in three-dimensional quantum gravity.
{"title":"Mapping class group representations and Morita classes of algebras","authors":"Iordanis Romaidis, Ingo Runkel","doi":"10.4171/qt/192","DOIUrl":"https://doi.org/10.4171/qt/192","url":null,"abstract":"A modular fusion category $mathcal{C}$ allows one to define projective representations of the mapping class groups of closed surfaces of any genus. We show that if all these representations are irreducible, then $mathcal{C}$ has a unique Morita class of simple non-degenerate algebras, namely, that of the tensor unit. This improves on a result by Andersen and Fjelstad, albeit under stronger assumptions. One motivation to look at this problem comes from questions in three-dimensional quantum gravity.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666552","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 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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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