Steven Duplij, Wladyslaw Marcinek
A proposal of the concept of $n$-regular obstructed categories is given. The corresponding regularity conditions for mappings, morphisms and related structures in categories are considered. An n-regular TQFT is introduced. It is shown the connection of time reversibility with the regularity.
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These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects', assuming only superficial familiarity with closed TQFTs in terms of commutative Frobenius algebras. The generalisation of this relation is a construction of pivotal 2-categories from defect TQFTs. We review this construction in detail, flanked by a range of examples. Furthermore we explain how open/closed TQFTs are equivalent to Calabi-Yau categories and the Cardy condition, and how to extract such data from pivotal 2-categories.
We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.
Regular and higher regular graded algebras (in simplest case satisfying Von Neumann regularity $Θ_{1}Θ_{2}Θ_{1}=Θ_{1}$ instead of anticommutativity) are introduced and their properties are studied. They are described in terms of obstructed categories with nonclosed invertible and noninvertible morphisms for which generalized (obstructed) functors and natural transformations are given. Regular algebras and bialgebras are considered with examples. Corresponding regularizations of the cross product and Wick algebras are made.
We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category $\mathcal{C}$, using the language of defect TQFT from [arXiv:1705.06085]: (i) spherical fusion categories give orbifolds for the "trivial" defect TQFT associated to vect, (ii) $G$-crossed extensions of $\mathcal{C}$ give group orbifolds for any finite group $G$, and (iii) we construct orbifolds from commutative $Δ$-separable symmetric Frobenius algebras in $\mathcal{C}$. We also explain how the Turaev-Viro state sum construction fits into our framework by proving that it is isomorphic…
We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category $\mathcal{C}$, there is a self enriched multi-fusion category $\mathfrak{C}$ giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the extended 3D TQFT given by the fully dualizable object $\mathfrak{C}$ extends the 1-2-3-dimensional Reshetikhin-Turaev TQFT associated to the modular tensor category $\mathcal{C}$ down to dimension zero.
We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of…
Constructing TQFTs Using Non-Hermitian Ribbon Fusion Categories
本文提出了一种从非厄米带环融合范畴构造拓扑量子场论的简单方法，并通过证明Müger中心、辫结构和球面结构的相关定理加以验证。
3-dimensional TQFTs from derived categories of quantum group representations
该研究证明有限模张量范畴A的导出∞-范畴D(A)支持三维拓扑量子场论，其无标记版本产生射影映射类群作用，相关导出场论可作为N=4超对称QFT拓扑A扭曲的数学形式化。
Unitary TQFTs, unitary disk-like $n$-categories, and higher Hilbert spaces
本文提出用完备有限酉圆盘状$n$-范畴定义有限$(n+1)$-希尔伯特空间，验证了$n=1,2$情形，并由此分类了低维酉TQFT。
Skein (3+1)-TQFTs from Non-Semisimple Ribbon Categories
本文基于绳结理论，定义了非半单有限单模绳结张量范畴的(3+1)维拓扑量子场论，提出'色非退化'条件，并推广了Crane-Yetter-Kauffman和Lyubashenko等不变量。
It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,λ)$, where $G$ is a compact Lie group, and $λ\in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the $E_6$ subfactor, and the other is the quantum double of the even sectors of the Haagerup subfactor. We cannot prove mathematically that the two TQFTs are indeed counterexamples because CSW TQFTs, while physically defined, are not yet mat…
Vladimir Turaev discovered in the early years of quantum topology that the notion of modular category was an appropriate structure for building 3-dimensional Topological Quantum Field Theories (TQFTs for short) containing invariants of links in 3-manifolds such as Witten-Reshetikhin-Turaev ones. In recent years, generalized notions of modular categories, which relax the semisimplicity requirement, have been successfully used to extend Turaev's construction to various non-semisimple settings. We report on these recent developments in the domain, showing the richness of Vladimir's lineage.
Non compact (2+1)-TQFTs from non-semisimple spherical categories
该研究提出曲面容许skein模与色范畴概念，构造了有限维非紧(2+1)-TQFT，证明其可延拓为真正TQFT的充要条件，并验证球面张量范畴均为色范畴，拓展了(2+1)-TQFT的构造范围。
Constructing new open-closed TQFTs from the interpolation of symmetric monoidal categories
本研究基于Lauda和Pfeiffer关于二维开闭TQFT与KFAs等价的结论，通过任意标量序列构建含KFA对象的对称幺半范畴，确定了生成半单阿贝尔范畴的序列条件，得到插值特定KFA自同构群表示范畴的新范畴族。
Levin-Wen models are microscopic spin models for topological phases of matter in (2+1)-dimension. We introduce a generalization of such models to (3+1)-dimension based on unitary braided fusion categories, also known as unitary premodular categories. We discuss the ground state degeneracy on 3-manifolds and statistics of excitations which include both points and defect loops. Potential connections with recently proposed fractional topological insulators and projective ribbon permutation statistics are described.
Braided categories of bimodules from stated skein TQFTs
本文通过陈述 skein TQFTs 构建了 braided 分类，并将其与 ribbon 单则代数的模分类联系起来，揭示了其与 Kerler-Lyubashenko TQFT 的关系。
We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($\text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($\text{TVBW}$) $\text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $\text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. It is shown that a …
Cochain valued TQFTs from nonsemisimple modular tensor categories
本文将向量空间值 TQFT 扩展为上链值拓扑场论，构造从 ribbon 配边范畴到线性上链范畴的对称幺半函子，并证明其保持同伦，从而局部化为取值于 dg 向量空间导出 ∞-范畴的理论，为导出量子群表示的导出 TQFT 奠定基础。
String nets for twisted pivotal categories
本文为具有扭曲主结构的幺半范畴开发了一种图形演算，并利用它构造了带Morse函数或Morse叶状结构的曲面上的扭曲弦网模，证明了这些模构成一个定向的范畴化2-TQFT，并研究了2-球面上扭曲弦网模的消失条件及其与有限张量范畴中可逆对象的联系。
These notes offer a lightening introduction to topological quantum field theory in its functorial axiomatisation, assuming no or little prior exposure. We lay some emphasis on the connection between the path integral motivation and the definition in terms symmetric monoidal categories, and we highlight the algebraic formulation emerging from a formal generators-and-relations description. This allows one to understand (oriented, closed) 1- and 2-dimensional TQFTs in terms of a finite amount of algebraic data, while already the 3-dimensional case needs an infinite amount of data. We evade these …
Using face algebras (i.e. algebras of L-operators of IRF models), we construct modular tensor categories with positive definite inner product, whose fusion rules and S-matrices are the same as (or slightly different from) those obtained by $U_q (\frak{sl}_{N})$ at roots of unity. Also we obtain state-sums of ABF models on framed links which give quantum SU(2)-invariants of corresponding 3-manifolds.
A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We compute this formulas and discuss reductions and refinements for modular categories related with SU(N).Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer.We introduce the notion of a sp…
Is Crane--Yetter fully extended?
本文利用稳定同伦论分类可逆四维 TQFT 的完全扩展，证明其由可逆对象等价类和六次单位根决定，并发现无穷多个完全扩展可逆 TQFT 重现 Crane-Yetter 配分函数，由 Witt 群的 Z/6-扩张参数化。
The spin-statistics theorem for topological quantum field theories
本文在Atiyah框架下通过引入费米子dagger紧致范畴，证明了拓扑量子场论的自旋-统计定理，并统一了自旋结构、超向量空间与幺正性。
We construct tensor and bitensor categories with given Grothedieck rig (fusion algebra) in simple cases. The results provide examples on which to test the conjectural construction of 4-D TQFT's proposed by Crane and Frenkel and shed light on several other constructions of TQFT's.
We propose a new mwthod of constructing 4D-TQFTs. The method uses a new type of algebraic structure called a Hopf Category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases.
Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua
本文研究Haagerup-Izumi范畴双范畴的模数据，通过双曲字构造Bethe真空，修正Q系统表条目，给出Bethe数据闭式解，明确相关模数据性质。
The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a "suspension" operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k >= n+2. We give evidence for this hypothesis and describe its relation to stable homotopy theory. We then propose a descri…
Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properties, including a higher dimensional generalization of Deligne's conjecture about the action of the little disks operad on Hochschild cochains. Along the way, we give a definition of a weak n-category with strong duality which is particularly well suited for work with TQFTs.
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