Norbert A'Campo
With the help of a new program, we do computations concerning the Witten-Reshetikhin-Turaev representations of mapping class groups. In particular we distinguish some mutant fibered knots. The program can be downloaded from http://www.geometrie.ch/TQFT
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We concretely construct a 2-categorically extended TQFT that extends the Reshetikhin-Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov-Voevodsky 2-vector spaces.
All once-extended 3D TQFTs are Reshetikhin--Turaev theories
该研究提出将Reshetikhin–Turaev TQFT扩展至圆范畴的直接几何构造，证明其与Bartlett等人的生成元-关系方法一致，替代了三维TQFT分类中的Cerf理论部分，得出所有一次扩展三维TQFT均为Reshetikhin–Turaev理论的结论
We derive an explicit formula for the Witten-Reshetikhin-Turaev SO(3)-invariants of lens spaces. We use the representation of the mapping class group of the torus corresponding to the Witten-Reshetikhin-Turaev SO(3)-TQFT to give such formula.
A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a p_1-structure, or an extended manifold structure. We perform the universal construction of Blanchet, Habegger, Masbaum and Vogel on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to t…
Equivalence of Extended $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs
本文证明了U(1) Chern-Simons理论与Reshetikhin-Turaev TQFTs在有限二次模结构下的等价性，展示了其在三维闭流形和 bordism 中的自然同构性。
Internal Reshetikhin-Turaev TQFT
该研究从带余端的ribbon范畴出发，推广Turaev的TQFT构造，通过特殊链环缠绕表示配边、结合容许着色与Kirby微积分推广，得到取值于透明对象对称幺半子范畴的内蕴TQFT，并验证其在模、预模范畴下的对应性质。
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 $Δ$-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 …
We work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Furthermore, we construct refined Turaev-Viro and Reshetikhin-Turaev TQFT's and study the relationship between them.
Equivalence of toral Chern-Simons and Reshetikhin-Turaev theories
本文证明了基于扭转Chern-Simons理论与由偶积分非退化对称双线性形式确定的有限二次模的Reshetikhin-Turaev理论之间存在自然同构，通过几何量子化方法建立扩展TQFT的自然同构。
Let C be a spherical fusion category. We prove that the Turaev-Viro-Barrett-Westbury state sum invariant of 3-manifolds derived from C is equal to the Reshetikhin-Turaev surgery invariant of 3-manifolds derived from Z(C), where Z(C) is the Drinfeld-Joyal-Street center of C.
On the Kashaev--Luo--Vartanov Partition Function for Cusped 3-Manifolds: Asymptotics and a squared norm property
本文研究尖点三维流形的KLV配分函数，证明其渐近性由双曲锥结构的体积和挠率决定，并引入Reshetikhin--Turaev型函数，将归一化KLV配分函数表示为该函数平方范数的加权积分，推广了Turaev--Viro与Reshetikhin--Turaev不变量之间的关系。
In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category $C$, to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center $Z(C)$. In the present paper we give a partial proof of this statement.
3d TQFTs and 3-manifold invariants
本文综述3维拓扑量子场论及其3-流形量子不变量，梳理融合范畴、WRT 与 Turaev-Viro 理论及其关系，并介绍扩展、缺陷、同伦和非半单推广。
We find decomposition series of length at most two for modular representations in positive characteristic of mapping class groups of surfaces induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at the p-th root of unity, where p is an odd prime. The dimensions of the irreducible factors are given by Verlinde-type formulas.
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.
We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to the p-modular representation theory of the symmetric groups S_n. We construct resolutions of simple p-modular TQFT's and S_n-modules over Z/pZ using powers of Lefschetz operators. These yield expansions of the irreducible p-modular S_n-characters in terms of the ordianry ones as …
We define a universal state sum construction which specializes to most previously known state sums (Turaev-Viro, Dijkgraaf-Witten, Crane-Yetter, Douglas-Reutter, Witten-Reshetikhin-Turaev surgery formula, Brown-Arf). The input data for the state sum is an n-category satisfying various conditions, including finiteness, semisimplicity and n-pivotality. From this n-category one constructs an n+1-dimensional TQFT, and applying the TQFT gluing rules to a handle decomposition of an n+1-manifold produces the state sum.
We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in (arXiv:1202.3553) including the Kashaev invariant of links. Here the modular category framework does not apply and we use the ``universal construction''. Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a roo…
Fixed-parameter tractable computation of Reshetikhin--Turaev knot polynomials via tensor networks
本文通过张量网络上的张量收缩分析Reshetikhin-Turaev扭结多项式的时间复杂度，证明其关于输入扭结图树宽的线性参数是固定参数可处理的，并结合树分解近似算法恢复亚指数时间界。
This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, $\mathrm{N}_r$ and $\mathrm{N}^0_r$, using non-semisimple categories of representations of a quantum version of $\mathfrak{sl}_2$ at a $2r$-th root of unity with $r \geqslant 2$. The secondary invariants $\mathrm{N}^0_r$ conjecturally extend the original Reshetikhin-Turaev quantum …
Towards a complexity-theoretic dichotomy for TQFT invariants
本文证明了任意固定复数域上$(2+1)$维TQFT计算闭3-流形不变量问题存在多项式时间可解或张量收缩$\\#\mathsf{P}$-困难的二分法，并指出未来可直接针对流形不变量改进该归约。
In this paper, we establish the general theory of (2+1)-dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev-Viro-Ocneanu unitary TQFT's obtained from subfactors satisfy the axioms of TQFT's with Verlinde bases. Hence, in a Turaev-Viro-Ocneanu TQFT, we have a Dehn surgery formula for 3-manifolds. It turns out that this Dehn surgery formula is nothing but the formula of the Reshetikhin-Turaev invariant constructed from a tube system,…
Density and unitarity of the Burau representation from a non-semisimple TQFT
本文利用四次单位根处的非半单 TQFT 构造 Burau 表示上的 Squier Hermitian 形式，并证明辫群在可能不定的酉表示空间中像稠密，同时探讨其拓扑量子计算应用。
A TQFT-based Platform for Efficient Computation of Knot Invariants
该平台统一费曼带状图构建、高阶陈 - 西蒙斯纽结不变量评估及类似FRD纽结识别，能将FRD构建为张量网络，评估相关不变量并识别纽结，是首个结合多种功能于统一工作流程的平台。
A geometrical description of untwisted 3d Dijkgraaf-Witten TQFT with defects
本文通过几何方法构建了带有所有维数缺陷的3维未扭Dijkgraaf-Witten理论，利用对称单子函子将缺陷cobordism范畴映射到复向量空间范畴，无需三角化状态和更高范畴的图示计算。
In this article, we discuss a (2+1)-dimensional topological quantum field theory, for short TQFT, with a Verlinde basis. As a conclusion of this general theory, we have a Dehn surgery formula. We show that Turaev-Viro-Ocneanu TQFT has a Verlinde basis. Several applications of this theorem are exposed. Based on Izumi's data of subfactors, we list several computations of Turaev-Viro-Ocneanu invariants for some 3-manifolds.
Zesting and the relative complexity of Reshetikhin-Turaev invariants
该研究证明 zesting 构造关联的 Reshetikhin-Turaev 不变量计算复杂性不变，开发了计算相关链环不变量的多项式时间算法，将其识别为 rack 上同调不变量，为拓扑量子场论等的复杂性层次结构提供了依据。
Computation of leading coefficients in asymptotics of relative quantum invariants
该研究提出相对Reshetikhin-Turaev等三类不变量的渐近展开猜想，用伴随扭曲Reidemeister挠率等表示首项系数，对指数项已知的特殊情况证明猜想，其展开项为几何不变量且含新的相对情形特有项。
In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjugate linear isomorphism preserving the products of the two algebras and commuting with the actions of SL(2,Z). Via this correspondence and the Dehn surgery formula, we compute Turaev-Viro-Ocneanu invariants from several subfactors for basic 3-manifolds including lens spaces and Bri…
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