Vishvajit V. S. Gautam
In this paper we propose a naive construction of 2-dimensional extended topological quantum field theories (TQFTs), which can be further generalized to the higher-dimension extended TQFTs.
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On the Higher Categorical Structure of Topological Defects in Quantum Field Theories
本文提出一个基于结构化高阶dagger范畴的统一数学框架，用于描述量子场论中带切向结构的拓扑缺陷的高阶范畴结构，并在分层配边假设下对具有稳定切向结构且允许直和的情形给出了证明。
It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z/2Z-orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Main exa…
Using a quiver algebra of a cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type A on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.
This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are viewed as symmetric monoidal functors from topological cobordism categories into the category of vector spaces. In two dimensions, they are classified by Frobenius algebras. In three dimensions, and under certain conditions, they are classified by modular categories. These are special kinds of categories in which topological notions such as braidings and twists play a prominent role. There is a powerful graphical calculus available for working in such categ…
This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topological quantum field theory. Our chronology ends around 2000, with just a taste of later developments such as open-closed topological string theory, the categorification of quantum groups, Khovanov homology, and Lurie's work on the classification of topological quantum field theories.
Topological proofs of categorical coherence
本研究利用单连通CW复形1-骨架等拓扑工具给出范畴化非对称操作子和对称幺半范畴Mac Lane一致性定理的拓扑证明，提出组合同伦概念并结合多面体复形得到几何证明，用Morse理论比较两种方法的普适性，探讨向高范畴推广的可能。
A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\infty)$. For a complete duality datum in the Hernandez-Leclerc category $\mathcal{C}^0_{\mathfrak{g}}$ of a quantum affine algebra $U_q'(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in $\mathcal{C}^0_{\mathfrak{g}}$ has an extended crystal structure isomorphic to the extended crystal $\widehat{B}_{\mathfrak{g}}(\inft…
Boundary symmetries of (2+1)D topological orders
本文提出代数框架描述(2+1)维拓扑序带隙边界的内禀拓扑对称性，并给出范畴论障碍，阻碍体态群对称性与边界对称性的相干性。
We construct a four dimensional topological Quantum Field Theory from a modular tensor category. We complete the proof in the case of SU(2)q at a root of unity. Our construction may be important in the physical interpretation of the Chern Simons state in the Ashtekar variables.
The Classification of 3+1d Symmetry Enriched Topological Order
该研究利用2 - 范畴的（去）等变对有有限\(G\) - 对称性的（3 + 1）维拓扑序分类，指出含\(G\) - 对称性的（3 + 1）维费米子拓扑序对应特定范畴，且相关范畴数据与特定构造一致，还说明了此方法可对所有辫合融合2 - 范畴分类。
Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
本研究针对带阿贝尔在位对称性的1+1维有能隙相，用富集融合范畴描述其时空可观测量，统一了对称相分类与格点对偶性，给出非平凡SPT相等实例，通过全息对偶关联二维拓扑序与一维有能隙量子液体，为量子相统一定义提供思路。
Mutual Influence of Symmetries and Topological Field Theories
研究在允许与TQFT堆叠作为等价关系时，费米子(2+1)d QFT的融合2-范畴对称性如何被修改，发现与Spin(n)_1堆叠会导致有限种不等价修改。
Fusion rule in conformal field theories and topological orders: A unified view of correspondence and (fractional) supersymmetry and their relation to topological holography
本文提出Z_N扩展的chiral和bulk共形场论中的融合规则及其对应的拓扑有序，通过构建非平凡的子代数结构揭示拓扑双影的对应关系，并统一了对称性和拉格朗日子代数。
Categorical spin-networks: state-sum invariants of 4-manifolds from higher-gauge theory
本文以枢轴融合2-范畴为输入定义(3+1)维TQFT及其格点实现，引入20j符号计算4-单纯形散射振幅，验证Pachner移动下拓扑不变性，并构造Levin-Wen型格点哈密顿模型，作为Turaev-Viro不变式的高维类比。
It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n> 1. Recently the physical meaning of these higher categorical structures has been recognized and concrete examples of Extended TFTs have been constructed. Some of these examples, like the Rozansky-Witten model, are of geometric nature, while others are related to representation theory. I outline two application of higher-dimensional TFTs. One is related to the problem of classifying monoidal deformations of the derived categ…
The Categorical 't Hooft Expansion
本文提出通过范畴方法推导QFT与弦理论对偶的策略，利用dg-TFT和$A_\infty$-范畴结构，研究大N展开下的D-膜变形。
Quantum cellular automata and categorical dualities of spin chains
本文研究了自旋链中范畴双射的性质，通过Doplicher-Haag-Roberts双模工具，提出了一种扩展双射到更大代数的准则，并分类了群情况下的双射。
We prove a decomposition formula for the dimensional reduction of an extended topological field theory that arises as an orbifold of an equivariant topological field theory. Our decomposition formula can be expressed in terms of a categorification of the integral with respect to groupoid cardinality. The application of our result to topological field theories of Dijkgraaf-Witten type proves a recent conjecture of Qiu-Wang.
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