2-categorical Poincare Representations and State Sum Applications

A 2-categorical state sum model

It has long been argued that higher categories provide the proper algebraic structure underlying state sum invariants of 4-manifolds. This idea has been refined recently, by proposing to use 2-groups and their representations as specific examples of 2-categories. The challenge has been to make these proposals fully explicit. Here we give a concrete realization of this program. Building upon our earlier work with Baez and Wise on the representation theory of 2-groups, we construct a four-dimensional state sum model based on a categorified version of the Euclidean group. We define and explicitly…

范畴自旋网络：来自高规范理论的4流形态和不变量的态和公式

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。

论量子场论中拓扑缺陷的高阶范畴结构

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。

Twisted Quantum Deformations of Lorentz and Poincaré algebras

We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deformations are obtained in explicit form. This work is an extended version of the paper \url{arXiv:0704.0081v1 [math.QA]}.

KLR algebras and the branching rule I: the categorical Gelfand-Tsetlin basis in type An

We define a quotient of the category of finitely generated modules over the cyclotomic Khovanov-Lauda-Rouquier algebra for type An and show it has a module category structure over a direct sum of certain cyclotomic Khovanov-Lauda-Rouquier algebras of type An-1, this way categorifying the branching rules for the inclusion of sl(n) in sl(n+1). Using this we give a new, elementary proof of Khovanov-Lauda cyclotomic conjecture. We show that continuing recursively gives the Gelfand-Tsetlin basis for type An. As an application we prove a conjecture of Mackaay, Stosic and Vaz concerning categorical W…

New twisted quantum deformations of D=4 super-Poincare algebra

We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called subordinated r-matrices of super-Abelian and super-Jordanian type. Corresponding twists describing quantum deformations are obtained in an explicit form. These twists are the super-extensions of twists obtained in the paper arXiv:0712.3962.