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.
In this article we review how categorical equivalences are realized by renormalization group flow in physical realizations of stacks, derived categories, and derived schemes. We begin by reviewing the physical realization of sigma models on stacks, as (universality classes of) gauged sigma models, and look in particular at properties of sigma models on gerbes (equivalently, sigma models with restrictions on nonperturbative sectors), and decomposition, in which two-dimensional sigma models on gerbes decompose into disjoint unions of ordinary theories. We also discuss stack structures on example…
We consider a higher gauge topological model in three spatial dimensions whose input datum is a 2-group encoding the mixing of a 0-form $\mathbb Z_2$- and 1-form $\mathbb Z_3$-symmetry. We study the excitation content of the theory on the symmetry-preserving boundary. We show that boundary operators are organised into the fusion 2-category of 2-representations of the 2-group. These can be interpreted as categorical charges for an effective boundary model that inherits a global 2-group symmetry from the bulk topological order. Interestingly, we find that certain simple 2-representations are phy…
Sigma models effectively describe ordered phases of systems with spontaneously broken symmetries. At low energies, field configurations fall into solitonic sectors, which are homotopically distinct classes of maps. Depending on context, these solitons are known as textures or defect sectors. In this paper, we address the problem of enumerating and describing the solitonic sectors of sigma models. We approach this problem via an algebraic topological method -- combinatorial homotopy, in which one models both spacetime and the target space with algebraic objects which are higher categorical gene…
We review the categorical approach to the BPS sector of a 4d $\mathcal{N}=2$ QFT, clarifying many tricky issues and presenting a few novel results. To a given $\mathcal{N}=2$ QFT one associates several triangle categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than on what we can learn from a single description. A second g…
Korea Advanced Institute of Science Technology（韩国科学技术院）；Peking University（北京大学）；East China Normal University（华东师范大学）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Kavli IPMU (WPI), UTIAS, The University of Tokyo（东京大学 Kavli 宇宙物理学与数学研究所（WPI）、UTIAS）
Open-Closed String Field Theory from Calabi-Yau Categories and its Applications to Enumerative Geometry
Université Sorbonne Paris Nord（索邦巴黎北大学）；University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；Sorbonne Université（索邦大学）；Université de Montpellier（蒙彼利埃大学）；University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）；Tsinghua University（清华大学）；Université de Strasbourg（斯特拉斯堡大学）
This paper develops a concept of 2-categorical algebraic quantum field theories (2AQFTs) that assign locally presentable linear categories to spacetimes. It is proven that ordinary AQFTs embed as a coreflective full 2-subcategory into the 2-category of 2AQFTs. Examples of 2AQFTs that do not come from ordinary AQFTs via this embedding are constructed by a local gauging construction for finite groups, which admits a physical interpretation in terms of orbifold theories. A categorification of Fredenhagen's universal algebra is developed and also computed for simple examples of 2AQFTs.
