Topological defects: from tensor categories to exotic phases of matter

A keystone in the understanding of nature, from quantum field theories to exotic states of matter, is the interplay between symmetry and topology. Topological defects provide crucial insight into this relationship. This project aims at deepening that insight, whereby it will also provide novel computational tools.

Specific issues to be examined include the construction of state-sum topological field theories (TFTs) with orientation-reversing defects, the extension of string-net models to unoriented surfaces and beyond semisimplicity and finiteness restrictions, and the use of TFTs with embedded defect networks for clarifying the behavior of fracton phases.

A long-term goal is a universal state-sum construction of so-called defect TFTs, in which defect networks form an integral part of the input data.The different tasks will largely be pursued in parallel, in varying collaborations.

The work will be based on a systematic exploitation of powerful mathematical methods and tools from the theory of tensor categories as well as higher-categorical structures, not making any simplifying assumptions that could restrict their applicability. Preparations will start during 2022 in combination with ongoing work.

The results of the project will be of intrinsic interest in mathematical physics as well as in pure mathematics, but also admit spin-offs in other areas.

In particular, substantial contributions to the understanding of topological phases, including fracton phases, are foreseen.