Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 2

String condensations in $3+1D$ and Lagrangian algebras

Pages: 583 – 622



Jiaheng Zhao (Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; and University of Chinese Academy of Sciences, Beijing, China)

Jia-Qi Lou (State Key Laboratory of Surface Physics, Fudan University, Shanghai, China; and Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai, China)

Zhi-Hao Zhang (Key Laboratory of Mathematics of C.A.S., School of Mathematical Sciences, University of Science & Technology of China, Hefei, China; and Shenzhen Institute for Quantum Science & Engineering, Southern University of Science & Technology, Shenzhen, China)

Ling-Yan Hung (State Key Laboratory of Surface Physics, Fudan University, Shanghai, China; Dept. of Physics & Center for Field Theory & Particle Physics, Fudan University, Shanghai, China; and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Liang Kong (Quantum Science & Engineering, Southern Univ. of Sci. & Tech., Shenzhen, China; Int’l Quantum Academy, Shenzhen, China; and Guangdong Provincial Key Lab. of Quantum Sci. & Engineering, Southern Univ. of Sci, & Tech., Shenzhen, China)

Yin Tian (Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, China)


We present three Lagrangian algebras in the modular $2$-category associated to the $3+1D \; \mathbb{Z}^2$ topological order and discuss their physical interpretations, connecting algebras with gapped boundary conditions, and interestingly, maps (braided autoequivalences) exchanging algebras with bulk domain walls. A Lagrangian algebra, together with its modules and local modules, encapsulates detailed physical data of strings condensing at a gapped boundary. In particular, the condensed strings can terminate at boundaries in non-trivial ways. This phenomenon has no lower dimensional analogue and corresponds to novel mathematical structures associated to higher algebras. We provide a layered construction and also explicit lattice realizations of these boundaries and illustrate the correspondence between physics and mathematics of these boundary conditions. This is a first detailed study of the mathematics of Lagrangian algebras in modular $2$-categories and their corresponding physics, that brings together rich phenomena of string condensations, gapped boundaries and domain walls in $3+1D$ topological orders.

Published 12 October 2023