Tensor networks and representation theory

• Applicant: Professor Dr. Christoph Schweigert
• Subject Area: Mathematics
• Term: from 2020 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 449480360

• Final Report Year: 2024

Final Report Abstract

In view of the long-term goal to develop modular functors based on pivotal Grothendieck-Verdier categories and to understand correlators in a large class of two-dimensional conformal field theories in this setting, the project contained two subprojects for two PhD students. In the first subproject, we developed a systematic approach to all correlators (including boundary fields and multi-pronged defect fields) in rational conformal field theories, based on string-nets. A deeper understanding of this approach could be obtained via a novel string-net construction for pivotal bicategories. The relation of these bicategorical string-nets to traditional string-nets in terms of a Frobenius functor captures the correlators in terms of a universal correlator. In the second subproject, we obtained a geometric understanding of the equivariance of Frobenius-Schur indicators in terms of state-sum models on three-manifolds with boundaries. Such models need a generalized graphical calculus which was developed as well.

Publications

• Frobenius–Schur indicators and the mapping class group of the torus. Letters in Mathematical Physics, 112(2).
  Farnsteiner, Julian & Schweigert, Christoph
  
• String-Net Construction of RCFT Correlators. SpringerBriefs in Mathematical Physics. Springer International Publishing.
  Fuchs, Jürgen; Schweigert, Christoph & Yang, Yang
  
• A G-Equivariant String-Net Construction. Annales Henri Poincaré, 25(1), 297-345.
  DeLazzer, Meunier Adrien; Schweigert, Christoph & Traube, Matthias
  
• The Evaluation of Graphs on Surfaces for State-Sum Models with Defects. Symmetry, Integrability and Geometry: Methods and Applications.
  Farnsteiner, Julian & Schweigert, Christoph
  
• Twisted Drinfeld centers and framed string-nets. Quantum Topology, 15(3), 537-566.
  Knötzele, Hannes; Schweigert, Christoph & Traube, Matthias
  
• Algebraic Structures in Two-Dimensional Conformal Field Theory. Encyclopedia of Mathematical Physics, 604-617. Elsevier.
  Fuchs, Jürgen; Schweigert, Christoph; Wood, Simon & Yang, Yang
  
• Duality structures for representation categories of vertex operator algebras and the Feigin–Fuchs boson. Selecta Mathematica, 31(2).
  Allen, Robert; Lentner, Simon; Schweigert, Christoph & Wood, Simon
  
• Grothendieck-Verdier duality in categories of bimodules and weak module functors. Contemporary Mathematics, 211-234. American Mathematical Society.
  Fuchs, Jürgen; Schaumann, Gregor; Schweigert, Christoph & Wood, Simon
  
• Spherical Morita contexts and relative Serre functors. Kyoto Journal of Mathematics, 65(3).
  Fuchs, Jürgen; Galindo, César; Jaklitsch, David & Schweigert, Christoph
  
• String-net models for pivotal bicategories. Theory and Applications of Categories, 44, 474-543.
  Fuchs, Jürgen; Schweigert, Christoph & Yang, Yang
  
• The Lyubashenko modular functor for Drinfeld centers via non-semisimple string-nets. Advances in Mathematics, 488, 110770.
  Müller, Lukas; Schweigert, Christoph; Woike, Lukas & Yang, Yang