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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2024 Volume 20, 102, 51 pp. (Mi sigma2104)

The Evaluation of Graphs on Surfaces for State-Sum Models with Defects

Julian Farnsteiner, Christoph Schweigert

Fachbereich Mathematik, Universität Hamburg, Bereich Algebra und Zahlentheorie, Bundesstraße 55, 20146 Hamburg, Germany

Abstract: The evaluation of graphs on 2-spheres is a central ingredient of the Turaev–Viro construction of three-dimensional topological field theories. In this article, we introduce a class of graphs, called extruded graphs, that is relevant for the Turaev–Viro construction with general defect configurations involving defects of various dimensions. We define the evaluation of extruded graphs and show that it is invariant under a set of moves. This ensures the computability and uniqueness of our evaluation.

Keywords: state-sum, vertex evaluation, topological quantum field theory, $6j$ symbols, defects.

MSC: 18M30, 18M20, 81T45

Received: December 10, 2023; in final form November 6, 2024; Published online November 18, 2024

Language: English

DOI: 10.3842/SIGMA.2024.102


ArXiv: 2312.01946


© Steklov Math. Inst. of RAS, 2025