Bulk Operator Reconstruction in Topological Tensor Network and Generalized Free Fields

TL;DR

This paper investigates operator reconstruction in holographic tensor networks derived from topological quantum field theories, revealing how the bulk operator behavior scales with group order and identifying when generalized free fields emerge.

Contribution

It introduces a detailed analysis of bulk operator reconstruction in topological tensor networks, highlighting the role of group structure and fusion categories in the emergence of generalized free fields.

Findings

Bulk operators scale with the order of the group in Dijkgraaf-Witten theories.

Generalized free fields appear in 2d and 3d bulk models based on certain groups.

No generalized free fields are found in models from more complex fusion categories.

Abstract

In this paper, we would like to study operator reconstruction in a class of holographic tensor networks describing renormalization group flows studied in arXiv:2210.12127. We study examples of 2d bulk holographic tensor networks constructed from Dijkgraaf-Witten theories and found that for both $\mathbb{Z}_n$ group and $S_3$ group the number of bulk operators behaving like a generalized free field in the bulk scales as the order of the group. We also generalize our study to 3d bulks and found the same scaling for $\mathbb{Z}_n$ theories. However, there is no generalized free field when the bulk comes from more generic fusion categories such as the Fibonacci model.