Tensor complex renormalization with generalized symmetry and topological bootstrap

TL;DR

This paper introduces a tensor complex renormalization algorithm for conformal field theories that leverages topological data to reconstruct fixed-point tensors, bridging topological and dynamical aspects of CFT.

Contribution

It develops a tensor complex renormalization method based on generalized symmetry and topological bootstrap, enabling reconstruction of fixed-point tensors from topological data alone.

Findings

Full fixed-point tensor can emerge from RG flow starting with three-point functions.

The topological bootstrap reconstructs the entire tensor from topological data.

Method suggests a pathway to algebraic descriptions of gapless quantum states.

Abstract

Recent progress in generalized symmetry and topological holography has shown that, in conformal field theory (CFT), topological data from one dimensional higher can play a key role in determining local dynamics. Based on this insight, a fixed-point (FP) tensor complex (TC) for CFT has recently been constructed. In this work, we develop a TC renormalization (TCR) algorithm adapted to this CFT-based structure, forming a renormalization-group (RG) framework with generalized symmetry. We show that the full FP tensor can emerge from the RG flow starting with only the three-point function of the primary fields. Remarkably, even when starting solely from topological data, the RG process can still reconstruct the full FP tensor--a method we call as topological bootstrap. This approach deepens the connection between the topological and dynamical aspects of CFT and suggests pathways toward a fully algebraic description of gapless quantum states, with potential extensions to higher dimensions.