Wilson Towers as Local Bulk Fields
Abstract
Multi-winding Wilson loops (Wilson spools) in 2+1-dimensional gravity can reproduce one-loop partition functions of local free fields in the bulk. Bulk free fields have a second-quantized Fock space, and the AdS/CFT correspondence suggests that the associated multi-particle sectors are dual to multi-trace primaries in the CFT. In this note, we use standard symmetric-function methods to show that the Wilson spool in thermal AdS3 can be recast as a sum over single-winding Wilson loops -- one for each multi-trace primary. In a companion paper to appear, we will view Wilson networks and TQFTs as the natural language of non-perturbative bulk quantum gravity. The present note illustrates how this can apply to local bulk fields, and not just defects: a bulk (generalized free) field is to be viewed as a full tower of multi-trace Wilson lines. We further show that the SL(2) descendants of each multi-trace primary, together with the boundary gravitons of the AdS3 background, correctly reproduce the full Virasoro character of each module. In this language, the role of a light insertion on a heavy primary is played by a topological Verlinde line. This allows us to obtain a "microscopic" description of one-loop determinants on the smooth BTZ handlebody. A spatially wound probe Wilson line on a torus with a contractible thermal cycle can be traded for a Verlinde line inserted on a dense family of heavy Polyakov loops -- with the roles of the two cycles exchanged, so that the spatial cycle is now contractible. The vacuum row of the modular S-kernel acts as the (approximate) density of the heavy primaries. This reinforces the case made in arXiv:2601.18775 that a smooth horizon is a stand-in for an ensemble of quantum states, each produced by a heavy Wilson line that appears as a singular horizon in the semi-classical limit.
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BibTeX
@misc{gayari2026wilson,
title = {{Wilson Towers as Local Bulk Fields}},
author = {Vishal Gayari and Chethan Krishnan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.21397},
eprint = {2607.21397},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21397},
}
RIS
TY - UNPB
TI - Wilson Towers as Local Bulk Fields
AU - Vishal Gayari
AU - Chethan Krishnan
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2607.21397
UR - https://arxiv.org/abs/2607.21397
AB - Multi-winding Wilson loops (Wilson spools) in 2+1-dimensional gravity can reproduce one-loop partition functions of local free fields in the bulk. Bulk free fields have a second-quantized Fock space, and the AdS/CFT correspondence suggests that the associated multi-particle sectors are dual to multi-trace primaries in the CFT. In this note, we use standard symmetric-function methods to show that the Wilson spool in thermal AdS_3 can be recast as a sum over single-winding Wilson loops -- one for each multi-trace primary. In a companion paper to appear, we will view Wilson networks and TQFTs as the natural language of non-perturbative bulk quantum gravity. The present note illustrates how this can apply to local bulk fields, and not just defects: a bulk (generalized free) field is to be viewed as a full tower of multi-trace Wilson lines. We further show that the SL(2) descendants of each multi-trace primary, together with the boundary gravitons of the AdS_3 background, correctly reproduce the full Virasoro character of each module. In this language, the role of a light insertion on a heavy primary is played by a topological Verlinde line. This allows us to obtain a "microscopic" description of one-loop determinants on the smooth BTZ handlebody. A spatially wound probe Wilson line on a torus with a contractible thermal cycle can be traded for a Verlinde line inserted on a dense family of heavy Polyakov loops -- with the roles of the two cycles exchanged, so that the spatial cycle is now contractible. The vacuum row of the modular S-kernel acts as the (approximate) density of the heavy primaries. This reinforces the case made in arXiv:2601.18775 that a smooth horizon is a stand-in for an ensemble of quantum states, each produced by a heavy Wilson line that appears as a singular horizon in the semi-classical limit.
ER -