Chromatic Spherical and Renormalized Reshetikhin-Turaev Invariants for Drinfeld Centers

Abstract

We give invariants of pairs (M,Γ) consisting of a closed connected oriented three-manifold and an oriented framed ribbon graph embedded in M colored by a the Drinfeld center of a unimodular chromatic category 𝒞. This invariant generalizes the chromatic spherical and renormalized Reshetikhin-Turaev invariants of three-manifolds. We use Heegaard-Graph diagrams which represent the pair (M,Γ) and use the data of a unimodular chromatic category and objects of the Drinfeld center to construct the invariant. We define a ribbon graph associated to embedded framed links to obtain an invariant of pairs (M,L). We show that if L is the empty link, then the invariant recovers the chromatic spherical invariant, and if M is the three-sphere, then the invariant recovers the renormalized Reshetikhin-Turaev invariant from the Drinfeld center. The relationship is proven by recovering a chromatic structure for the center induced by that from 𝒞. The main theorem of the paper demonstrates that if the graph is a link L colored by a generator in the Drinfeld center, then we recover the chromatic spherical invariant of the surgery manifold M(L), contributing to a proof of the relationship between the chromatic spherical and renormalized Reshetikhin-Turaev invariants for unimodular chromatic categories, generalizing a recent result for unimodular Hopf algebras by Reina. To this end, we give a Heegaard diagram for M(L) coming from the Heegaard-Link diagram representing the pair (M,L).

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BibTeX RIS

BibTeX

@misc{bridges2026chromatic,
  title = {{Chromatic Spherical and Renormalized Reshetikhin-Turaev Invariants for Drinfeld Centers}},
  author = {Nicolas Bridges and Shawn Cui and Nathan Geer},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06708},
  eprint = {2611.06708},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06708},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Chromatic Spherical and Renormalized Reshetikhin-Turaev Invariants for Drinfeld Centers
AU  - Nicolas Bridges
AU  - Shawn Cui
AU  - Nathan Geer
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06708
UR  - https://arxiv.org/abs/2611.06708
AB  - We give invariants of pairs (M,Γ) consisting of a closed connected oriented three-manifold and an oriented framed ribbon graph embedded in M colored by a the Drinfeld center of a unimodular chromatic category 𝒞. This invariant generalizes the chromatic spherical and renormalized Reshetikhin-Turaev invariants of three-manifolds. We use Heegaard-Graph diagrams which represent the pair (M,Γ) and use the data of a unimodular chromatic category and objects of the Drinfeld center to construct the invariant. We define a ribbon graph associated to embedded framed links to obtain an invariant of pairs (M,L). We show that if L is the empty link, then the invariant recovers the chromatic spherical invariant, and if M is the three-sphere, then the invariant recovers the renormalized Reshetikhin-Turaev invariant from the Drinfeld center. The relationship is proven by recovering a chromatic structure for the center induced by that from 𝒞. The main theorem of the paper demonstrates that if the graph is a link L colored by a generator in the Drinfeld center, then we recover the chromatic spherical invariant of the surgery manifold M(L), contributing to a proof of the relationship between the chromatic spherical and renormalized Reshetikhin-Turaev invariants for unimodular chromatic categories, generalizing a recent result for unimodular Hopf algebras by Reina. To this end, we give a Heegaard diagram for M(L) coming from the Heegaard-Link diagram representing the pair (M,L).
ER  -

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