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 the Drinfeld center of a unimodular finite chromatic category $\mathcal C$. 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 $\mathcal C$. 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 $(M,L)$.