Field Theory Phase Factor vs Anomaly In this paper on topological quantum field theories the authors discuss something called the anomaly in section 5. In Witten's paper on field theory and the Jone's polynomial he discusses something called the phase ambiguity on page 390. Are the anomaly and the phase ambiguity related? Edit: I originally referenced the wrong page and term in Witten's paper. Fixed it.

Yes, the "phase ambiguity" in the Witten's paper is the so-called _framing anomaly_ ; The Witten-Reshetikhin-Turaev invariant gains an extra phase factor once you change the framing of a 3-manifold, but this problem is fixed by using the canonical 2-framing.

It seems that in the paper you referenced, the $\theta$ map is a Dehn twist, and the "anomaly" is a sort of framing anomaly in this sense.