Presentation Holonomy in Chern–Simons Theory Flat line bundles over the 2–framing torsor and the framing anomaly

In the Witten–Reshetikhin–Turaev (WRT) formulation of Chern–Simons theory, the closed
3–manifold invariant is not canonically a scalar: it depends on a choice of 2–framing. Equiv
alently, the partition function is a section of a one–dimensional complex vector space varying
over the Z–torsor F2(M) of 2–framings. We prove that the WRT partition function defines a
flat unitary line bundle over F2(M), and that its holonomy under the canonical generator is
exp 2πi 24 c(G,k), where c(G,k) is the WZW central charge. We state a precise theorem at the level of WRT
TQFT and record the SU(2)k check on S3 in terms of the modular data (S,T).