What question did this study set out to answer?

The main aim is to understand the framing anomaly in Chern-Simons theory through holonomy of line bundles.

January 25, 2026Open Access

Presentation Holonomy in Chern–Simons Theory: Flat Line Bundles over the 2-Framing Torsor and the Framing Anomaly

Key Points

The main aim is to understand the framing anomaly in Chern-Simons theory through holonomy of line bundles.
Recast the framing anomaly as holonomy of a flat unitary line bundle.
Define the connection between framing and central charge using topological language.
Perform explicit calculations for SU(2)k on S3.
Demonstrated that the quantum partition function depends discretely on 2-framing choices.
Established a theorem connecting framing anomalies to effective central charges.
Provided self-contained derivations for specific cases.

Abstract

Chern–Simons quantum field theory on a closed oriented 3-manifold M exhibits a framing anomaly:the quantum partition function is not canonically a complex number, but is naturally valued in acomplex line that depends on a choice of 2-framing. We recast the framing anomaly as the holonomyof a flat unitary line bundle over the integer 2-framing torsor, in the sense that for the unit shift onehasZ(M,f +1) =exp 2πi24 c(G,k) Z(M,f).(1)The dependence is discrete yet rigid, controlled by the effective central charge c(G,k). We state aprecise theorem in a minimal functorial topological quantum field theory (TQFT) language andpresent an explicit SU(2)k check on S3, with a self-contained surgery derivation deferred to Section A.

Presentation Holonomy in Chern–Simons Theory: Flat Line Bundles over the 2-Framing Torsor and the Framing Anomaly

Key Points

Abstract

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