Kirby Problem 3.64

Definitions and mathematical statement

Let $M$ be a closed oriented smooth three-manifold. In a chosen trivialization of its product principal $\operatorname{SU}(2)$ bundle, a connection is an $\mathfrak{su}(2)$-valued one-form $A$. Here $\mathfrak{su}(2)$ consists of traceless skew-Hermitian complex matrices. Flatness means that its curvature vanishes: \[ F_A=dA+A\wedge A=0. \] Products of matrix-valued forms combine exterior multiplication with matrix multiplication, and $\operatorname{Tr}$ is the ordinary trace in the defining two-dimensional representation.

Using the source's normalization, set \[ \operatorname{CS}(M,A)=\frac1{4\pi^2} \int_M\operatorname{Tr}\left(A\wedge dA+\frac23A\wedge A\wedge A\right). \] Is this real number rational for every such $M$ and every flat connection $A$? The statement quantifies over all flat holonomies, including both reducible and irreducible ones, and imposes no restriction on the manifold's geometry.

A gauge transformation is a smooth map $M\to\operatorname{SU}(2)$ changing the trivialization. It changes the displayed number by an integer multiple of a fixed integral period, so rationality is independent of this choice. One can equivalently ask whether the gauge-invariant class modulo that period is represented by a rational number.

This question concerns the Chern--Simons functional of a flat compact-group connection. It does not substitute the Levi-Civita connection of a chosen Riemannian metric. A decimal approximation, even of high accuracy, cannot distinguish a rational value from an irrational one.

Short English statement

Are all Chern--Simons values of flat SU(2) connections rational?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2862 (KP-3.64), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.64. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.