Expected properties of Yang–Mills plus Chern–Simons theory

Let $G$ be a compact Lie group, let $e\to\text{?}$ be the coupling constant, and let $\lambda\in H^4(BG;\mathbb{Z})$ be a nondegenerate level. Let $F_{e,\lambda}$ be the Wick-rotated Yang–Mills plus Chern–Simons field theory determined semiclassically by the Lagrangian

$$L_{e,\lambda}(A)=\frac{1}{4e^2}F_A\wedge *F_A+\Gamma_\lambda(A).$$

Write $F_\lambda$ for the singular limit as $e\to\infty$, $\overline{F}_\lambda$ for its projectivization, and $\alpha_\lambda$ for the resulting projectivity (anomaly).

Expected properties of Yang–Mills plus Chern–Simons theory. The following assertions are expected: (1) the Lagrangian determines a family of Wick-rotated field theories; (2) the singular limit $e\to\infty$ exists and defines a field theory; (3) the projectivization $\overline{F}_\lambda$ factors through the bordism category without the Riemannian metric and is therefore a projective topological field theory; and (4) the anomaly extends to a 4-dimensional invertible field theory whose partition function on a closed oriented 4-manifold $W$ is

$$\widetilde{\alpha}_\lambda(W)=\exp\left(\frac{2\pi i c(\lambda)}{24}\langle p_1(W),[W]\rangle\right),$$

where $c(\lambda)\in\mathbb{Q}$ is the central charge associated to $\lambda$.

These are expectations motivated by the anticipated mass gap and singular long-distance limit of the gapped theory, with evidence from the abelian case; the general assertions are presented as conjectural properties of three-dimensional Yang–Mills plus Chern–Simons theories.