Four-dimensional Chern–Simons theory

A heuristic puts strong restrictions on the ${\displaystyle C}$ to be considered. This theory is studied perturbatively, in the limit that the Planck constant ${\displaystyle \hbar <<1}$. In the path integral formulation, the action will contain a ratio ${\displaystyle \omega /\hbar }$. Therefore, zeroes of ${\displaystyle \omega }$ naïvely correspond to points at which ${\displaystyle \hbar \rightarrow \infty }$, at which point perturbation theory breaks down. So ${\displaystyle \omega }$ may have poles, but not zeroes. A corollary of the Riemann–Roch theorem relates the degree of the canonical divisor defined by ${\displaystyle \omega }$ (equal to the difference between the number of zeros and poles of ${\displaystyle \omega }$, with multiplicity) to the genus ${\displaystyle g}$ of the curve ${\displaystyle C}$, giving

$${\displaystyle {\text{number of zeros of }}\omega -{\text{number of poles of }}\omega =2g-2}$$

Then imposing that ${\displaystyle \omega }$ has no zeroes, ${\displaystyle g}$ must be ${\displaystyle 0}$ or ${\displaystyle 1}$. In the latter case, ${\displaystyle \omega }$ has no poles and ${\displaystyle C=\mathbb {C} /\Lambda }$ a complex torus (with ${\displaystyle \Lambda }$ a 2d lattice). If ${\displaystyle g=0}$, then ${\displaystyle C}$ is ${\displaystyle \mathbb {CP} ^{1}}$ the complex projective line. The form ${\displaystyle \omega }$ has two poles; either a single pole with multiplicity 2, in which case it can be realized as ${\displaystyle \omega =dz}$ on ${\displaystyle \mathbb {C} }$, or two poles of multiplicity one, which can be realized as ${\displaystyle \omega ={\frac {dz}{z}}}$ on ${\displaystyle \mathbb {C} ^{\times }\cong \mathbb {C} /\mathbb {Z} }$. Therefore ${\displaystyle C}$ is either a complex plane, cylinder or torus.

There is also a topological restriction on ${\displaystyle \Sigma }$, due to a possible framing anomaly. This imposes that ${\displaystyle \Sigma }$ must be a parallelizable 2d manifold, which is also a strong restriction: for example, if ${\displaystyle \Sigma }$ is compact, then it is a torus.