Extension phases with period 2πi are independent exactly at integral level

Let $B$ be a nonempty set of boundary data. For each $b\in B$, let $E_b$ be a nonempty set of admissible extensions and let ${\mathrm{\Gamma }}_b:E_b\to i\mathbb{R}$ be their imaginary-valued action term. Assume every extension difference is an integral period, meaning ${\mathrm{\Gamma }}_b(e_1)-{\mathrm{\Gamma }}_b(e_0)\in 2\pi i\mathbb{Z}$ for every $b\in B$ and $e_0,e_1\in E_b$. Assume also that some pair $e_0,e_1\in E_b$ has difference $2\pi i$. For real $k$, define the unit circle $U(1):=\{u\in \mathbb{C}:\mathrm{\mid }u\mathrm{\mid }=1\}$ and the phase $W_{b,k}(e)=\exp (-k{\mathrm{\Gamma }}_b(e))\in U(1)$. Then $W_{b,k}(e)$ is independent of the chosen extension for every $b$ if and only if $k\in \mathbb{Z}$.

This lemma proves the level criterion only from the stated period-group hypotheses. In a compact WZW model those period hypotheses are part of the chosen normalization of the Wess–Zumino datum; the lemma does not prove that such a datum exists for every Lie group.