This question is related with Polyakov, "Gauge Fields and Strings" section 4.2
In section 4.2, partition function is \begin{equation} Z=\sum_{n_{x,\delta}}\int_{-\pi}^{\pi}\prod_x\frac{d\varphi_x}{2\pi}\exp(-\frac{\beta}{2}\sum_{x,\delta}(\varphi_x-\varphi_{x+\delta}+2\pi n_{x,\delta})^2). \end{equation}
Where any set $n_{x,\delta}$ can be represented by
\begin{equation} n_{x,\delta}=m_x-m_{x+\delta}+\alpha_x-\alpha_{x+\delta}+\varepsilon_{\delta\beta}\left(\phi_{x_*}-\phi_{x_*-\beta} \right). \end{equation}
How $n$ is written in terms of $m$, $\alpha$ and $\phi$?
