Suppose that a given field configuration has N punctures located at x i , i = 1 , … … , N each with index n i = ± ± 1 . Then, ϕ ϕ decomposes into the sum of a field configuration with no punctures, ϕ ϕ 0 and ∑ ∑ i = 1 N n i arg ⁡ ⁡ ( z − − z i ) , where we have switched to the complex plane coordinates for convenience. The complex argument function has a branch cut, but, because ϕ ϕ is defined modulo 2 π π , it has no physical consequences.