To render the theory well-defined, it is only defined up to some energetic cut-off scale Λ Λ , so that we can puncture the plane at the points where the vortices are located, by removing regions with size of order 1 / Λ Λ . If γ γ winds counter-clockwise once around a puncture, the contour integral ∮ ∮ γ γ d ϕ ϕ is an integer multiple of 2 π π . The value of this integer is the index of the vector field ∇ ∇ ϕ ϕ .