This is a standard particle-vortex duality transformation. The idea is there exists a "dual" description of a superfluid, in which the bosons (to be precise, the Goldstone mode) is described by a 2-form(tensor) gauge field, and the vortices are now string-like "matter" charged under the gauge field. The effective action for phase fluctuations (i.e. Goldstone mode) is just

$ L=\frac{1}{2}\rho^2 (\partial_\mu\theta)^2$

Here $\rho$ is the superfluid density, roughly your $f$ at infinity. $\theta$ can be divided into two parts: first the smooth fluctuations, and the singular ones (vortices). We will write $\theta=\eta + \theta_v$. $\eta$ satisfies $[\partial_x,\partial_y]\eta=0$.

The next thing is to rewrite the Lagrangian using Hubbard-Stratonovich transformation, by introducing an auxiliary vector field $\zeta_\mu$:

$L=\partial_\mu \theta \zeta^\mu -\frac{1}{\rho^2} \zeta^2=\partial_\mu \eta\zeta^\mu+\partial_\mu \theta_v\zeta^\mu -\frac{1}{\rho^2} \zeta^2$.

Since $\eta$ is a smooth function, the functional integration over $\eta$ can be easily performed, yielding a constraint $\partial_\mu \zeta^\mu=0$ which can be resolved by writing $\zeta^\mu=\varepsilon^{\mu\nu\lambda\rho}\partial_\nu B_{\lambda\rho}$. Notice that $B$ is a 2-form gauge field, since this expression has exactly the gauge redundancy. Then the Lagrangian can be casted into a gauge theory of 2-form tensor gauge field coupled to string-like sources (i.e. the $\theta_v$).

This post imported from StackExchange Physics at 2015-07-01 14:33 (UTC), posted by SE-user Meng Cheng