My question is, during the duality map, real physical quantities seem to acquire a prefactor of $i$ and become purely imaginary. And I feel uncomfortable.
Take bosonic current for example. Consider the Villian representation of the 2D square lattice XY model, with an external source field $\vec{s}$ coupled to the current:
$$ Z = \int \mathcal{D}\phi \sum_{\lbrace\vec{m}\rbrace} \exp\left[ -\frac{1}{2T}\sum \left(\vec{\Delta}\phi - 2\pi \vec{m}\right)^2 - i\sum\vec{s}\cdot \left(\vec{\Delta}\phi - 2\pi \vec{m}\right)\right]. $$
The sum is over all links of the lattice, and that imaginary $i$ in front of the coupling term is needed to ensure convergence in duality transformation.
Please consult your favorite reference for the full duality transformation; I am only going to do part relevant to my question. First step of the duality is the Hubbard-Stratonovich decoupling of the quadratic part: $$ Z \rightarrow \int \mathcal{D}\phi \mathcal{D}\vec{h} \sum_{\lbrace\vec{m}\rbrace} \exp\left[ -\frac{T}{2}\sum h^2 - i\sum(\vec{s}+\vec{h})\cdot \left(\vec{\Delta}\phi - 2\pi \vec{m}\right)\right]. $$ The $\phi$ integral now forces $\vec{\Delta}\cdot(\vec{s}+\vec{h})$ to vanish. We can define $(\vec{\Delta}\times\theta) = \vec{h} + \vec{s}$, and the $h^2$ term becomes $$ \sum h^2 = \sum (\vec{\Delta}\times\theta - \vec{s})^2 = \sum \left[ (\Delta\theta)^2 - 2\vec{s}\cdot(\vec{\Delta}\times\theta) + \dots\right] $$
And the other part eventually becomes the $\cos(2\pi\theta)$ part of the sine-Gordon model.
Now we can identify the current in the XY model as the quantity linearly coupled to $-i\vec{s}$. Oops, that $i$. So we get: $$ (\Delta_i \phi - 2\pi m_i) \leftrightarrow i T(\epsilon_{ij} \Delta_j \theta) $$
By definition $\theta$ is real. So I am identifying a real quantity on the left with a purely imaginary one on the right. How do I make sense of this?
--edit-- Let me add two more observations:
Vorticity in XY model is just the curl of current; it also gets the same $i$.
Sachdev in "Quantum phases of matters" has the $i$, but makes no comment whatsoever how the current becomes imaginary.
