As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like
$$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{pmatrix}, $$
which means for the components $\psi_{1,2}$ of the spinor that they transform like
\begin{align} 
\psi_1 \to \psi_1' &= \text e^{\text i\epsilon}\, \psi_1, \\
\psi_2 \to \psi_2' &= \text e^{-\text i\epsilon}\, \psi_2. 
\end{align}
If the symmetry is realized, then the divergence of the axial current $j_A^{\,\mu} = \overline\psi\gamma^\mu\gamma^5\psi$ vanishes: $\partial_\mu \,j_A^{\,\mu}=0$. If there appears an anomaly, then $\partial_\mu\, j_A^{\,\mu} =\mathcal A \neq 0$ and the action transforms as 
$$ S\to S+\delta_\epsilon S = S+\int \text d^2x\,\, \epsilon\, \mathcal A .$$

> How do I calculate this anomaly $\mathcal A$?

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From [this paper $[1]$][1] Eq. (2.4), [this paper $[2]$][2] Eq. (14) and the [Wiki article on the Fujikawa method $[3]$][3], I deduce that the divergence of the axial current in $d=1+1$ dimensions is proportional to the field strength tensor:
$$ \partial_\mu \,j_A^{\,\mu} \propto \epsilon_{\mu\nu}F^{\mu\nu}. $$
Despite the references listed above, I am not able to reproduce this result. Here are my thoughts for each reference:

$\bullet$ [Ref. $[1]$][1] motivates the result by some kind of momentum shift that I cannot connect to the axial transformation. 

$\bullet$ [Ref. $[2]$][2] gets the result by calculating some kind of vacuum polarization (why?). 

$\bullet$ [Ref. $[3]$][3] has a complete calculation of the Fujiwara method (looking at the path integral measure $\mathcal D\psi$ and how it transforms. This is done in $d$ dimensions, which I cannot "simplify" to the case $d=2$. 

Any help is much appreciated! 

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My choice for the two-dimensional gamma matrices is 
$$ \gamma^0=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad \gamma^1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad  \gamma^5=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. $$

  [1]: https://arxiv.org/abs/0912.2560
  [2]: https://arxiv.org/abs/hep-th/0509097v1
  [3]: https://en.wikipedia.org/wiki/Fujikawa_method