The integrated form of the anomalous nonconservation equation in two dimensions
I have questions in the section 19.1 of <PERSON> and <PERSON>.
\begin{equation} \psi = \begin{pmatrix} \psi_+ \ \psi_- \tag{19.7} \end{pmatrix} \end{equation}
The subscripts indicates the $\gamma^5$ eigenvalue.
Below we'll show that there is situtions where the axial current conservation law $$ \partial_{\mu} j^{\,\mu 5} =0 . $$ is violated, and the integrated version of $$ \partial_{\mu} j^{\,\mu 5} = \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu} \tag{19.18} $$ holds. where the totally antisymmetric symbol $\epsilon^{\mu \nu}$ is defined as $\epsilon^{01}=+1$ on page 653.
Let us analyze this problem by thinking about fermions in one space dimension in a background $A^1$ field that is constant in space and has a very slow time dependence. We will assume that the system has a finite length $L$, with the periodic boundary conditions.
So $A^1(t,0)=A^1(t,L) $, and also $ \psi(t,0)=\psi(t,L) $.
the single-particle eigen-states of $H$ have energies \begin{alignat}{2} \psi_+ : \quad \quad & E_n && =+(k_n -eA^1), \ \psi_+ : \quad \quad & E_n && =-(k_n -eA^1). \tag{19.36} \end{alignat}
Now we consider the slow shift of $A^1$.
If $A^1$ changes by the finite amount
$$ \Delta A^1 = \frac{2\pi}{eL} \quad \quad \quad \quad \quad \quad \quad \quad (19.37)$$
Where $ \Delta A^1 \gt 0 $.
the spectrum of $H$ returns to its original form. 346
In this process, each level of $\psi_+$ moves down to the next poston, and each level of $\psi_-$ moves up to the next position, The occupation numbers of levels should be maintained in this adiabatic process. Thus, remarkably, one right-moving fermion($\psi_+$) disappears from the vacuum and one extra left-moving fermion($\psi_-$) appears.
So $$ \quad \Delta N_R=-1 ,\quad \Delta N_L=1 $$.
At the same time, \begin{alignat}{2} \int d^2x \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu} &=&& \int dt dx \frac{e}{\pi} \partial_0 A^1 \ &=&& \frac{e}{\pi} L(-\Delta A^1) \ &=&& -2 \tag{19.38} \end{alignat}
where I've rectified a misprint on the LHS of the first line.
where we have inserted (19.37) in the last line. Thus the integrated form of the anomalous nonconservation equation (19.18) is indeed satisfied: $$ \Delta N_R - \Delta N_L = \int d^2x \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu} . \tag{19.39} $$
For $ \Delta N_R - \Delta N_L =-2 $.
From $$ \int d^2x \partial_{\mu} j^{\,\mu 5} =\Delta N_R - \Delta N_L \tag{19.30} $$ we obtain $$ \int d^2x \partial_{\mu} j^{\,\mu 5} = \int d^2x \frac{e}{2\pi} \epsilon^{\mu\nu}F_{\mu\nu}. $$
Question 1:
I've checked all the calculations leading to (19.38). But I cannot understand the minus sign in the second anf the third line of (19.38).
Where does it come from? 677