I'm reading that many articles are using the "axial anomaly equation" (e.g. Fermion number fractionization in quantum field theory pag.142 or eq (2.27) of Spectral asymmetry on an open space), and I know what axial anomaly in QFT is, but I'm not sure to what equation they are referring to. Can someone help me?
The articles said that the following identity is equivalent to the axial anomaly equation for D even,
$Tr[(x|\frac{H_k}{H_k^2 + \omega_k^2}|y)]= \frac{k}{2\sigma^2}(\frac{\partial}{\partial x_i}\frac{\partial}{\partial y_i})Tr[(x|i\Gamma^i\Gamma^c\frac{1}{H + i\sigma}|y)] + \frac{k}{2\sigma^2}Tr[[K(x) - K(y)](x|\Gamma^c\frac{1}{H + i\sigma}|y)]$ (1)
where $H_k$ is a D-dimensional Dirac operator,
\begin{bmatrix} k & D \\ D^+ & -k \\ \end{bmatrix}
with $D=iP_i\partial_i + Q(x)$ and $H=H_{k=0}$. $\Gamma^i$ and $\Gamma^c$
\begin{bmatrix} 0 & P_i \\ P_i^+ & 0 \\ \end{bmatrix}
\begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}
and K(x),
\begin{bmatrix} 0 & Q(x) \\ Q(x)^+ & 0 \\ \end{bmatrix}
Also why if D (dimension of the Euclidean space on which $H_k$ is defined) odd the last term of eq.(1) vanishes?
