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Let $D$ be the Dirac operator.

The equation \begin{equation}\tag{1} \mathrm{ind}\,D_+=-\frac{1}{8\pi^2}\int_M\text{tr}\,F^2=-\frac{1}{8\pi^2}\int_MF^a\wedge F^b\ \mathrm{tr}(T_aT_b) \end{equation} is a special case of the Atiyah-Singer index theorem (the RHS is equal to an integer, the instanton number) - see section $12.6.2$ of Mikio Nakahara's book . He also derives the equation in section $13.2.1$ (Abelian anomalies - Fujikawa's method).

I have to understand the derivation for my Bachelor thesis, but I find it very challenging to understand Nakahara's book - I prefer modern mathematical treatments of Gauge Theory (e.g. Mark Hamilton's book).

Do you know any other book containing the derivation of $(1)$? I would prefer a proper mathematical derivation, but all additional references would help. (I am not looking for a proof of the Atiyah-Singer index theorem in its most general form).

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Do you know any other book containing the derivation of (1)? I would prefer a proper mathematical derivation, but all additional references would help.

Here's one in the "additional references" category: Fujikawa and Suzuki (2004), Path Integrals and Quantum Anomalies, Oxford University Press.

It includes detailed derivations of chiral anomalies using different methods, including the one that made Fujikawa famous (using a smooth mode cut-off to define the path integral Jacobian) and another one using lattice QFT, which is the first time I've seen a lattice QFT calculation of chiral anomalies in a book. It's written more for an audience of physicists than for mathematicians, but it's relatively detailed.

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You can try the proof by Getzler: "A short proof of the local Atiyah-Singer Index theorem" Topology, Vol 25, pages 111-117 (1986). That give the general case, including curved space, and in any dimension.

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