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I am a physicist reading Ezra Getzler's paper A short proof of the local Atiyah-Singer index theorem (Topology 25 (1986) 111-117). The clever step in his proof is his rescaling so as to focus of the short distance behaviour of the heat kernel for the Lichnerowicz laplacian $$ {\rm Dirac}^2=g^{\mu\nu} (D_\mu D _\nu- {\Gamma^\lambda}_{\nu\mu}D_\lambda) +{\textstyle\frac 12} F_{\mu\nu} \gamma^\mu\gamma^\nu - {\textstyle\frac 14} R. $$ A key equation is the rescaled version of Lichnerowicz which appears as the equation before his numbered equation (5). His version reads $$ D^\epsilon = g^{ij}(\epsilon x)\left( (\partial_i + \epsilon^{-1} \Gamma_i(\epsilon x)\circ_\epsilon +\epsilon A_i(\epsilon x)) (\partial_j + \epsilon^{-1} \Gamma_j(\epsilon x)\circ_\epsilon+\epsilon A_\nu(\epsilon x))\right. \\ \left . + {\Gamma^k}_{ij}(\epsilon x) (\partial_k + \epsilon^{-1} \Gamma_j(\epsilon x)\circ_\epsilon +\epsilon^{-1} A_k(\epsilon x))\right)+ {\frac 14} F_{ab}(\epsilon x)(e^a\wedge e^b)\circ_\epsilon- \frac {\epsilon^2} {4}R(\epsilon x). $$ I get a slightly different expression: $$ D^\epsilon = g^{ij}(\epsilon x)\left( (\partial_i + \epsilon^{-1} \Gamma_i(\epsilon x)\circ_\epsilon +\epsilon A_i(\epsilon x)) (\partial_j + \epsilon^{-1} \Gamma_j(\epsilon x)\circ_\epsilon+\epsilon A_\nu(\epsilon x))\right. \\ \left . -\epsilon\, {\Gamma^k}_{ij}(\epsilon x) (\partial_k + \epsilon^{-1} \Gamma_j(\epsilon x)\circ_\epsilon +\epsilon A_k(\epsilon x))\right)+ {\frac 14} F_{ab}(\epsilon x)(e^a\wedge e^b)\circ_\epsilon- \frac {\epsilon^2} {4}R(\epsilon x). $$ The difference of sign before the ${\Gamma^k}_{ij}$ is insignificant and comes from his quoted version of Lichnerowicz, but the $\epsilon$ multiplying my ${\Gamma^k}_{ij}$ is absent in his expression. Its presence seems to be necessary for the ${\Gamma^k}_{ij}$ term to vanish in the $\epsilon\to 0$ limit. Also, although not so important, is that I do not see why there are $\epsilon A_i$'s in his first two gauge connection terms but an $\epsilon^{-1} A_k$ in its third appearence in his expression. I get $\epsilon A$'s in all three places. I find his notation hard to translate into my physics language, and I wonder if I have misunderstood something. Have I made an error somewhere? --- or are there typos in his text?

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    $\begingroup$ Just out of curiosity, have you tried contacting Getzler? $\endgroup$
    – Filippo
    Apr 29, 2021 at 20:41

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