# Typos in Getzler's "Short proof of local Atiyah-Singer index theorem"?

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?

• Just out of curiosity, have you tried contacting Getzler? Apr 29, 2021 at 20:41