Abstract
We give a heat kernel proof of the algebraic index theorem for deformation quantization with separation of variables on a pseudo-Kähler manifold. We use normalizations of the canonical trace density of a star product and of the characteristic classes involved in the index formula for which this formula contains no extra constant factors.
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Notes
If \(\alpha = f(z,\bar{z})\hbox {d}z^1 \wedge \cdots \wedge \hbox {d}z^m \wedge \hbox {d} \bar{z}^1 \wedge \cdots \wedge \hbox {d}\bar{z}^m\) is a compactly supported volume form on U, denote by \(\hat{\alpha }= f(z,\bar{z}) \theta ^1 \ldots \theta ^m \bar{\theta }^1 \ldots \bar{\theta }^m\) the corresponding function on \(\Pi TU\). Then,
$$\begin{aligned} \int _{\Pi TU} \hat{\alpha }\, \hbox {d}\beta = \int _U \alpha . \end{aligned}$$
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This paper is dedicated to my teacher Alexandre Aleksandrovich Kirillov.
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Karabegov, A. A heat kernel proof of the index theorem for deformation quantization. Lett Math Phys 107, 2093–2145 (2017). https://doi.org/10.1007/s11005-017-0980-9
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DOI: https://doi.org/10.1007/s11005-017-0980-9