An Abstract Index Theorem via Rees Algebras

Abstract

We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant $C^\omega_{\mathcal{R}}A$, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions to Rees algebras. The main result (Corollary 3.11) is an abstract index theorem: given a compatible datum of filtered differential graded associative algebras with traces $A$, $B$, $G$ with a morphism $\phi\colon A\to B$, an action $\rho\colon G\otimes A\to A$ and a morphism $i\colon G\to B$, together with a graded-central element, one has $\mathrm{tr}_B(e^{f_0+f_1}) = \mathrm{tr}_{\widehat{\mathrm{gr}}A}(e^{\gamma}h)$, where the left-hand side is the trace in $B$ and the right-hand side the trace in the Laurent series associated graded $\widehat{\mathrm{gr}}A$ of $A$. Here $f_0+f_1$ is a curvature-type element, i.e., an element of the form $d_B\beta+\beta^2$ for some odd-degree element $\beta$, while $\gamma$ and $h$ are certain elements in $\widehat{\mathrm{gr}}A$. The formalism is modelled on the Getzler rescaling technique and on the derivation of the localization formula for the loop space Chern character by Ludewig and Yi.