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In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the index theorem for Dirac operators here, where $\hat A$ is a bit more prominent. It typically appears in the description of the topological index, but - depending on your source and its method of proof- for different reasons.

  1. If you follow Shanahan or Lawson-Michelsohn, then the $\hat A$-class measures the failure of commutativity between the Chern character and the Thom isomorphisms in $K$-theory and cohomology. The Todd class is explained similarly in this answer.

  2. Bismut, following an idea of Atiyah, obtains the $\hat A$-class by equivariant cohomology on the loop space.

  3. If you give a proof using heat kernels, then the $\hat A$-class appears as a correction factor in Mehler's formula for the Getzler rescaled Dirac Laplacian, see chapter 4 of Berline-Getzler-Vergne.

  4. If you follow Berline-Vergne's proof, the $\hat A$-class is introduced though the Jacobian of the exponential map on a $G$-principal bundle, where $G$ is now a compact Lie group, see chapter 5 of Berline-Getzler-Vergne.

I would like to know if one can see a more direct connection between these descriptions than via a detour through the index theorem? I would in particular be interested in a link between the topological characterisation in 1 and one of the others, which all more or less give the Chern-Weil description.

And if there are other interesting ways to cook up the $\hat A$-class that I have forgotten?

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    $\begingroup$ Similar to 1: The Dirac operator defines a class in K-homology. Apply now the Chern-Connes character to go to de Rham homology. What you will end up with is the Poincare dual of the \hat{A}-class. $\endgroup$
    – AlexE
    Commented Feb 8, 2016 at 20:29
  • $\begingroup$ FWIW, in the context of Atiyah-Singer, I liked to think of the Chern character as encoding analytic properties, and the Todd class as encoding geometrical properties. $\endgroup$
    – Steve Huntsman
    Commented Feb 8, 2016 at 20:50
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    $\begingroup$ @SebastianGoette Yes, the Dirac operator of a spin manifold is a KO-fundamental class (and for a spin^c manifold the corresponding Dirac type operator is a K-fundamental class). That it can not be compatible in the naiv way with the homological fundamental class of the induced orientation of the manifold can be seen by noting the following: different spin structures (and there may be many) give rise to different Poincare duality maps, but the homological Poincare duality only depends on the induced orientation (of which there are only two). $\endgroup$
    – AlexE
    Commented Feb 9, 2016 at 9:30
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    $\begingroup$ Another topological description: the $\hat{A}$ class is the genus associated to the $E_\infty$ ring homomorphism $M Spin \to KO$. This might somehow mediate between your $1$ and $2$. $\endgroup$
    – Paul Siegel
    Commented Feb 9, 2016 at 9:54
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    $\begingroup$ Also, I thought quite a bit about the relation between 1 and 3 in graduate school, but all I was left with was a question: can one prove Bott periodicity using heat kernels? I think this would have to be the first step, though I'm not sure. $\endgroup$
    – Paul Siegel
    Commented Feb 9, 2016 at 9:59

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