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?

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?