The n-Category Café

Just heard [that was last Friday] a very nice talk by Dan Freed at Bonn University on Dirac operators and differential K-theory. Below is my transcript of my lecture notes. Probably I won’t make it beyond the review part and into the new research stuff before I have to run again – to his next talk. So here is what you migth probably regard as the condensation of an extended review of Atiyah-Singer index theory. Of course that means we will also encounter the Quillen superconnections that we recently talked about.

Dan Freed began with a remark how he had been before in Bonn long ago in the 80s of last century, then hearing Quillen talk about Dirac operators and superconnections. The following picks that thread up, enriched meanwhile by the concept of differential K-theory.

So, here is roughly my transcript:

Let $X$ be a compact Riemannian spin manifold of even dimension.

Let $$\array{ (E,\nabla) \\ \downarrow \\ X }$$

be a complex hermitian vector bundle with covariant derivative $\nabla$ (encoding the connection). Denote by $D_X(\nabla)$ the corresponding Dirac operator.

$E$ determines of course a class in K-theory $[E] \in K^0(X)$. And the index of $D_X(\nabla)$ depends only on that class. This follows simply from the fact that the index is a continuous function with values in integers on the space of all connections on $E$, which however is an affine space, hence in particular connectes. So the index has to be constant function of the connection.

Define the notation

$$\pi_*^{analytic} : K^0(X) \to \mathbb{Z}$$ for the index map $$[E] \mapsto index D_X(\nabla) \,.$$

This is going to be interpreted, of course, as push-forward to the point in K-theory. The point being, however, that there is another prescription for that push-forward and the Atiyah-Singer theorem saying that it coincides with the one just given.

That other procedure, push-forward in $K$-theory, is the homomorphism denoted $$\pi_*^{top} : K^0(X) \to \mathbb{Z}$$

Theorem (Atiyah-Singer): $$\pi_*^{analytic} = \pi_*^{top} \,.$$

There is famously an explicit cohomological formula for $\pi_*$, which expresses it in terms of an integral over the product of the $\hat A$-genus and the Chern-class of $E$: $$\pi_*^{top}([E]) = \int_X \hat A(X) \; ch(E) \,.$$ That integral, in turn, is to be interpreted as push-forward in rational cohomology.

A few years after that theorem the “families”-version was proved:

Let now $$\array{ X \\ \downarrow \\ S }$$ be a family of compact spin manifolds of dimension $n$ with Riemannian structure on the relative tangent bundle $T(X/S)$ (i.e. the bundle of vertical vectors). A metric along the fibers together with a horizontal distribution canonically defines a Levi-Civita conection $$\nabla^{X/S}$$ on $T(X/S)$.

Given now a vector bundle $$\array{ (E,\nabla) \\ \downarrow \\ X \\ \downarrow \\ S }$$ on the total space, we obtain a family $D_{X/S}(\nabla)$ of Dirac operators parameterized by $S$. And again, we have two different prescriptions for doing push-forward $$\pi_*^{analytic}, \pi_*^{top} _ K^0(X) \to K^{-n}(S)$$ now taking values in the $K$-theory over the base, with a shift in degree given by the dimension that we “integrate over”.

Again

Family index theorem. (Atiyah-Singer): $$\pi_*^{analytic} = \pi_*^{top} \,.$$

Later other geometric invariants of Dirac operators were found, which encoded more non-discrete information:

Atixah-Patodi-Singer looked at odd dimensional $X$ (still compact Riemannian spin manifolds) and vector bundles $$\array{ (E,\nabla) \\ \downarrow \\ X }$$ over these. Again we form the Dirac operator $D_X(E)$. Being self-adjoint its spectrum is real. It is also discrete, since $X$ is compact. Using this one defines a quantity $$\eta(D_X(E))$$ which is, heuristically, the number of positive minus the number of negative Eigenvalues of the Dirac operator. Since there are infinitely many of both this doesn’t make sense literally, bbut gives a well-defined result after some regularization procedure.

Then define the quantity $$\xi(D_X(E)) = \frac{\eta(D_X(E)) + dim ker D_X(\nabla)}{2} \in \mathbb{R}/\mathbb{Z} \,.$$ This is not an “integral” invariant as before, but now takes values on the circle.

[Here the typescript of my notes terminates for the moment, though this is just the first fourth of the talk. To be continued when time permits.]