The Hodge-Dirac operator on $\bS^2$

If $M$ is a compact, oriented Riemannian manifold that has no structures, ¿can one define Dirac-like operators on an $\sB$-$\sA$-bimodule $\sE$ that is not pointwise irreducible under the action of $\sB$? It turns out that one can do so, if $\sE$ carries a ``Clifford connection'', that is, a connection $\nabla^\sE$ such that

$$\nabla^\sE(c(\al)s) = c(\nabla\al) s + c(\al) \nabla^\sE s,$$

for $\al \in \sA^1(M)$, $s \in \sE$, and which is Hermitian with respect to a suitable $\sA$-valued sesquilinear pairing on $\sE$. For instance, we may take $\sE = \sA^\8(M)$, the full algebra of differential forms on $M$, which we know to be a left $\sB$-module under the action generated by $c(\al) = \eps(\al) + \iota(\al^\3)$. The Clifford connection is just the Levi-Civita connection on all forms, obtaining by extending the one on $\sA^1(M)$ with the Leibniz rule (and setting $\nabla f := df$ on functions). The pairing $\pairing{\al}{\bt} := g(\bar\al,\bt)$ extends to a pairing on $\sA^\8(M)$; by integrating the result over $M$ with respect to the volume form $\nu_g$, we get a scalar product on forms, and we can then complete $\sA^\8(M)$ to a Hilbert space.

If $\{E_1,\dots,E_n\}$ and $\{\th^1,\dots,\th^n\}$ are local orthonormal sections for $\sX(M)$ and $\sA^1(M)$ respectively, compatible with the given orientation, so that $c(\th^j) = \eps(\th^j) + \iota(E_j)$ locally, then

$$\hstar := c(\ga) = (-i)^m c(\th^1) c(\th^2) \dots c(\th^n)$$

is globally well-defined as an $\sA$-linear operator taking $\sA^\8(M)$ onto itself, such that $\hstar^2 = 1$. This is the Hodge star operator, and it exchanges forms of high and low degree.

(Actually, our sign conventions differ from the usual ones in differential geometry books, that do not include the factor $(-i)^m$. With the standard conventions, $\hstar^2 = \pm 1$ on each $\sA^k(M)$, with a sign depending on the degree $k$.)

The codifferential $\dl$ on $\sA^\8(M)$ is defined by

$$\dl := - \hstar d \hstar.$$

This operation lowers the form degree by $1$. The Hodge-Dirac operator is defined to be $-i(d + \dl)$ on $\sA^\8(M)$. One can show that, on the Hilbert-space completion, the operators $d$ and $-\dl$ are adjoint to one another, so that $-i(d + \dl)$ extends to a selfadjoint operator. (With the more usual sign conventions, $d$ and $+\dl$ are adjoint, so that the Hodge-Dirac operator is written simply $d + \dl$.)

Now we take $M = \bS^2$, the $2$-sphere of radius $1$. The round (i.e., rotation-invariant) metric on $\bS^2$ is written $g = d\th^2 + \sin^2\th d\phi^2$ in the usual spherical coordinates, which means that $\{d\th, \sin\th d\phi\}$ is a local orthonormal basis of $1$-forms on $\bS^2$. The area form is $\nu = \sin\th d\th \w d\phi$. The Hodge star is specified by defining it on $1$ and on $d\th$:

$$\hstar(1) := -i \nu, \qquad \hstar(d\th) := i\sin\th d\phi.$$

To find the eigenforms of the Hodge-Dirac operator, it is convenient to use another set of coordinates, obtained form the Cartesian relation $(x^1)^2 + (x^2)^2 + (x^3)^2 = 1$ by setting $\ze := x^1 + ix^2 = e^{i\phi} \cos\th$, along with $x^3 = \cos\th$; the pair $(\ze,x^3)$ can serve as coordinates for $\bS^2$, subject to the relation $\ze\bar\ze + (x^3)^2 = 1$. (The extra variable $\bar\ze$ gives a third coordinate, extending $\bS^2$ to $\bR^3$.)

These commutation relations show that if $L_\pm =: L_1 \pm i L_2$, then $L_1,L_2,L_3$ generate a representation of the Lie algebra of the rotation group $\SO(3)$. One obtains representation spaces of $\SO(3)$ by finding functions $f_0$ (``highest weight vectors'') such that $L_3 f_0$ is a multiple of $f_0$, $L_+ f_0 = 0$, and $\set{(L_-)^r f_0 : r \in \bN}$ spans a space of finite dimension. To get spaces of differential forms with these properties, one extends each vector field $L_j$ to an operator on $\sA^\8(\bS^2)$, namely its Lie derivative $\sL_j$, just by requiring that $\sL_j d = d \sL_j$. Since the Hodge star operator is unchanged by applying a rotation to an orthonormal basis of $1$-forms, one can also show that $\sL_j \hstar = \hstar \sL_j$, so that the Hodge-Dirac operator $-i(d + \dl)$ commutes with each $\sL_j$. This gives a method of finding subspaces of joint eigenforms for each eigenvalue of the Hodge-Dirac operator.

We introduce the following families of forms:

$$\begin{aligned} \phi^+_l &:= i\ze^l(1 - i\nu), \quad l = 0... ...}(d\ze - \hstar(d\ze)), \quad l = 1,2,3,\dots. \end{aligned}$$

Clearly, $\hstar(\phi^\pm_l) = \pm \phi^\pm_l$ and $\hstar(\psi^\pm_l) = \pm \psi^\pm_l$. Thus $\phi^+_l$ and $\psi^+_l$ are even, while $\phi^-_l$ and $\psi^-_l$ are odd, with respecting to the $\bZ_2$-grading on forms given by $\sA^\8(\bS^2) = \sA^+(\bS^2) \oplus \sA^-(\bS^2)$, where $\sA^\pm(\bS^2) := \half(1 \pm \hstar) \sA^\8(\bS^2)$.

With some more works, it can be shown that all these eigenforms span a dense subspace of the Hilbert-space completion of $\sA^\8(\bS^2)$, so that these eigenvalues in fact give the full spectrum of the Hodge-Dirac operator.