Dirac equation on surfaces

\[ \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\H}{\mathbb{H}} \newcommand{\Cl}{\textrm{Cl}} \newcommand{\cCl}{\C\textrm{l}} \newcommand{\p}{\partial} \newcommand{\End}{\textrm{End}} \]

This is a quick recollection of how the Dirac equation on spinor bundles over flat space looks. We'll recall the two dimensional case with Riemannian signature.

Riemannian surface

We consider \((M,g)\) to be \(\R^2\) with the Euclidean metric. The tangent space is spanned by \(\{e_1, e_2\}=\{\p_x, \p_y\}\) and the associated Clifford algebra is \(\Cl_2\) which is isomorphic as an algebra to the quaternions \(\H\). The Dirac operator acting on \(\Cl_2\) is

\[D = e_1 \cdot \p_x + e_2 \cdot \p_y.\]

The Clifford algebra decomposes into even and odd elements, \(\Cl_2^0 \oplus \Cl_1^0\), and the Dirac operator respects this decomposition

\[ D = \left[ \begin{array}{cc} 0 & D^1 \\ D^0 & 0 \end{array}\right] \]

We identify the Clifford algebra with two copies of \(\C\) via \( \varphi : \Cl_2^0 \oplus \Cl_1^0 \to \C\oplus\C \) sending \(\{1, e_1e_2\}\mapsto\{1, -i\}\) and \(\{e_1, e_2\}\mapsto\{1, i\}\). Under this identification, the Dirac operator behaves like the operators providing the Cauchy-Riemann equations:

\[ \varphi D \varphi^{-1} = \left[ \begin{array}{cc} 0 & -\p_z \\ \p_{\bar z} & 0 \end{array}\right] \]

We check one part of the preceding statement (restricting to even sections):

\[ \begin{align*} \left(\left.\varphi D \varphi^{-1}\right|_{\C\oplus 0} \right)(u+iv) &= \varphi \left(( e_1\cdot \p_x + e_2 \cdot p_y ) (u - ve_1e_2)\right) \\ &= \varphi \left( (u_x - v_y)e_1 + (u_y + v_x)e_2\right) \\ &= (u_x-v_y) + i(u_y + v_x) \\ &= \p_{\bar z}(u+iv). \end{align*} \]

Keeping this identification, we see \(\varphi D^2\varphi^{-1}\) gives two copies of the Laplacian \(\Delta = -\p_x^2 - \p_y^2\).

The original physics version of spinors

There is a lot of choice in the bundle we choose. We shall now highlight a choice of bundle that is more in line with the spirit of the bundle associated with electrons in 3+1 dimensions. Consider the complexification \(\cCl_2\) which is isomorphic as an algebra to \(\C(2)\). One possible identification is

\[ \{ 1, e_1, e_2, e_1e_2 \} \mapsto \left\{ \begin{bmatrix} 1&0\\0&1 \end{bmatrix}, \begin{bmatrix} 0&-1\\1&0\end{bmatrix}, \begin{bmatrix} i&0\\0&-i\end{bmatrix}, \begin{bmatrix} -i&0\\0&-i\end{bmatrix} \right\}. \]

And it is wise to introduce the complex volume element \(\omega_\C = ie_1e_2\), such that \(\omega_\C^2=1\). We can see this isomorphism as a representation of \(\cCl_2\) on \(\C^2\). The spinor bundle \(\Sigma\) above \(M\) is then \(\C^2\). The complex volume element decomposes \(\Sigma\) via projections \(\pi_{\pm} = \frac12(1\pm \omega_\C)\). In the 3+1 physics setting, these are chiral projection operators giving left and right chiral components.

There are a couple of headaches when trying to port over the physics world to this setting.

• We should have started with the Lorentzian version of the Dirac equation. This would have required \(e_1e_1=1\). The squared operator would then give a d'Alembertian (rather than a Laplacian) in the sense of the Klein-Gordon equation.

• The vector bundle for spinors has rank \(4\) in the standard setting. The four degrees of freedom are associated with two spin-1/2 particles (e.g. the electron and the positron). With only two degrees of freedom, this interpretation is no longer accessible.