Question

Consider the spin^c structure on the flat standard 3-torus, which you get from the trivial (or any other) spin structure. Its associated vector bundle can be identified with a trivial bundle with fibre $\mathbb{C}^2$. spin^c Dirac operators on this bundle are parametrized by one-forms and look like $D_\alpha = D_0+ic_\alpha$, where $D_0$ is the spin Dirac operator and the $c$ means Clifford multiplication.

My aim is now to find a spectral decomposition for $D_\alpha$. If α is closed, this can be easily done by reducing everything to the case where α is harmonic. The case where α is not closed seems to be more tricky, so I would like to ask the community:

1. What is the spectrum of $D_\alpha$?
2. How can you construct eigenvectors?

Thank you!

Answer 1

Here is a possible unpromising start that hints at probable headaches. Square the Dirac to get

$$ D_\alpha^2= \Delta+ c(d\alpha)$$

where $c(d\alpha)$ denotes the Clifford multiplication by the $2$-form $d\alpha$. Note that

$$ {\rm spec}(D_\alpha^2)= \bigl(\; \mathrm{spec}(D_\alpha)\;\bigr)^2 $$

To find ${\rm spec}(D_\alpha^2)$ you need to understand spectrum of ordinary differential operators of the form

$$ -\partial^2_\theta + A(\theta) $$

acting on functions $u: S^1 \to \mathbb{C}^2$ where $A(\theta)$ is a $2\times 2$ complex hermitian matrix depending smoothly on $\theta\in S^1$. I don't know how to find the spectrum of such an operator but maybe you can find something in the literature.