Consider the spinc 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 $\mathbb{C}^2$. spinc 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!

  • $\begingroup$ A suggestion, can you gauge transform so that $\alpha$ is nice? Gauge transformations tend to preserve spectral decompositions. $\endgroup$
    – Paul
    Commented Feb 17, 2010 at 1:36
  • 1
    $\begingroup$ If you decompose α as $d\beta + d^*\gamma + \delta$, where $\delta$ is harmonic, then you can "gauge away" the d\beta-Part by the usual $U(1)$-gauge. Since δ is constant, this gives you the solution for α closed. I don't know what kind of gauge should work on the $d^*\gamma$-part. $\endgroup$ Commented Feb 17, 2010 at 10:52

1 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.


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