$Spin^c$-Dirac-operator on the 3-torus - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:46:49Z http://mathoverflow.net/feeds/question/14269 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14269/spinc-dirac-operator-on-the-3-torus $Spin^c$-Dirac-operator on the 3-torus J. Fabian Meier 2010-02-05T13:57:55Z 2012-02-08T17:41:48Z <p>Consider the spin<sup>c</sup> 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<sup>c</sup> 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.</p> <p>My aim is now to find a spectral decomposition for $D_\alpha$. If &alpha; is closed, this can be easily done by reducing everything to the case where &alpha; is harmonic. The case where &alpha; is not closed seems to be more tricky, so I would like to ask the community:</p> <ol> <li>What is the spectrum of $D_\alpha$?</li> <li>How can you construct eigenvectors?</li> </ol> <p>Thank you!</p> http://mathoverflow.net/questions/14269/spinc-dirac-operator-on-the-3-torus/87906#87906 Answer by Liviu Nicolaescu for $Spin^c$-Dirac-operator on the 3-torus Liviu Nicolaescu 2012-02-08T17:21:24Z 2012-02-08T17:41:48Z <p>Here is a possible unpromising start that hints at probable headaches. Square the Dirac to get</p> <p>$$D_\alpha^2= \Delta+ c(d\alpha)$$</p> <p>where $c(d\alpha)$ denotes the Clifford multiplication by the $2$-form $d\alpha$. Note that </p> <p>$${\rm spec}(D_\alpha^2)= \bigl(\; \mathrm{spec}(D_\alpha)\;\bigr)^2$$</p> <p>To find ${\rm spec}(D_\alpha^2)$ you need to understand spectrum of ordinary differential operators of the form</p> <p>$$-\partial^2_\theta + A(\theta)$$</p> <p>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.</p>