Transforming the Dirac Operator on $S^1$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:40:28Z http://mathoverflow.net/feeds/question/116306 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116306/transforming-the-dirac-operator-on-s1 Transforming the Dirac Operator on $S^1$ mkreisel 2012-12-13T18:35:26Z 2012-12-20T15:24:19Z <p>This is related to my question <a href="http://math.stackexchange.com/questions/252742/transforming-the-dirac-operator-on-s1" rel="nofollow">http://math.stackexchange.com/questions/252742/transforming-the-dirac-operator-on-s1</a> on stack exchange which has not yet received an answer. For the purposes of this question, fix the spin structure over $S^1$ to be given by the connected double cover. </p> <p>Sections of the bundle of spinors on $S^1$ can be thought of in two ways. First, by trivializing the bundle over $U_1 = S^1 \setminus {i}$ and $U_2 = S^1 \setminus {-i},$ sections can be considered as pairs of real valued functions $f_1, f_2$ defined on $\mathbb{R}$ such that $f_1(\frac{1}{x}) = f_2(x)$ when $x> 0,$ and $f_1(\frac{1}{x}) = -f_2(x)$ when $x&lt; 0.$ Following Lawson/Michelson's "Spin Geometry," we can compute the Dirac operator locally. For example, $f_1 \rightarrow i\frac{df_1}{dx}.$</p> <p>We can also identify the sections as function $f: S^1 \rightarrow \mathbb{C}$ such that $f(-\theta) = -f(\theta)$ by the procedure described here <a href="http://math.stackexchange.com/questions/250835/expressing-the-sections-of-the-mobius-bundle-on-s1-as-globally-defined-real-v" rel="nofollow">http://math.stackexchange.com/questions/250835/expressing-the-sections-of-the-mobius-bundle-on-s1-as-globally-defined-real-v</a>. When we identify sections this way, I have seen the Dirac operator expressed as $f \rightarrow -i\frac{df}{d\theta}.$ However I have never seen a derivation of this fact from the local expression given above. My attempt at the computation is contained in the stack exchange link at the top of the page. </p> <p>Can anyone tell me how to translate between these 2 points of view?</p>