Topological index and Dirac operator with a non compact group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:22:16Z http://mathoverflow.net/feeds/question/100140 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100140/topological-index-and-dirac-operator-with-a-non-compact-group Topological index and Dirac operator with a non compact group Gian 2012-06-20T15:24:17Z 2012-06-20T15:24:17Z <p>A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold $M$ and $E$ is an associated vector bundle of $P(M,G)$ in an appropriate representation. A Dirac operator $D: \Delta^+ \otimes E\rightarrow \Delta^- \otimes E$ in this case is</p> <p>$D=\gamma^\mu(\partial_\mu+A_\mu)$</p> <p>where $A_\mu$ is a $SO(p,q)$ connection on $E$.</p> <p><strong>With this non-compact special orthogonal Lie group, can I always define a topological index for the above Dirac operator?</strong> </p> <p>$ind (D)= -\frac{1}{8 \pi^{2}}\int F\wedge F$</p> <p>with $F=dA+A\wedge A$ ?</p> <p>Is it well defined the Atiyah-Singer index theorem in this case?</p>