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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

$D=\gamma^\mu(\partial_\mu+A_\mu)$

where $A_\mu$ is a $SO(p,q)$ connection on $E$.

With this non-compact special orthogonal Lie group, can I always define a topological index for the above Dirac operator?

$ind (D)= -\frac{1}{8 \pi^{2}}\int F\wedge F$

with $F=dA+A\wedge A$ ?

Is it well defined the Atiyah-Singer index theorem in this case?

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