geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
On a Kähler manifold $(X,\omega)$ with a choice of spin structure given by a Theta characteristic $\sqrt{\Omega^{0,n}}$, the sum of the Dolbeault operator $\overline{\partial}$ with its adjoint $\overline{\partial}^\ast$ are identified with the corresponding Dirac operator
Here the action of $\overline{\partial} + \overline{\partial}^\ast$ on $\Omega^{0,\bullet}$ is the canonical one. For the action on $\sqrt{\Omega^{n,0}}$ choose any connection $\nabla$ on this line bundle. Then on a local coordinate patch the action is given by differentiating along a coordinate vector, multiplying with the corresponding Clifford element and projecting on the antiholomorphic part, then summing this over all coordinate vectors (e.g. Friedrich 97, p. 79).
The index is the Todd genus (see there).
Textbook accounts include
section 3.4 (specifically around p. 79) of
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