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A real-linear Cauchy–Riemann operator on a complex rank-$n$ vector bundle over a closed genus-$g$ Riemann surface is a Fredholm operator between the usual Sobolev spaces and has the displayed index. Zero-order real-linear terms do not change this index, since they form a continuous deformation of Fredholm operators. For the bundle pulled back by a sphere, subtracting the six-dimensional reparametrization group gives the dimension formula for regular J-holomorphic spheres.

On a spin manifold, tensor the spinor bundle $S$ with a Hermitian vector bundle $W$ carrying a unitary connection. Composing the induced tensor-product connection on a vector bundle with Clifford multiplication defines the twisted Dirac operator. In even dimension it maps $S^{+}\otimes W$ to $S^{-}\otimes W$ and conversely. On a closed manifold its chiral restriction $D_W^{+}:H^1(S^{+}\otimes W)\to L^2(S^{-}\otimes W)$ is a Fredholm operator.