Riemann-Roch index for a real Cauchy-Riemann operator (${\mathrm{ind}}_{\mathbb{R}}D=2n(1-g)+2c_1(E)[\Sigma ]$)

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.