Automorphismes r\'eels d'un fibr\'e et op\'erateurs de Cauchy-Riemann

Let $(N,c_N)$ be a complex vector bundle equipped with a real structure over a real curve $(\Sigma,c_\Sigma)$ of genus $g \geq 0$. We compute the sign of the action of the automorphisms of $(N,c_N)$ lifting the identity of $\Sigma$ on the orientations of the determinant line bundle over the space of real Cauchy-Riemann operators on $(N,c_N)$. This sign can be obtained as the product of two terms. The first one computes the signature of the permutations induced by the automorphisms acting on the $Pin^\pm$ structures of the real part of $(N,c_N)$. The second one comes from the action of the automorphisms of $(N,c_N)$ on the bordism classes of real Spin structures on $(\Sigma,c_\Sigma)$.