The boundary value problem for the super-Liouville equation

We study the boundary value problem for the - conformally invariant - super-Liouville functional E (u,psi) = integral(M) {1/2 vertical bar del u vertical bar(2) + K(g)u + <(D + e(u)) psi,psi > - e(2u) } dz that couples a function u and a spinor psi on a Riemann surface. The boundary condition that we identify (motivated by quantum field theory) couples a Neumann condition for u with a chirality condition for psi. Associated to any solution of the super-Liouville system is a holomorphic quadratic differential T (z), and when our boundary condition is satisfied, T becomes real on the boundary. We provide a complete regularity and blow-up analysis for solutions of this boundary value problem. (C) 2013 Published by Elsevier Masson SAS.