The boundary value problem for the super-Liouville equation

We study the boundary value problem for the — conformally invariant — super-Liouville functionalE(u,ψ)=∫M{12|∇u|2+Kgu+〈(D̸+eu)ψ,ψ〉−e2u}dz that couples a function u and a spinor ψ 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 ψ  . Associated to any solution of the super-Liouville system is a holomorphic quadratic differential T(z)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.