MiS Preprint Repository

We study the boundary value problem for the -- conformally invariant -- super Liouville functional $$E(u,\psi )={\int }_M\{\frac{1}{2}{\left|\mathrm{\nabla }u\right|}^2+K_{g}u+⟨(D+e^u)\psi ,\psi ⟩-e^{2u}\}dz$$} 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.