# The boundary value problem for the super-Liouville equation

### Jürgen Jost

Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany### Guofang Wang

Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany### Chunqin Zhou

Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, China### Miaomiao Zhu

Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

## Abstract

We study the boundary value problem for the — conformally invariant — super-Liouville functional

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 $T(z)$, 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.

## Cite this article

Jürgen Jost, Guofang Wang, Chunqin Zhou, Miaomiao Zhu, The boundary value problem for the super-Liouville equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 685–706

DOI 10.1016/J.ANIHPC.2013.06.002