The boundary value problem for the super-Liouville equation

Jürgen Jost; Guofang Wang; Chunqin Zhou; Miaomiao Zhu

doi:10.1016/j.anihpc.2013.06.002

JournalsaihpcVol. 31, No. 4pp. 685–706

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

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Abstract

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

E (u, ψ) = M \int {\frac{1}{2} ∣ \nabla u ∣^{2} + K_{g} u + ⟨ (/ D + e^{u}) ψ, ψ ⟩ - e^{2 u}} d z

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