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

E(u,\psi ) = \int \limits_{M}\left\{\frac{1}{2}|\mathrm{∇}u|^{2} + K_{g}u + 〈\left(D̸ + e^{u}\right)\psi ,\psi 〉−e^{2u}\right\}\: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.

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