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MiS Preprint
60/2011
The Boundary Value Problem for the Super-Liouville Equation
Jürgen Jost, Guofang Wang, Chunqin Zhou and Miaomiao Zhu
Abstract
We study the boundary value problem for the -- conformally invariant -- super Liouville functional \begin{equation*} E\left( u,\psi \right) =\int_{M}\{\frac 12 \left| \nabla u\right| ^2+K_gu+\left\langle (D+e^u)\psi ,\psi \right\rangle -e^{2u}\}dz \end{equation*}} 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.
