The boundary value problem for the super-Liouville equation
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 4, pp. 685-706.

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

E(u,ψ)= M{1 2|u| 2 +K g u+(D̸+e u )ψ,ψ-e 2u }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), 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.

@article{AIHPC_2014__31_4_685_0,
     author = {Jost, J\"urgen and Wang, Guofang and Zhou, Chunqin and Zhu, Miaomiao},
     title = {The boundary value problem for the {super-Liouville} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {685--706},
     publisher = {Elsevier},
     volume = {31},
     number = {4},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.06.002},
     mrnumber = {3249809},
     zbl = {1319.30028},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2013.06.002/}
}
TY  - JOUR
AU  - Jost, Jürgen
AU  - Wang, Guofang
AU  - Zhou, Chunqin
AU  - Zhu, Miaomiao
TI  - The boundary value problem for the super-Liouville equation
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2014
SP  - 685
EP  - 706
VL  - 31
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2013.06.002/
DO  - 10.1016/j.anihpc.2013.06.002
LA  - en
ID  - AIHPC_2014__31_4_685_0
ER  - 
%0 Journal Article
%A Jost, Jürgen
%A Wang, Guofang
%A Zhou, Chunqin
%A Zhu, Miaomiao
%T The boundary value problem for the super-Liouville equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2014
%P 685-706
%V 31
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2013.06.002/
%R 10.1016/j.anihpc.2013.06.002
%G en
%F AIHPC_2014__31_4_685_0
Jost, Jürgen; Wang, Guofang; Zhou, Chunqin; Zhu, Miaomiao. The boundary value problem for the super-Liouville equation. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 4, pp. 685-706. doi : 10.1016/j.anihpc.2013.06.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.06.002/

[1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623 -727 | MR | Zbl

[2] C. Ahn, C. Rim, M. Stanishkov, Exact one-point function of N=1 super-Liouville theory with boundary, Nucl. Phys. B 636 no. FS (2002), 497 -513 | MR | Zbl

[3] H. Baum, T. Friedrich, R. Grunewald, I. Kath, Twistor and Killing Spinors on Riemannian Manifolds, Humboldt Universität, Berlin (1990) | MR | Zbl

[4] H. Brezis, F. Merle, Uniform estimates and blow up behavior for solutions of -Δu=V(x)e u in two dimensions, Comm. Partial Differential Equations 16 (1991), 1223 -1253 | Zbl

[5] Q. Chen, J. Jost, G. Wang, M.M. Zhu, The boundary value problem for Dirac-harmonic maps, J. Eur. Math. Soc. 15 (2013), 997 -1031 | EuDML | MR | Zbl

[6] T. Fukuda, K. Hosomichi, Super-Liouville theory with boundary, Nucl. Phys. B 635 (2002), 215 -254 | MR | Zbl

[7] G.W. Gibbons, S.W. Hawking, G.T. Horowitz, M.J. Perry, Positive mass theorems for black holes, Comm. Math. Phys. 88 (1983), 295 -308 | MR

[8] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1 -55 | MR | Zbl

[9] O. Hijazi, S. Montiel, A. Roldán, Eigenvalue boundary problems for the Dirac operator, Comm. Math. Phys. 231 (2002), 375 -390 | MR | Zbl

[10] J. Jost, Riemannian Geometry and Geometric Analysis, Springer (2011) | MR | Zbl

[11] J. Jost, G. Wang, Analytic aspects of the Toda system: I. A Moser–Trudinger inequality, Comm. Pure Appl. Math. 54 (2001), 1289 -1319 | MR | Zbl

[12] J. Jost, G. Wang, C.Q. Zhou, Super-Liouville equations on closed Riemann surfaces, Comm. Partial Differential Equations 32 (2007), 1103 -1128 | MR | Zbl

[13] J. Jost, G. Wang, C.Q. Zhou, Metrics of constant curvature on a Riemann surface with two corners on the boundary, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009), 437 -456 | EuDML | Numdam | MR | Zbl

[14] Katrin Wehrheim, Uhenbeck Compactness, European Mathematical Society (2004) | MR | Zbl

[15] H.B. Lawson, M. Michelsohn, Spin Geometry, Princeton Math. Ser. vol. 38 , Princeton University Press, Princeton, NJ (1989) | MR | Zbl

[16] A.M. Polyakov, Quantum geometry of fermionic strings, Phys. Lett. B 103 (1981), 211 | MR

[17] J.N.G.N. Prata, The super-Liouville equation on the half-line, Phys. Lett. B 405 (1997), 271 -279 | MR

Cité par Sources :