Generality of electromagnetic AdS-invariant conservative boundary conditions

Establish whether the boundary conditions \(\left[\tilde{E}_I+\alpha\tilde{B}_I\right]_{\mathcal I}=0\), for arbitrary real \(\alpha\) including \(\alpha=\infty\), are the most general local, AdS-invariant boundary conditions for the electromagnetic field in \(\mathrm{AdS}_{1,3}\) that admit well-posed evolution and a conserved positive energy.

Background

The paper identifies a family of local, AdS-invariant Maxwell boundary conditions by imposing a linear relation between the rescaled electric and magnetic fields at conformal infinity. The authors show that these conditions yield well-posed and stable dynamics for all values of the parameter α\alpha, and they construct a covariant phase space for the resulting solutions.

What remains unresolved is whether this family exhausts all local, AdS-invariant, conservative boundary conditions satisfying the stated dynamical and energy requirements, independently of the Ishibashi–Wald prescription.

References

We conjecture that---independently of the IW prescription---the boundary conditions eq:EMdualrot are the most general, local, AdS-invariant boundary conditions on the electromagnetic field in AdS$_{1, 3}$ that admit a well-posed evolution and are ``conservative'' in the sense that it has a conserved positive energy of the type discussed in .

— Alternative Boundary Conditions in Asymptotically Anti-de Sitter Spacetimes  (2609.10805 - Wald et al., 9 Sep 2026) in Section 3.2, Maxwell theory in AdS_{1,3} (equation (\ref{eq:EMdualrot}))

We conjecture that they give rise to well-posed evolution.

— Alternative Boundary Conditions in Asymptotically Anti-de Sitter Spacetimes  (2609.10805 - Wald et al., 9 Sep 2026) in Section 3.3, Yang–Mills Theory in AdS_{1,3} (equations (\ref{eq:YMEtq})–(\ref{eq:YMBtq}))

We conjecture that, for all $\alpha$ (including $\alpha = \infty$), the Yang-Mills equations with the boundary condition eq:YMdualrot yield well-posed, stable evolution in AdS$_{1, 3}$. As we shall show in section \ref{sec:phase}, we will be able to define a covariant phase space structure on the solutions to the Yang-Mills equations with boundary conditions eq:YMdualrot, and we conjecture that eq:YMdualrot are the most general AdS-invariant boundary conditions for which this is possible.

— Alternative Boundary Conditions in Asymptotically Anti-de Sitter Spacetimes  (2609.10805 - Wald et al., 9 Sep 2026) in Section 3.3, Yang–Mills Theory in AdS_{1,3} (equation (\ref{eq:YMdualrot}))