Complete non-local sector of the effective action

Characterize the complete non-local sector of the effective brane action W[γ,Φ], including the possible traceless contribution at the leading derivative order and higher-order non-local terms, and determine to what extent it is fixed by the radial Hamilton–Jacobi equation and the requirement that W[γ,Φ] arise from a functional W[g] of the induced metric.

Background

The paper introduces a leading non-local term to account for part of the derivative expansion and determines only its non-zero-trace contribution. The consistency relation between metric and Weyl-factor variations may require an additional traceless term at the same derivative order.

Further non-local contributions are expected at higher derivative orders. The unresolved issue is whether the radial Hamilton–Jacobi equation, together with the fact that the effective theory must depend on the induced metric g, uniquely determines these terms.

References

An additional traceless contribution at the same derivative order may be required to enforce the consistency condition ⟨T_{μν}γ{μν}⟩=−(4π/√−γ)δW[γ,Φ]/δΦ, and further non-local contributions are expected at higher derivative orders. It would therefore be important to characterize the complete non-local sector of the effective action and understand to what extent it is fixed by the radial Hamilton--Jacobi equation together with the requirement that W[γ,Φ] originates from a functional W[g] of the induced metric.

— Brane effective actions and their island rule from $T\overline T$ flows  (2608.30533 - Callebaut et al., 31 Aug 2026) in Section 'Discussion and outlook', paragraph 'Completing the effective brane theory'; see also Appendix 'Non-local contributions'

It remains unclear to what extent the derivative corrections obtained from the radial flow in section \ref{sec:braneeffective} can be matched uniquely to those obtained from the fluctuating-cutoff calculation of section \ref{sec:fluctuating}.

— Brane effective actions and their island rule from $T\overline T$ flows  (2608.30533 - Callebaut et al., 31 Aug 2026) in Section 'Discussion and outlook', paragraph 'Completing the effective brane theory'