Complete derivative expansion of the effective brane theory

Determine closed-form expressions for the kinetic and higher-derivative terms in the effective brane action W[γ,Φ], and characterize how the generalized T̄T-like trace-flow equation constrains those terms.

Background

The paper derives the effective brane action W[γ,Φ] by solving a generalized T̄T-like trace-flow equation in a derivative expansion. The potential and curvature-coupling terms are obtained in closed form, but the kinetic term is determined only asymptotically and the higher-derivative local sector only at leading subleading order.

The trace-flow equation constrains the trace of the stress tensor rather than the full effective action, leaving undetermined derivative contributions. Completing these terms is necessary for a fully specified effective theory at finite cutoff and beyond the lowest derivative orders.

References

For general values of the parameters and energy, it is a genuinely new theory. It is natural to ask what that theory is, i.e. what is the boundary interpretation of the spectrum Eb, E0b for general values of the parameters. We will leave a full understanding of this question to future work.

— Interpolating between single and double trace $T\bar T$ by moving LST into the bulk  (2609.10823 - Chu et al., 9 Sep 2026) in Section 4.1, Dirichlet boundary conditions

A first open question concerns the derivative and non-local sectors of the effective brane theory. While the derivative expansion provides a controlled strategy to determine W[γ,Φ], the trace-flow equation directly constrains only the part of the effective action that contributes to the trace of the stress tensor. In particular, our analysis determines only the leading contribution in an asymptotic expansion of the kinetic term and the first subleading contribution of the higher derivative terms of the local sector. Determining a closed form for the kinetic and higher-derivative terms and understanding how they are constrained by the trace-flow equation is therefore an important open problem.

— 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'