Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

===== We place a thin shell Σ\SigmaΣ with surface Lagrangian density =====

Sbdy=∫Σdτ d2Ω −γ  [−14λΣ fabfab+χˉ(iℏ \slashedDΣ−mΣc)χ+JΣμAμ+ΠabKab].S_{\text{bdy}}=\int_{\Sigma}\mathrm{d}\tau\,\mathrm{d}^2\Omega\,\sqrt{-\gamma}\;
\Big[
-\tfrac{1}{4}\lambda_\Sigma\, f_{ab}f^{ab}
+ \bar{\chi}\big(i\hbar\,\slashed{D}_\Sigma - m_\Sigma c\big)\chi
+ J^\mu_{\Sigma} A_\mu
+ \Pi^{ab} K_{ab}
\Big].Sbdy=∫Σdτd2Ω−γ[−41λΣfabfab+χˉ(iℏ\slashedDΣ−mΣc)χ+JΣμAμ+ΠabKab].
Meanings to audit:
* γ\gammaγ: det. of boundary metric; fabf_{ab}fab is the boundary field strength if a tangential gauge d.o.f. is used (optional).
* χ\chiχ: boundary (surface) charge carrier field (can be scalar for first audit).
* JΣμJ^\mu_{\Sigma}JΣμ: physical surface current coupling to AμA_\muAμ (minimal coupling).
* ΠabKab\Pi^{ab}K_{ab}ΠabKab: coupling to extrinsic curvature KabK_{ab}Kab (gives surface stress-energy), with Πab\Pi^{ab}Πab the canonical momentum conjugate to habh_{ab}hab on Σ\SigmaΣ.

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