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

==== One-loop Euclidean effective action: ====

W=12ln⁡det⁡ΔB−ln⁡det⁡ΔF+(ghost determinant)+(surface operator contribution),W = \tfrac{1}{2}\ln\det\Delta_{\rm B} - \ln\det\Delta_{\rm F} + (\text{ghost determinant}) + (\text{surface operator contribution}),W=21lndetΔB−lndetΔF+(ghost determinant)+(surface operator contribution),
where Δ\DeltaΔ are second-order elliptic operators (after gauge fixing; the Dirac field is handled by D2D^2D2). Use

ln⁡det⁡Δ=−∫ε∞dtt Tr e−tΔ.\ln\det\Delta = -\int_{\varepsilon}^{\infty}\frac{dt}{t}\,\mathrm{Tr}\,e^{-t\Delta}.lndetΔ=−∫ε∞tdtTre−tΔ.
In 4 dimensions, small-ttt asymptotic expansion with boundary reads:

Tr e−tΔ∼(4πt)−2 ⁣∑k=0∞tk/2Ak(Δ),\mathrm{Tr}\,e^{-t\Delta} \sim (4\pi t)^{-2}\!\sum_{k=0}^{\infty} t^{k/2}A_k(\Delta),Tre−tΔ∼(4πt)−2k=0∑∞tk/2Ak(Δ),
with AkA_kAk = integrated bulk + boundary invariants. The coefficient that produces the curvature term is A2A_2A2 (the t−1t^{-1}t−1 piece). After integrating ttt with a UV cutoff Λ\LambdaΛ (e.g. lower bound t=Λ−2t=\Lambda^{-2}t=Λ−2), we obtain the induced Einstein–Hilbert contribution.

Schematic extraction:

W⊃12(∑isiA2(i))∫Λ−2dt t−1∫d4xg R  ⇒  1Gind∝(∑isiA2(i))×F(Λ),W \supset \frac{1}{2}\Big(\sum_i s_i A_2^{(i)}\Big)\int_{\Lambda^{-2}} dt\,t^{-1} \int d^4x\sqrt{g}\,R
\;\Rightarrow\;
\frac{1}{G_{\rm ind}} \propto \Big(\sum_i s_i A_2^{(i)}\Big)\times F(\Lambda),W⊃21(i∑siA2(i))∫Λ−2dtt−1∫d4xgR⇒Gind1∝(i∑siA2(i))×F(Λ),
with si=+1s_i=+1si=+1 for bosons and −1-1−1 for fermions. For a sharp UV cutoff the dominant scaling is ∼Λ2 \sim\Lambda^2∼Λ2 (Sakharov-style), while for zeta regularization it appears as logarithmic pieces and finite parts.