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

==== We want the Euclidean one-loop effective action ====

W  =  12ln⁡det⁡ΔB−ln⁡det⁡ΔF+[ghosts]+[surface determinants]W\;=\;\frac{1}{2}\ln\det\Delta_{\rm B} - \ln\det\Delta_{\rm F} + [\text{ghosts}] + [\text{surface determinants}]W=21lndetΔB−lndetΔF+[ghosts]+[surface determinants]
where ΔB\Delta_{\rm B}ΔB is the second-order elliptic operator for bosonic fluctuations (vector) after gauge fixing and ΔF\Delta_{\rm F}ΔF the Dirac (squared) operator for fermions.

Use the heat-kernel representation (Euclidean):

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
and the small-t asymptotic expansion for the trace (in 4D with boundary):

Tr e−tΔ∼∑k=0∞t(k−4)/2 Ak(Δ)\mathrm{Tr}\,e^{-t\Delta} \sim \sum_{k=0}^{\infty} t^{(k-4)/2}\,A_k(\Delta)Tre−tΔ∼k=0∑∞t(k−4)/2Ak(Δ)
The A2A_2A2 coefficient (the term that gives t−1t^{-1}t−1) carries the integrated curvature invariants that feed the induced ∫R\int R∫R term.

Schematic relation: the induced Einstein–Hilbert term scales with heat-kernel coefficient A2A_2A2 and the UV regulator (e.g. Λ2\Lambda^2Λ2 or ln⁡Λ\ln\LambdalnΛ).