The Seeley–DeWitt Bridge: From Nambu--Jona-Lasinio Dynamics to the Einstein--Hilbert Action - V1

Starting from a microscopic NJL-type fermionic theory, we perform a Hubbard–Stratonovich transformation to introduce auxiliary bosonic fields and rewrite the four-fermion interactions. This yields an effective Dirac operator in a background that encodes the relevant condensates and mean fields. The square of this Dirac operator admits a Lichnerowicz–type decomposition, which puts it in Laplace form and makes the underlying geometric structure explicit. Using the Schwinger proper-time / heat-kernel representation of the resulting Laplace-type operator, we obtain a systematic Seeley–DeWitt expansion, whose coefficients {a0,a2,a4,a6,… } encode the effective action and, in particular, the emergent Einstein–Hilbert and higher-curvature terms.