Dirac Spectral Calculus: Heat Kernels, Spectral Actions, Zeta Determinants, Index Theory, Determinant Lines, and Quillen Geometry

This paper presents a unified analytic and geometric framework for the principal invariants associated with Dirac-type operators on closed manifolds. Starting from functional calculi of the squared Dirac operator, the paper derives heat kernel expansions, spectral actions, zeta functions, zeta-regularized determinants, the McKean–Singer index formula, determinant line constructions, and the Quillen metric. The framework clarifies the relationship between ordinary trace invariants governing the nonzero spectrum and graded supertrace invariants governing the kernel sector and index theory.