Spectral and Fredholm Structure of the Wilsonian Renormalization Group

Spectral and Fredholm Structure of the Wilsonian Renormalization Group develops a rigorous operator-theoretic formulation of the Wilsonian renormalization group (RG) near a fixed point. The RG flow is modeled as a dynamical system on a Banach manifold of effective theories, and its linearization is analyzed using sectorial operator theory, analytic semigroups, and Fredholm analysis. Under natural structural assumptions—sectoriality, Fredholm property, and a spectral gap—the paper proves that the resolvent of the linearized RG generator is meromorphic in appropriate regions, with finite-rank residues. As a consequence, unstable directions correspond to isolated eigenvalues of finite algebraic multiplicity, and the generated analytic semigroup admits an exponential dichotomy. The nonlinear RG flow is then shown to possess a finite-dimensional local unstable manifold tangent to the unstable spectral subspace. The number of relevant directions is determined entirely by spectral data and is invariant under smooth scheme transformations, which act by similarity on the generator. Conceptually, the work reframes universality classes as equivalence classes of sectorial Fredholm generators classified by intrinsic spectral invariants, placing renormalization group theory within the broader framework of infinite-dimensional dynamical systems and analytic spectral theory.