UV Structure and Asymptotic Freedom of the 4D Sigma Model on Sp(2N,R)/U(N) with Fisher–Bures Metric

We establish the ultraviolet structure of the four-dimensional nonlinear sigma model with target manifold M = Sp(2N,R)/U(N) equipped with the Fisher–Bures metric. The model arises in the Vacuum Time Geometry programme, where spacetime geometry emerges from the entanglement structure of the vacuum via a Page–Wootters mechanism, and the spacetime metric is the pullback of the quantum Fisher–Bures metric through an entanglement map λ : M4 →M.
We prove three main results. Theorem 1 gives the exact two-loop coefficient c1(N) = [N(N + 1)2 − 4]/[8(N + 1)2(N − 1)] for all N ≥ 2, with c1 → 1/8 as N → ∞. Theorem 2
establishes asymptotic freedom: the beta function satisfies β(T) < 0 for T ∈ (0, T∗), where T∗(d = 4,N = 28) ≈ 6.305, and the asymptotic freedom window does not collapse as N varies. Theorem 3 reports that three independent non-perturbative truncations of the Wetterich exact renormalization group equation yield β(T) < 0 for all T > 0, with no UV fixed point at finite coupling — a result strictly stronger than perturbative asymptotic freedom.
Asymptotic freedom follows from the negativity of the Ricci curvature, Ric < 0, which is itself a consequence of the non-negative sectional curvature KFB ≥ 0 of the Fisher–Bures metric on the symmetric space of noncompact type. The UV limit T → 0 corresponds, in the Page–Wootters framework, to the unentangled pure-state vacuum with flat spacetime geometry.