Asymptotics of Analytic Torsion for Hitchin Fibrations: Vanishing Cohomology and Limiting Mixed Hodge Structures

This paper analyzes the asymptotic behavior of Ray–Singer analytic torsion for degenerating Hitchin fibers in the context of algebraic geometry and Hodge theory. We establish an explicit connection between differential-geometric invariants (torsion) and algebro-geometric structures (weight filtrations on vanishing cohomology). Main Results: We prove that near discriminant loci, the analytic torsion develops a logarithmic singularity of the form: $$\tau(X_t) \sim C \cdot |t|^{-\delta/6}$$ where $\delta$ is the number of nodes in the degenerate fiber. This coefficient $\lambda(N) = \delta/6$ is obtained via the Bismut–Bost curvature theorem and admits an algebraic reformulation through the weight filtration on vanishing cohomology $H_{\text{van}}$. Key Innovation: We resolve the apparent paradox that the monodromy logarithm $N$ is nilpotent with $\text{Tr}(N) = 0$, yet produces a non-trivial singularity coefficient. The resolution involves: Introduction of vanishing cohomology $H_{\text{van}} := \text{Gr}^W_0 \oplus \text{Gr}^W_2$, excluding non-collapsing cycles from the normalization Distinction between the nilpotent operator $N$ and the semisimple grading operator $Y^W$ Reformulation: $\lambda(N) = \kappa \sum_p (-1)^p \text{Tr}(Y^W|H^p_{\text{van}})$ with $\kappa = -1/12$ Verification: For the Legendre family of elliptic curves $E_\lambda: y^2 = x(x-1)(x-\lambda)$, we explicitly verify: Bismut–Bost direct calculation: $\lambda(N) = 1/6$ Weight filtration reformulation: $\lambda(N) = 1/6$ Conformal field theory anomaly: $c/12 = 1/12$ consistent with $\kappa = -1/12$ We prove genus-independence: for a single node, $\lambda(N) = 1/6$ regardless of genus $g \geq 1$, because $\dim(H_{\text{van}}) = 2$ is constant while $\dim(\text{Gr}^W_1) = 2(g-1)$ varies but does not contribute to metric collapse. Connection to P = W: We propose a conjectural relationship between torsion asymptotics and the perverse filtration arising from the P = W phenomenon (de Cataldo–Hausel–Migliorini). Under the P = W correspondence $\text{Gr}^P_k \cong \text{Gr}^W_{2k}$, the torsion singularity may provide an analytic measure of perverse complexity. Mathematical Framework: Geometric setup: Hitchin fibration $h: M_{\text{Dol}} \to A_{\text{par}}$ for moduli of Higgs bundles Analytic tool: Ray–Singer torsion $\log \tau_{\text{RS}}(X,E,g) = -\frac{1}{2}\sum_p (-1)^p p \zeta'_p(0)$ Hodge theory: Schmid's nilpotent orbit theorem, limiting mixed Hodge structures Curvature formulas: Bismut–Gillet–Soulé variation, Bismut–Bost logarithmic singularity Technical Contributions: Rigorous separation of vanishing vs. non-vanishing cohomology via weight filtration Explicit calculation of the normalization constant $\kappa = -1/12$ relating spectral and Hodge-theoretic perspectives Computational verification for concrete families (Legendre, higher genus) Framework for extending to higher-dimensional varieties and multiple singularities Keywords: Hitchin fibration, Ray–Singer torsion, limiting mixed Hodge structure, weight filtration, vanishing cohomology, P = W phenomenon, Bismut–Bost curvature, monodromy, conformal field theory