[Paper Review] Asymptotics for general connections at infinity

Carlos Simpson|arXiv (Cornell University)|Nov 28, 2003

Geometric and Algebraic Topology29 references3 citations

TL;DR

This paper establishes that the Laplace transform of monodromy matrices for connections approaching infinity in the de Rham moduli space of a compact Riemann surface admits analytic continuation with locally finite singularities. The key result is that the convex hull of exponential growth rates—determined by these singularities—is a polygon, whose vertices are tied to integrals of the tautological 1-form over piecewise homotopy liftings on the spectral curve, though the precise nature of the singularities remains undetermined.

ABSTRACT

For a standard path of connections going to a generic point at infinity in the moduli space $M_{DR}$ of connections on a compact Riemann surface, we show that the Laplace transform of the family of monodromy matrices has an analytic continuation with locally finite branching. In particular the convex subset representing the exponential growth rate of the monodromy is a polygon, whose vertices are in a subset of points described explicitly in terms of the spectral curve. Unfortunately we don't get any information about the size of the singularities of the Laplace transform, which is why we can't get asymptotic expansions for the monodromy.

Motivation & Objective

• To understand the asymptotic behavior of monodromy matrices for connections tending to infinity in the de Rham moduli space.
• To analyze the structure of exponential growth rates of monodromy under degeneration to a generic Higgs bundle.
• To determine the geometric and analytic nature of singularities in the Laplace transform of monodromy functions.
• To clarify the role of the spectral curve and its tautological 1-form in governing the asymptotic structure of monodromy.

Proposed method

• Use meromorphic gauge transformations to reduce the family of connections to the form $ d + B + tA $, where $ B $ may have poles.
• Apply techniques from previous work on singular perturbations, adapted to handle the presence of poles in $ B $.
• Define the Laplace transform $ f(\zeta) $ of the monodromy matrix family $ m(t) $, and study its analytic continuation.
• Characterize the set of singularities of $ f(\zeta) $ as lying in the set $ \Sigma(\gamma) $, of complex integrals of the tautological 1-form $ \alpha $ over piecewise homotopy liftings on the spectral curve.
• Show that the convex hull of exponential growth rates, $ \mathbf{hull}(m) $, is a polygon by analyzing the location of singularities in the Laplace transform.
• Use specialization arguments and Hartogs' theorem to prove that for generic Higgs bundles, at least one monodromy matrix has multiple singularities, implying nontrivial growth.

Experimental results

Research questions

• RQ1 What is the structure of the singularities in the Laplace transform of monodromy matrices for connections approaching infinity in $ M_{\mathrm{DR}} $?
• RQ2 How are the exponential growth rates of monodromy matrices encoded in the analytic properties of their Laplace transform?
• RQ3 Can the convex hull of exponential growth rates be characterized geometrically in terms of the spectral curve and its tautological 1-form?
• RQ4 Are the singularities of the Laplace transform confined to continuous homotopy liftings, or do they include more general piecewise liftings?
• RQ5 Does the monodromy representation exhibit strictly exponential growth for generic Higgs bundles, and can this be deduced from the location of singularities?

Key findings

• The Laplace transform $ f(\zeta) $ of the monodromy matrix family $ m(t) $ admits an analytic continuation with locally finite branching over $ \mathbb{C} $, as defined in Definition 6.2.
• The set of singularities of $ f(\zeta) $ is contained in $ \Sigma(\gamma) $, the set of integrals of the tautological 1-form $ \alpha $ over piecewise homotopy liftings on the spectral curve.
• The convex hull of exponential growth rates, $ \mathbf{hull}(m) $, is a polygon, whose vertices are subsets of points in $ \Sigma(\gamma) $, explicitly related to the spectral curve data.
• For generic Higgs bundles, at least one monodromy matrix has multiple singularities in its Laplace transform, implying that the monodromy is not of polynomial or single-exponential type.
• The Procesi coordinates $ R_i(t) $ of the monodromy representation $ \rho_t $ also have polygonal hulls, and for generic $ (E,\theta) $, at least one $ R_i $ is semistrictly exponential, implying $ |\rho_t| \geq c e^{a|t|} $ in most directions.
• Despite these results, the behavior of $ f(\zeta) $ near its singularities remains unknown—no bounds on growth (e.g., polynomial or otherwise) are established, leaving asymptotic expansions out of reach.