[Paper Review] Complex Chern--Simons bundles in the relative setting

Dennis Eriksson, Gerard Freixas i Montplet+1 more|arXiv (Cornell University)|Sep 5, 2021

Geometry and complex manifolds38 references4 citations

TL;DR

This paper introduces complex Chern–Simons bundles in the relative setting for families of Riemann surfaces, using a functorial intersection theory inspired by Deligne–Elkik and Bott–Chern classes. It establishes the crystalline nature of the bundle, proves its curvature equals the Atiyah–Bott–Goldman form, and resolves two major conjectures: on the comparison between Bergman and Bers projective structures, and on the Riemann–Roch formula for Cappell–Miller torsion.

ABSTRACT

Complex Chern-Simons bundles are line bundles with connection, originating in the study of quantization of moduli spaces of flat connections with complex gauge groups. In this paper we introduce and study these bundles in the families setting. The central object is a functorial direct image of characteristic classes of vector bundles with connections, for which we develop a formalism. Our strategy elaborates on Deligne-Elkik's intersection bundles, and a refined Chern-Simons theory which parallels the use of Bott-Chern classes in Arakelov geometry. In the context of moduli spaces, we are confronted with flat relative connections on families of Riemann surfaces. To be able to rely on the functorial approach, we prove canonical extension results to global connections, inspired by the deformation theory of harmonic maps in non-abelian Hodge theory. The relative complex Chern-Simons bundle $\mathcal{L}_{CS}$ is then defined as a functorial direct image of the second Chern class on the relative moduli space of flat vector bundles. We establish the crystalline nature of $\mathcal{L}_{CS}$, and the existence of a holomorphic extension of natural metrics from Arakelov geometry. The curvature of $\mathcal{L}_{CS}$ can be expressed in terms of the Atiyah-Bott-Goldman form, in agreement with the classical topological approach. To highlight a few applications, we first mention a characterization of projective structures of Riemann surfaces in terms of connections on intersection bundles. In particular, we settle a conjecture of Bertola-Korotkin-Norton on the comparison between the Bergman and the Bers projective structures. This is the problem of determining the accessory parameters of quasi-Fuchsian uniformizations. A conjecture of Cappell-Miller is also established, to the effect that their holomorphic torsion satisfies a Riemann-Roch formula.

Motivation & Objective

• To develop a functorial formalism for complex Chern–Simons bundles in the relative setting of families of compact Riemann surfaces.
• To resolve the conjecture of Bertola–Korotkin–Norton on the relationship between Bergman and Bers projective structures.
• To establish the Riemann–Roch formula for Cappell–Miller holomorphic torsion, extending Deligne’s Riemann–Roch isometry.
• To provide a new characterization of projective structures via connections on intersection bundles using the Chern–Simons transform.
• To prove canonical extensions of flat relative connections, enabling global metric and curvature analysis in non-abelian Hodge theory.

Proposed method

• Adapts Deligne–Elkik’s intersection bundle formalism to construct a functorial direct image of the second Chern class on relative moduli spaces of flat bundles.
• Develops a refined Chern–Simons theory using Bott–Chern classes to handle secondary invariants in Arakelov geometry.
• Introduces the Chern–Simons transform as a kernel that maps families of projective structures to connections on Deligne pairings.
• Applies deformation theory of harmonic maps to prove canonical extensions of flat relative connections to global connections.
• Uses the universal moduli space of flat $ SL_r(bC)$-bundles over a family of curves to define the relative complex Chern–Simons bundle $ L_{ m CS}$.
• Establishes holomorphic extension of metrics and proves the curvature of $ L_{ m CS}$ matches the Atiyah–Bott–Goldman form via transgression classes.

Experimental results

Research questions

• RQ1 How can complex Chern–Simons bundles be defined in a functorial, relative setting for families of Riemann surfaces?
• RQ2 What is the relationship between the Bergman and Bers projective structures on a Riemann surface, and can it be characterized via Chern–Simons theory?
• RQ3 Does the Cappell–Miller holomorphic torsion satisfy a Riemann–Roch formula, and if so, how is it related to Deligne’s Riemann–Roch isometry?
• RQ4 Can the complex Chern–Simons bundle be used to characterize projective structures via connections on intersection bundles?
• RQ5 What is the curvature of the relative complex Chern–Simons bundle, and how does it relate to classical topological invariants like the Atiyah–Bott–Goldman form?

Key findings

• The relative complex Chern–Simons bundle $ L_{ m CS}$ is shown to be crystalline, with curvature equal to the Atiyah–Bott–Goldman form.
• The conjecture of Bertola–Korotkin–Norton on the comparison between Bergman and Bers projective structures is proven using the Chern–Simons transform.
• The Cappell–Miller torsion satisfies a Riemann–Roch formula, confirming a conjecture and extending Deligne’s Riemann–Roch isometry to holomorphic torsion.
• A new proof of Wolpert’s result relating the first tautological class to the Weil–Petersson form on Teichmüller space is obtained via the formalism.
• The complex Chern–Simons bundle provides a simple construction of the Takhtajan–Zograf Liouville action on Schottky space.
• The existence of a holomorphic extension of metrics from Arakelov geometry to the relative complex Chern–Simons bundle is established.