Cheeger–Chern–Simons

Cheeger–Chern–Simons
Name	Cheeger–Chern–Simons
Field	Differential geometry, Algebraic topology
Introduced	1970s
Creators	Jeff Cheeger; Shiing-Shen Chern; James Simons

Cheeger–Chern–Simons Cheeger–Chern–Simons classes are refined secondary invariants in differential geometry linking analytic, topological, and geometric data first developed by Jeff Cheeger, Shiing-Shen Chern, and James Simons. They extend the theory of Chern classs and Chern–Simons invariants and connect to index theory, Atiyah–Singer index theorem, and regulators in algebraic K-theory through constructions related to the work of André Weil, Alexander Grothendieck, and Armand Borel. These classes play roles in problems considered by researchers affiliated with institutions such as Institute for Advanced Study, Princeton University, and Harvard University.

Introduction

The Cheeger–Chern–Simons construction refines characteristic classes introduced by Chern–Weil theory and the differential form representatives used by Hermann Weyl and Élie Cartan, while incorporating torsion phenomena studied by John Milnor and Jean-Pierre Serre, and analytic torsion considered by D. B. Ray and Isadore Singer. It arose in the context of geometric analysis pursued by Michael Atiyah, Raoul Bott, and Isadore Singer and interfaces with arithmetic questions explored by Pierre Deligne and Spencer Bloch. The invariant lives naturally in groups akin to Deligne cohomology and the real cohomology groups considered by Serre duality contexts and the work of Grothendieck on regulators.

Definition and Construction

The construction uses a principal bundle with connection over a smooth manifold studied by Shiing-Shen Chern and uses curvature forms via Chern–Weil theory and transgression forms akin to those in Chern–Simons theory developed by James Simons. Given a principal G-bundle with structure group related to Lie groups such as SU(n), SO(n), or GL(n,C), one chooses an invariant polynomial from the list studied by Cartan, Killing form contexts, and forms a differential character in the sense of Cheeger and Simons. The resulting class sits in a group analogous to Deligne cohomology as in work by Pierre Deligne, Alexander Beilinson, and Spencer Bloch, matching topological Chern class data of Steenrod algebra flavor while recording connection-dependent secondary data reminiscent of constructions by Michel Kervaire and Bott periodicity phenomena.

Secondary Characteristic Classes

Cheeger–Chern–Simons invariants are examples of secondary characteristic classes, following the lineage of secondary invariants studied by Raoul Bott, John Milnor, and Michel Atiyah in contexts such as the Atiyah–Patodi–Singer index theorem. These classes generalize secondary classes like the Pontryagin class corrections used by William Browder and Kirby–Siebenmann theory and incorporate torsion information analogous to the phenomena studied by Serre and Borel. They can detect subtle geometric structures on manifolds investigated by William Thurston and Grigori Perelman in geometric-topological applications and refine regulators considered by Beilinson and Bloch–Kato.

Relation to Chern–Simons and Cheeger–Simons Theory

Cheeger–Chern–Simons theory refines Chern–Simons theory introduced by Shiing-Shen Chern and James Simons and relates directly to the differential characters of Cheeger–Simons differential characters constructed by Jeff Cheeger and James Simons. It interfaces with quantum field theoretic formulations explored by Edward Witten and Anton Kapustin and with the modular and arithmetic perspectives studied by Don Zagier and Maxim Kontsevich. The formalism parallels regulator maps in algebraic K-theory developed by Daniel Quillen and Andrei Suslin, and is compatible with index-theoretic viewpoints advanced by Atiyah–Singer collaborators and Bismut in analytic torsion studies.

Applications and Examples

Concrete computations appear for flat bundles over manifolds studied by John Milnor and for lens spaces and Seifert fibered spaces in the tradition of Raymond Weeks and William Thurston, as well as in the study of mapping tori arising in the work of Dennis Sullivan and William Thurston. In arithmetic geometry, Cheeger–Chern–Simons classes connect to regulator computations in the spirit of Beilinson and Bloch for motives that figure in the programs of Pierre Deligne and Alexander Grothendieck. In mathematical physics they contribute to anomaly cancellation and quantization conditions in models considered by Edward Witten, Michael Green, and John Schwarz, and interact with topological quantum field theories studied by Graeme Segal and Kevin Costello.

Properties and Functoriality

The invariants are natural under pullback along smooth maps as in functorial frameworks developed by Grothendieck and respect transgression properties analogous to those in Chern–Simons theory and Bott periodicity. They admit suspension and product formulas reflecting multiplicative behavior studied by Hirzebruch and satisfy compatibility with spectral sequence calculations used by Jean Leray and J. W. Milnor in fibration contexts. Functoriality with respect to group homomorphisms reflects the structure theory of Lie groups such as SU(n), SO(n), and Sp(n), and their representation-theoretic aspects investigated by Hermann Weyl and Élie Cartan.

Category:Differential geometry