Bismut–Cheeger eta form

Bismut–Cheeger eta form
Name	Bismut–Cheeger eta form

Bismut–Cheeger eta form is a differential form introduced in the analytic study of families of elliptic operators by Jean-Michel Bismut and Jeff Cheeger, arising in the context of the families index theorem and adiabatic limits. It refines the spectral eta invariant of Atiyah, Patodi and Singer into a differential form on the base of a fibration, connecting index theory, characteristic classes, and global analysis. The construction plays a central role in the study of anomalies in mathematical physics, and in geometric formulations of the Riemann–Roch theorem for families.

Definition and Context

The Bismut–Cheeger eta form was defined by Jean-Michel Bismut and Jeff Cheeger within the framework developed by Michael Atiyah, Isadore Singer, and Richard Melrose for spectral asymmetry and boundary value problems. It sits alongside objects studied by Daniel Quillen, Raoul Bott, and Henri Cartan in relations between curvature and characteristic classes, and it refines invariants related to the work of Sir Michael Atiyah and Isadore Singer on the Index Theorem and the contributions of Vladimir Arnold in symplectic geometry. The eta form associates to a Riemannian fibration with fibre Dirac operator a differential form on the base first systematically explored in papers by Bismut and Cheeger and later used by Edward Witten in quantum field theory contexts.

Construction and Properties

Bismut–Cheeger construction uses heat kernel techniques pioneered by Paul Dirac, Spencer Bloch, and Daniel Freed, and relies on superconnection formalism introduced by Quillen and elaborated by Bismut, Berline, Getzler, and Vergne. Given a smooth fibration with compact fibre and a family of Dirac-type operators, the eta form arises from the rescaled Bismut superconnection and the corresponding transgression forms studied by Jean-Pierre Serre, Alexander Grothendieck, and David Mumford. Key properties mirror those of Chern–Simons forms in work by Shiing-Shen Chern and James Simons: the exterior derivative of the eta form equals the difference between the families index density and a local characteristic form identified by René Thom and Friedrich Hirzebruch, and it behaves naturally under pullback by maps considered by John Milnor and René Thom. Analytic regularization methods related to John Tate and Igor Krichever control convergence and spectral flow aspects explored by Daniel Freed and John Lott.

Relation to Eta Invariants and Index Theory

The eta form refines the scalar eta invariant introduced by Atiyah, Patodi and Singer in their study of spectral asymmetry on manifolds with boundary and links to the Atiyah–Singer index theorem for families as developed by Atiyah and Singer and furthered by Michael Hopkins and Isadore Singer. In the adiabatic limit studied by Witten and Cheeger, the eta form captures the transgression between local index densities described by Hirzebruch and Bott and global spectral invariants related to the work of Don Zagier and Edward Witten. Connections to analytic torsion studied by Ray and Singer and to determinant line bundles examined by Quillen and Bismut–Freed clarify anomaly cancellation phenomena investigated by Alberto S. Cattaneo and Constantin Teleman. The eta form also interfaces with K-theory constructions of Atiyah and Joshua Lurie and with cobordism theories associated to René Thom and John Milnor.

Applications in Differential Geometry and Topology

Applications span geometric analysis, topological quantum field theory, and index-theoretic formulations of Riemann–Roch type results associated to Grothendieck and Hirzebruch. The eta form appears in anomaly formulas considered by Edward Witten and Gregory Moore, in determinant bundle curvature computations related to Quillen and Bismut–Freed, and in spectral flow analyses pursued by Atiyah and Patodi. In differential K-theory developed by Hopkins and Singer and in the study of gerbes and Ramond–Ramond fields influenced by Juan Maldacena and Maxim Kontsevich, the eta form provides correction terms ensuring well-definedness of pushforward maps studied by Gerd Faltings and Pierre Deligne. It also contributes to problems studied by Richard Schoen, Karen Uhlenbeck, and Shing-Tung Yau in geometric analysis and to topological applications related to John Milnor and Michael Freedman.

Examples and Computations

Concrete computations of the eta form have been carried out for circle fibrations like those appearing in the work of Claude Chevalley and Élie Cartan, torus bundles studied by William Thurston and John Milnor, and mapping tori considered by Dennis Sullivan and Robion Kirby. Explicit formulas in low-dimensional situations employ techniques from heat kernel asymptotics developed by Lars Hörmander and Victor Guillemin and use characteristic classes in the spirit of Chern and Pontryagin as in Hirzebruch’s signature theorem. Computations relevant to anomalies in string theory reference contributions by Edward Witten and Andrew Strominger, while examples involving families with symmetry exploit representation-theoretic tools from Hermann Weyl and Harish-Chandra.

Generalizations and Extensions

Extensions of the Bismut–Cheeger eta form appear in equivariant settings studied by Michèle Vergne and Nicole Berline, in higher eta invariants explored by John Lott and Matthias Lesch, and in noncommutative geometry contexts initiated by Alain Connes and Masoud Khalkhali. Relations to elliptic cohomology researched by Haynes Miller and Graeme Segal, to derived algebraic geometry advanced by Jacob Lurie and Maxim Kontsevich, and to index theory on manifolds with corners developed by Richard Melrose and Boris Tsygan broaden its applicability. Active research continues at the interface with mathematical physics by Edward Witten, Gregory Moore, and Anton Kapustin, and with homotopy-theoretic refinements pursued by Mike Hopkins and Isadore Singer.

Category:Index theory