NDLI: Cheeger–Chern–Simons Theory and Differential String Classes

Content Provider	Springer Nature Link
Author	Becker, Christian
Copyright Year	2016
Abstract	We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class $${\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}$$ , we recover isomorphism classes of geometric string structures on Spin n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class $${\frac{1}{2} p_1}$$ together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.
Starting Page	1529
Ending Page	1594
Page Count	66
File Format	PDF
ISSN	14240637
Journal	Annales Henri Poincaré
Volume Number	17
Issue Number	6
e-ISSN	14240661
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-04-22
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Theoretical, Mathematical and Computational Physics Dynamical Systems and Ergodic Theory Quantum Physics Mathematical Methods in Physics Classical and Quantum Gravitation, Relativity Theory Elementary Particles, Quantum Field Theory
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Nuclear and High Energy Physics Mathematical Physics

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