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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bär, C., Becker, C.: Differential Characters. Lecture Notes in Mathematics, vol. 2112. Springer, Berlin (2014)
Baez J., Crans A., Stevenson D., Schreiber U.: From loop groups to 2-groups. Homol. Homotopy Appl. 9, 101–135 (2007)
Becker, C., Schenkel, A., Szabo, R.J.: Differential cohomology and locally covariant quantum field theory (2014) (preprint). arXiv:1406.1514
Borel A.: Sur la Cohomologie des Esapces Fibrés Principaux et des Espaces Homogènes des Groupes de Lie Compacts. Ann. Math. (2) 57, 115–207 (1953)
Borel A., Hirzebruch F.: Characteristic classes and homogeneous spaces I. Am. J. Math. 80, 458–538 (1958)
Biswas I., Hurtubise J., Stasheff J.: A construction of a universal connection. Forum Math. 24, 365–378 (2012)
Brightwell M., Turner P.: Relative differential characters. Commun. Anal. Geom. 14, 269–282 (2006)
Brylinski, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Progress in Mathematics. Birkhäuser, Boston (1993)
Bunke, U.: Differential cohomology (2012) (preprint). arXiv:1208.3961
Bunke U., Kreck M., Schick T.: A geometric description of differential cohomology. Ann. Math. Blaise Pascal 17, 1–16 (2010)
Bunke U., Schick T.: Real secondary index theory. Algebraic Geom. Topol. 8, 1093–1139 (2008)
Carey A., Johnson S., Murray M., Stevenson D., Wang B.-L.: Bundle gerbes for Chern–Simons and Wess–Zumino–Witten theories. Commun. Math. Phys. 259, 577–613 (2005)
Cartan, H.: La transgression dans un groupe de Lie et dans un espace fibré principal. Colloque de topologie, (Espace fibrés), pp. 57–71. Liege (1950)
Cheeger J., Simons J.: Differential characters and geometric invariants. Geom. Topol. Lect. Notes Math. 1167, 50–80 (2006)
Chern S.S., Simons J.: Characteristic forms and geometric invariants. Ann. Math. 99, 48–69 (1974)
Coquereaux R., Pilch K.: String structures on loop bundles. Commun. Math. Phys. 120, 353–378 (1989)
Davis J.F., Kirk P.: Lecture Notes in Algebraic Topology. American Mathematical Society, Providence (2001)
Dijkgraaf R., Witten E.: Topological gauge theories and group cohomology. Commun. Math. Phys. 129, 393–429 (1990)
Dupont, J.L.: Curvature and Characteristic Classes. Lecture Notes in Mathematics, vol. 640. Springer, Berlin (1978)
Dupont J.L., Ljungmann R.: Integration of simplicial forms and Deligne cohomology. Math. Scand. 97, 11–39 (2005)
Freed D., Hopkins J.M.: Chern–Weil forms and abstract homotopy theory. Bull. Am. Math. Soc. 50, 431–468 (2013)
Garland H., Murray M.K.: Kac–Moody monopoles and Periodic instantons. Commun. Math. Phys. 120, 335–351 (1988)
Gomi K., Terashima Y.: A fiber integration formula for the smooth Deligne cohomology. Int. Math. Res. Not. 13, 699–708 (2000)
Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7, 65–222 (1982)
Harrington B.J., Shephard H.K.: Periodic Euclidean solutions and the finite temperature Yang–Mills gas. Phys. Rev. D 17, 2122–2125 (1978)
Harvey R., Lawson B., Zweck J.: The de Rham-Federer theory of differential characters and character duality. Am. J. Math. 125, 791–847 (2003)
Heitsch J.L., Lawson H.B.: Transgressions, Chern–Simons invariants and the classical groups. J. Differ. Geom. 9, 423–434 (1974)
Hekmati P., Murray M.K., Vozzo R.F.: The general caloron correspondence. J. Geom. Phys. 62, 224–241 (2012)
Henriques A.: Integrating \({L_\infty}\)-algebras. Compos. Math. 144, 1017–1045 (2008)
Ho, M.H.: On differential characteristic classes (2013) (preprint). arXiv:1311.3927
Hochschild G.: The Structure of Lie Groups. Holden-Day, San Francisco (1965)
Hopkins M.J., Singer I.M.: Quadratic functions in geometry, topology, and M-theory. J. Differ. Geom. 70, 329–452 (2005)
Killingback T.P.: World-sheet anomalies and loop geometry. Nucl. Phys. B 288, 578–588 (1987)
Kriegl A., Michor P.W.: The Convenient Setting of Global Analysis. American Mathematical Society, Providence (1997)
Kumar S.: A remark on universal connections. Math. Ann. 260, 453–462 (1982)
Ljungmann, R.: Secondary invariants for families of bundles. Ph.D. thesis, Aarhus (2006)
Meinrenken E.: The basic gerbe over a compact simple Lie group. Enseign. Math. 49, 307–333 (2003)
Milnor J.: Construction of universal bundles. II. Ann. Math. 63, 430–436 (1956)
Mostow M.A.: The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations. J. Differ. Geom. 14, 255–293 (1979)
Murray M.K., Vozzo R.F.: The caloron correspondence and higher string classes for loop groups. J. Geom. Phys. 60, 1235–1250 (2010)
Nahm W.: Self-dual monopoles and calorons. Group theoretical methods in physics. Lect. Notes Phys. 201, 189–200 (1984)
Narasimhan M.S., Ramanan S.: Existence of universal connections. Am. J. Math. 83, 563–572 (1961)
Nikolaus T., Sachse C., Wockel C.: A smooth model for the string group. Int. Math. Res. Not. 16, 3678–3721 (2013)
Pressly A., Segal G.: Loog Groups. Oxford University Press, New York (1986)
Ramadas T.R.: On the space of maps inducing isomorphic connections. Ann. Inst. Fourier 32, 263–276 (1982)
Redden C.: String structures and canonical 3-forms. Pac. J. Math. 249, 447–484 (2011)
Redden, C.: Differential Borel equivariant cohomology via connections (2016) (preprint). arXiv:1602.06921
Sati H., Schreiber U., Stasheff J.: Twisted differential string and fivebrane structures. Commun. Math. Phys. 315, 169–213 (2012)
Schlafly R.: Universal connections. Invent. Math. 58, 59–65 (1980)
Schommer-Pries C.: Central extensions of smooth 2-groups and a finite-dimensional string 2-group. Geom. Topol. 15, 609–676 (2011)
Stevenson D.: Bundle 2-gerbes. Proc. Lond. Math. Soc. (3) 88, 405–435 (2004)
Stolz S.: A conjecture concerning positive Ricci curvature and the Witten genus. Math. Ann. 304, 785–800 (1996)
Stolz, S., Teichner, P.: What is an Elliptic Object? Topology, Geometry and Quantum Field Theory. London Mathematical Society Lecture Note Series, vol. 308, pp. 247–343. Cambridge University Press, Cambridge (2004)
Waldorf, K.: Spin structures on loop space that characterize string manifolds (2012) (preprint). arXiv:1209.1731
Waldorf K.: String connections and Chern–Simons theory. Trans. Am. Math. Soc. 365, 4393–4432 (2013)
Waldorf K.: String geometry vs. spin geometry on loop spaces. J. Geom. Phys. 97, 190–226 (2015)
Witten, E.: The index of the Dirac operator in loop space. In: Elliptic Curves and Modular Forms in Algebraic Topology. Lecture Notes in Mathematics, vol. 1326, pp. 161–181. Springer, Berlin (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Boris Pioline.
Rights and permissions
About this article
Cite this article
Becker, C. Cheeger–Chern–Simons Theory and Differential String Classes. Ann. Henri Poincaré 17, 1529–1594 (2016). https://doi.org/10.1007/s00023-016-0485-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-016-0485-6