In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold , to the free loop space , and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of , in much the same way that differential K-theory can be defined using the Chern-Simons form [14]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle.
@article{AMBP_2015__22_1_121_0,
author = {Tradler, Thomas and Wilson, Scott O. and Zeinalian, Mahmoud},
title = {Loop differential K-theory},
journal = {Annales math\'ematiques Blaise Pascal},
volume = {22},
year = {2015},
pages = {121-163},
doi = {10.5802/ambp.348},
language = {en},
url = {http://dml.mathdoc.fr/item/AMBP_2015__22_1_121_0}
}
Tradler, Thomas; Wilson, Scott O.; Zeinalian, Mahmoud. Loop differential K-theory. Annales mathématiques Blaise Pascal, Tome 22 (2015) pp. 121-163. doi : 10.5802/ambp.348. http://gdmltest.u-ga.fr/item/AMBP_2015__22_1_121_0/
[1] Index theorem and equivariant cohomology on the loop space, Comm. Math. Phys., Tome 98 (1985) no. 2, pp. 213-237 http://projecteuclid.org/euclid.cmp/1103942357 | Article | MR 786574 | Zbl 0591.58027
[2] Differential cohomology theories as sheaves of spectra (http://arxiv.org/abs/1311.3188)
[3] Smooth -theory, Astérisque (2009) no. 328, p. 45-135 (2010) | MR 2664467 | Zbl 1202.19007
[4] Uniqueness of smooth extensions of generalized cohomology theories, J. Topol., Tome 3 (2010) no. 1, pp. 110-156 | Article | MR 2608479 | Zbl 1252.55002
[5] Characteristic forms and geometric invariants, Ann. of Math. (2), Tome 99 (1974), pp. 48-69 | Article | MR 353327 | Zbl 0283.53036
[6] On Ramond-Ramond fields and -theory, J. High Energy Phys. (2000) no. 5, pp. Paper 44, 14 | Article | MR 1769477 | Zbl 0990.81624
[7] The uncertainty of fluxes, Comm. Math. Phys., Tome 271 (2007) no. 1, pp. 247-274 | Article | MR 2283960 | Zbl 1126.81045
[8] Differential forms on loop spaces and the cyclic bar complex, Topology, Tome 30 (1991) no. 3, pp. 339-371 | Article | MR 1113683 | Zbl 0729.58004
[9] The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), Tome 7 (1982) no. 1, pp. 65-222 | Article | MR 656198 | Zbl 0499.58003
[10] Supersymmetric QFT, Super Loop Spaces and Bismut-Chern Character, University of California, Berkeley (2005) (Ph. D. Thesis) | MR 2712304
[11] Quadratic functions in geometry, topology, and M-theory, J. Differential Geom., Tome 70 (2005) no. 3, pp. 329-452 http://projecteuclid.org/euclid.jdg/1143642908 | MR 2192936 | Zbl 1116.58018
[12] The fixed point theorem in equivariant cohomology, Trans. Amer. Math. Soc., Tome 322 (1990) no. 1, pp. 35-49 | Article | MR 1010411 | Zbl 0723.55003
[13] index theory, Comm. Anal. Geom., Tome 2 (1994) no. 2, pp. 279-311 | MR 1312690 | Zbl 0840.58044
[14] Structured vector bundles define differential -theory, Quanta of maths, Amer. Math. Soc., Providence, RI (Clay Math. Proc.) Tome 11 (2010), pp. 579-599 | MR 2732065 | Zbl 1216.19009
[15] Supersymmetric field theories and generalized cohomology, Mathematical foundations of quantum field theory and perturbative string theory, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 83 (2011), pp. 279-340 | Article | MR 2742432 | Zbl 1257.55003
[16] Equivariant holonomy for bundles and abelian gerbes, Comm. Math. Phys., Tome 315 (2012) no. 1, pp. 39-108 | Article | MR 2966940 | Zbl 1254.53048
[17] A Chern character in cyclic homology, Trans. Amer. Math. Soc., Tome 331 (1992) no. 1, pp. 157-163 | Article | MR 1044967 | Zbl 0762.55004