Loop differential K-theory

Tradler, Thomas; Wilson, Scott O.; Zeinalian, Mahmoud

Annales mathématiques Blaise Pascal, Tome 22 (2015) no. 1, pp. 121-163.

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 $M$, to the free loop space $LM$, 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 $M$, 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.

[1] Bismut, Jean-Michel Index theorem and equivariant cohomology on the loop space, Comm. Math. Phys., Volume 98 (1985) no. 2, pp. 213-237 http://projecteuclid.org/euclid.cmp/1103942357 | DOI | MR | Zbl

[2] Bunke, U; Nikolaus, T; Voelkl, M Differential cohomology theories as sheaves of spectra (http://arxiv.org/abs/1311.3188)

[3] Bunke, Ulrich; Schick, Thomas Smooth $K$-theory, Astérisque (2009) no. 328, p. 45-135 (2010) | Numdam | MR | Zbl

[4] Bunke, Ulrich; Schick, Thomas Uniqueness of smooth extensions of generalized cohomology theories, J. Topol., Volume 3 (2010) no. 1, pp. 110-156 | DOI | MR | Zbl

[5] Chern, Shiing Shen; Simons, James Characteristic forms and geometric invariants, Ann. of Math. (2), Volume 99 (1974), pp. 48-69 | DOI | MR | Zbl

[6] Freed, Daniel S.; Hopkins, Michael On Ramond-Ramond fields and $K$-theory, J. High Energy Phys. (2000) no. 5, pp. Paper 44, 14 | DOI | MR | Zbl

[7] Freed, Daniel S.; Moore, Gregory W.; Segal, Graeme The uncertainty of fluxes, Comm. Math. Phys., Volume 271 (2007) no. 1, pp. 247-274 | DOI | MR | Zbl

[8] Getzler, Ezra; Jones, John D. S.; Petrack, Scott Differential forms on loop spaces and the cyclic bar complex, Topology, Volume 30 (1991) no. 3, pp. 339-371 | DOI | MR | Zbl

[9] Hamilton, Richard S. The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), Volume 7 (1982) no. 1, pp. 65-222 | DOI | MR | Zbl

[10] Han, F. Supersymmetric QFT, Super Loop Spaces and Bismut-Chern Character, University of California, Berkeley (2005) (Ph. D. Thesis) | MR

[11] Hopkins, M. J.; Singer, I. M. Quadratic functions in geometry, topology, and M-theory, J. Differential Geom., Volume 70 (2005) no. 3, pp. 329-452 http://projecteuclid.org/euclid.jdg/1143642908 | MR | Zbl

[12] Jones, J. D. S.; Petrack, S. B. The fixed point theorem in equivariant cohomology, Trans. Amer. Math. Soc., Volume 322 (1990) no. 1, pp. 35-49 | DOI | MR | Zbl

[13] Lott, John $\mathbf{R}/\mathbf{Z}$ index theory, Comm. Anal. Geom., Volume 2 (1994) no. 2, pp. 279-311 | MR | Zbl

[14] Simons, James; Sullivan, Dennis Structured vector bundles define differential $K$-theory, Quanta of maths (Clay Math. Proc.), Volume 11, Amer. Math. Soc., Providence, RI, 2010, pp. 579-599 | MR | Zbl

[15] Stolz, Stephan; Teichner, Peter Supersymmetric field theories and generalized cohomology, Mathematical foundations of quantum field theory and perturbative string theory (Proc. Sympos. Pure Math.), Volume 83, Amer. Math. Soc., Providence, RI, 2011, pp. 279-340 | DOI | MR | Zbl

[16] Tradler, Thomas; Wilson, Scott O.; Zeinalian, Mahmoud Equivariant holonomy for bundles and abelian gerbes, Comm. Math. Phys., Volume 315 (2012) no. 1, pp. 39-108 | DOI | MR | Zbl

[17] Zamboni, Luca Quardo A Chern character in cyclic homology, Trans. Amer. Math. Soc., Volume 331 (1992) no. 1, pp. 157-163 | DOI | MR | Zbl