A loop group extension of the odd Chern character

JOURNAL OF GEOMETRY AND PHYSICS

卷:102

Article

Wilson, Scott O.¹ 

[1] CUNY, Queens Coll, Dept Math, 65-30 Kissena Blvd, Flushing, NY 11367 USA

关键词: Chern character;    K-theory;    Loop groups;   

DOI  :  10.1016/j.geomphys.2016.01.001

来源: Elsevier

【 摘 要 】

We show that the universal odd Chern form, defined on the stable unitary group U, extends to the loop group LU as an equivariantly closed differential form. This provides an odd analogue to the Bismut Chern form that appears in supersymmetric field theories. We also describe the associated transgression form, the so-called Bismut Chern Simons form, and explicate some properties it inherits as a differential form on the space of maps of a cylinder into the stable unitary group. As one corollary, we show that in a precise sense the spectral flow of a loop of self adjoint Fredholm operators equals the lowest degree component of the Bismut Chern Simons form, and the latter, when restricted to cylinders which are tori, is an equivariantly closed extension of spectral flow. As another corollary, we construct the Chern character homomorphism from odd K-theory to the periodic cohomology of the free loop space, represented geometrically on the level of differential forms. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free