A loop group extension of the odd Chern character

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• 作者：Scott O. Wilson ^{scott.wilson@qc.cuny.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：19L10 ; 58J28 ; 22E67
• 刊名：Journal of Geometry and Physics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：102
• 期：Complete
• 页码：32-43
• 全文大小：432 K

文摘

We show that the universal odd Chern form, defined on the stable unitary group U$U$, extends to the loop group LU$LU$ 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$K$-theory to the periodic cohomology of the free loop space, represented geometrically on the level of differential forms.