Mathematics
Equivariant APS index for Dirac operators of non-product type near the boundary
Document Type
Article
Abstract
We consider a generalized Atiyah-Patodi-Singer boundary problem for a G-invariant Dirac-type operator, which is not of product type near the boundary. We establish a delocalized version (a so-called Kirillov formula) of the equivariant index theorem for this operator. We obtain more explicit formulas for different geometric Dirac-type operators. In particular, we get a formula for the equivariant signature of a local system over a manifold with boundary. In case of a trivial local system, our formula can be viewed as a new way to compute the infinitesimal equivariant eta-invariant of S. Goette. We explicitly compute all the terms in this formula, which involve the equivariant Hirzebruch L-form and its transgression, for four-dimensional Kähler manifolds admitting a special Kähler-Ricci potential (SKR metrics), a class including many Kähler conformally Einstein manifolds, in the case where the boundary is given as the zero level set of this Killing potential. In the case of SKR metrics which are local Kähler products, these terms are zero, and we obtain a vanishing result for the infinitesimal equivariant eta invariant.
Publication Title
Indiana University Mathematics Journal
Publication Date
2019
Volume
68
Issue
2
First Page
435
Last Page
501
ISSN
0022-2518
DOI
10.1512/iumj.2019.68.7621
Repository Citation
Braverman, Maxim and Maschler, Gideon, "Equivariant APS index for Dirac operators of non-product type near the boundary" (2019). Mathematics. 31.
https://commons.clarku.edu/mathematics/31
