2020/07/02 by Lin, Dung-Cheng, Tsai, I-Hsun
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2007.00944
We study an index of a transversal Dirac operator on an odd-dimensional manifold X with locally free \mathbbS1-action. One difficulty of using heat kernel method lies in the understanding of the asymptotic expansion as t→ 0+. By a probabilistic approach via the Feynman-Kac formula, the transversal heat kernel on X can be linked to the ordinary heat kernel for functions on the orbifold M=X/\mathbbS1 which is more tractable. After some technical results for a uniform bound estimate as t→ 0+, we are reduced from the transversal, orbifold situation to the classical situation particularly at points of the principal stratum. One application asserts that for a certain class of spin orbifolds M, to the classical index problem of Kawasaki in the Riemannian setting the net contributions arising from the lower-dimensional strata beyond the principal one vanish identically.