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We study an index of a transversal Dirac operator on an odd-dimensionalmanifold $X$ with locally free $\mathbb{S}^1$-action. One difficulty of usingheat kernel method lies in the understanding of the asymptotic expansion as$t\to 0^+$. By a probabilistic approach via the Feynman-Kac formula, thetransversal heat kernel on $X$ can be linked to the ordinary heat kernel forfunctions on the orbifold $M=X/\mathbb{S}^1$ which is more tractable. Aftersome technical results for a uniform bound estimate as $t\to 0^+$, we arereduced from the transversal, orbifold situation to the classical situationparticularly at points of the principal stratum. One application asserts thatfor a certain class of spin orbifolds $M$, to the classical index problem ofKawasaki in the Riemannian setting the net contributions arising from thelower-dimensional strata beyond the principal one vanish identically.

A Local Index Theorem of Transversal Type on Manifolds with Locally Free $\mathbb{S}^1$-action