Riemannian embeddings in codimension one as unbounded $KK$-cycles

Given a codimension one Riemannian embedding of Riemannian spin${}^c$-manifolds $\mathrm{ı}:X\to Y$ we construct a family $\{{\mathrm{ı}}_{!}^{\epsilon }{\}}_{0<\epsilon <{\epsilon }_0}$ of unbounded $KK$-cycles from $C(X)$ to $C_0(Y)$, each equipped with a connection ${\mathrm{\nabla }}^{\epsilon }$ and each representing the shriek class ${\mathrm{ı}}_{!}\in KK(C(X),C_0(Y))$. We compute the unbounded product of ${\mathrm{ı}}_{!}^{\epsilon }$ with the Dirac operator $D_Y$ on $Y$ and show that this represents the $KK$-theoretic factorization of the fundamental class $[X]={\mathrm{ı}}_{!}\otimes [Y]$ for all $\epsilon$. In the limit $\epsilon \to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}{\epsilon }T+D_X+\mathcal{O}(\epsilon )$ where the divergent'' part $T$ is an index cycle representing the unit in $KK(\mathbb{C}, \mathbb{C})$ and the constant renormalized’’ term is the Dirac operator $D_X$ on $X$. The curvature of $({\mathrm{ı}}_{!}^{\epsilon },{\mathrm{\nabla }}^{\epsilon })$ is further shown to converge to the square of the mean curvature of $\mathrm{ı}$ as $\epsilon \to 0$.