Real embeddings, eta invariant and Chern-Simons current

We present an alternate proof of the Bismut-Zhang localization formula for $\eta$-invariants without using the analytic techniques developed by Bismut-Lebeau. A Riemann-Roch property for Chern-Simons currents, which is of independent interest, is established in due course.

💡 Research Summary

The paper “Real embeddings, eta invariant and Chern‑Simons current” offers a new proof of the Bismut‑Zhang localization formula for η‑invariants that completely avoids the sophisticated analytic machinery introduced by Bismut‑Lebeau. The authors focus on a smooth real embedding i : M ↪ N of a closed oriented manifold M into a higher‑dimensional closed oriented manifold N, and on Dirac‑type operators D_M on M and D_N on N equipped with Atiyah‑Patodi‑Singer (APS) boundary conditions when a boundary is present. The classical Bismut‑Zhang theorem relates the difference η(D_N) − η(D_M) to a transgression form built from the heat kernel of a hypoelliptic Laplacian; this approach, while powerful, requires deep analytic estimates and is technically demanding.

The authors replace this analytic route with a purely topological‑geometric argument based on super‑connections and Chern‑Simons currents. First, they construct the K‑theoretic push‑forward i_! : K(M) → K(N) using Quillen’s super‑connection formalism. For a complex vector bundle E → M, the super‑connection A_t = ∇^E + √t c(·) on the pull‑back bundle i^*E is considered; as t → ∞, the associated Chern character form ch(A_t) stabilizes. The push‑forward bundle i_!E is then defined as a virtual bundle on N whose Chern character satisfies

 ch(i_!E) = i_* ch(E) + d CS(i,E).

Here CS(i,E) is a globally defined Chern‑Simons current (a differential form of odd degree) given explicitly by the transgression integral

 CS(i,E) = ∫_0^1 Tr