# Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2.

I was wondering if there exists any reference concerning the calculation of the Atiyah-Patodi-Singer relative eta invariant for the RS operator on three-dimensional spin manifolds. My issue concerns the calculation of the Chern-Simons form in the case of RS fermions. This calculation in the case of the Dirac operator was derived, for instance, in the following reference

http://www.sciencedirect.com/science/article/pii/0003491685903835

through the relative eta invariant.

1) Is there any similar result for the RS operator?

2) Can I derive the relative eta invariant/Chern-Simons form in 3D from the index theorem in 4D spin manifolds? In this case, the value of the topological index for the RS operator is well known. See, for instance

http://www.sciencedirect.com/science/article/pii/0550321379905169

• Could you recall what a relative $\eta$-invariant is? – Sebastian Goette May 3 '16 at 8:49
• The relative eta-invariant has been nicely explained by Paul Siegel in one of my previous questions: mathoverflow.net/questions/85104/… – Gian May 3 '16 at 14:55
• Ah, it's the $\rho$ or $\xi$ invariant. – Sebastian Goette May 3 '16 at 17:31