In the paper "Quantum Field Theory and Jones Polynomial", (equation 2.16, page 359), as well as in "Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups" (equation 4.2 in page 33), the authors used a formula which is derived from the APS index theorem.
For example, in Witten's paper, equatin 2.16 says that
14(η(A)−η(0))=c2(G)2πI[A]
(the eta invariant in Witten's paper is half of the usual eta invariant) where G is a compact simple gauge group, c2(G) is its quadratic Casimir element, η(A) is the APS eta-invariant of a Dirac type operator twisted by the gauge field A, η(0) is the APS eta-invariant of trivial gauge A=0, and I[A] is the Chern-Simons action on a three dimensional manifold Y. i.e.
I[A]=14π∫YTr(A∧dA+23A∧A∧A)
However, the APS index theorem is saying that
−18π2∫MTr(F∧F)+dimE192π2∫MTr(R∧R)−12η(A)
is a topological invariant, where M is a compact four dimensional manifold such that ∂M=Y, F=dA+A∧A is the curvature 2-form on the G-bundle E over M, and R is the Riemann tensor on M.
I want to prove Witten's formula (2.16). Let me assume a simple case when M is flat so that R=0. By using the Stoke's theorem, the APS index theorem implies that
−12πI[A]−12η(A)
is a topological invariant. Therefore, I expect that
14(η(A)−η(0))=−14πI[A]
Question: Where does the second order Casimir element come from?