In this paper, the author’s main focus is on \(E_8\) gauge theory. In his joint paper [V. Mathai and the author, “Some relations between twisted \(K\)-theory and \(E_8\) gauge theory”, J. High Energy Phys. 0403, 016 (2004), arXiv:hep-th/0312033], they found an expression for the phase of \(M\)-theory by using the adiabatic limit of the eta invariant. The resulting expression was an integral over the ten-dimensional base of the circle bundle which related the \(M\)-theory data on the nontrivial circle bundle to the data of type IIA on a ten-dimensional manifold in the context of the \(E_8\) principal bundle. But the expression was not evaluated and, since it involved the eta forms, the desire to find an interpretation of the components of the eta form was expressed in [loc. cit.]. As a main purpose of this paper, the author proposes such an interpretation for the first nontrivial eta forms of degree two. He does this by comparing the expression of the adiabatic limit with the one-loop term in type IIA. The author distinguishes between Dirac operators on the circle part and Dirac operators on the base part of the circle bundle. The study of the former uses J. Mickelsson’s construction [in: T. Wurzbacher (ed.), Infinite dimensional groups and manifolds. Based on the 70th meeting of theoretical physicists and mathematicians at IRMA, Strasbourg, France, May 2004. Berlin: de Gruyter. IRMA Lectures in Mathematics and Theoretical Physics 5, 93–107 (2004; Zbl 1058.81067), arXiv:hep-th/0206139]. This then leads one to suspect the possibility of having a Wess-Zumino-Witten (WZW) construction. Indeed, we make the connection to such a construction, which suggests viewing spacetime as part of a generalized WZW model with \(E_8\) as target. From the the \(LE_8\) point of view (\(LE_8\) is bundle associated with a projective representation), the loop bundles coupled to the Dirac operator give contributions to the index. The author also investigates the reduction of the \(LE_8\) bundle down to finite dimensional bundles using [R. L. Cohen and A. Stacey, in: P. Goerss (ed.) et al., Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic \(K\)-theory. Papers from the international conference on algebraic topology, Northwestern University, Evanston, IL, USA, March 24–28, 2002. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 346, 85–95 (2004; Zbl 1068.55016), arXiv:math.AT/0210351] and interprets the corresponding Higgs field à la [M. K. Murray and D. Stevenson, Commun. Math. Phys. 243, No. 3, 541–555 (2003; Zbl 1085.53019), arXiv:math.DG/0106179]. The author utilizes the interesting example that the eta form is the index grebe [J. Lott, Commun. Math. Phys. 230, No. 1, 41–69 (2002; Zbl 1027.58026), arXiv:math.DG/0106177] related to the families index theorem [A. L. Carey and B.-L. Wang, “On the relationship of berges to the odd families index theorem”, arXiv:math.DG/0407243] to the problem of the paper and discusses the implications for string theory and \(M\)-theory, the latter being elaborated in the last section.