The vanishing conjecture for complex valued Ray–Singer torsion

Let $E$ be a flat complex vector bundle over $M$, and let $b$ be a fiber-wise non-degenerate symmetric bilinear form on $E$. Let $\mathcal S_{E,[b]}$ denote the invariant obtained from the Mathai–Quillen form, the Kamber–Tondeur form, and the associated analytic and combinatorial torsion constructions.

Vanishing conjecture. We have

$$\mathcal S_{E,[b]}=1$$

for every flat complex vector bundle $E$ and every fiber-wise non-degenerate symmetric bilinear form $b$ on $E$.

This conjecture is an analogue of results of Cheeger, Müller, and Bismut–Zhang concerning the comparison of analytic and combinatorial torsions. It has been verified in several non-trivial situations, but is not established in full generality.