Generic-spin functional-determinant formula

Prove the conjectured functional-determinant representation of the one-loop four-sphere path integral for every complex, strictly massless totally symmetric fermionic gauge field of spin $s\geq\tfrac{3}{2}$, namely the ratio of the transverse-traceless spin-$s$ and spin-$(s-1)$ Dirac-type determinants with the stated ultraviolet prefactor and omitted lowest modes.

Background

The paper derives the determinant formula directly for spin 5/2 and relies on an earlier result for spin 3/2. It then proposes a formula for arbitrary strictly massless fermionic spin s≥32s\geq\tfrac{3}{2} as the ratio of transverse-traceless functional determinants in the spin-ss and spin-(s−1)(s-1) sectors, multiplied by a spin-dependent power of the ultraviolet cutoff.

The authors explicitly label this general formula a conjecture and state that it has been verified only for spins 3/2 and 5/2. Thus, a derivation or proof for all higher spins remains unresolved.

References

In this section, we will begin with a conjecture for the functional determinant form of the $S4$ path integral for spin-$s$ fermionic gauge fields (for $s =\tfrac{3}{2}$ and $s =\tfrac{5}{2}$ the conjecture has been verified).

— Sphere path integrals for fermionic gauge fields  (2609.19275 - Baracco et al., 16 Sep 2026) in Section 3, “Character integral transform for strictly massless fermions of any spin,” opening paragraphs and equation (assumption for generic spin strictly massless)