Higher-point antipodal symmetry of folded form-factor quivers

Determine whether an antipodal symmetry exists for folded periodic Grassmannian quivers at higher multiplicities, and whether such a symmetry distinguishes even from odd multiplicities.

Background

At four points, the antipodal map is realized only after restricting the kinematics to the parity-preserving surface and applying a further folding to the \widetilde{\mathrm{VI}} cluster algebra. The two-period folding alone does not yield an analogous antipodal action at general multiplicity.

The five-point folded Gr(4,10) quiver has the expected multi-collinear boundary subalgebras, but the paper does not identify an antipodal symmetry for the full quiver. Establishing whether such a symmetry exists at higher points would extend the four-point cluster-algebraic realization of antipodal duality and could reveal a distinction between even- and odd-point form factors.

References

The higher-point extension of antipodal duality remains open. At four points, the cluster symmetry emerges only after restricting to the parity-preserving surface and performing a further folding of the $\widetilde{\mathrm{VI}}$ quiver. The two-period folding by itself does not provide an analogous antipodal action at general multiplicity. In particular, although the folded $Gr(4,10)$ quiver has the expected multi-collinear boundary subalgebras, we have not identified an antipodal symmetry of the full five-point quiver. Whether such a symmetry exists at higher points, and whether it distinguishes even from odd multiplicities, remains an open question.

— Form Factor Alphabets and Antipodal Duality from Cluster Algebras  (2609.04839 - He et al., 4 Sep 2026) in Section 5, “Summary and outlook”

More generally, it remains to determine for which values of $n$ folding the periodic $Gr(4,2n)$ quiver produces a consistent cluster algebra.

— Form Factor Alphabets and Antipodal Duality from Cluster Algebras  (2609.04839 - He et al., 4 Sep 2026) in Section 5, “Summary and outlook”

When the resulting algebra is infinite, one must also explain how its finite physical alphabet is selected.

— Form Factor Alphabets and Antipodal Duality from Cluster Algebras  (2609.04839 - He et al., 4 Sep 2026) in Section 5, “Summary and outlook”