Unified explanation of the different interpretations of q

Derive a transparent common explanation for the parameter q as the crossing weight of chord diagrams, gravitational coupling, ASEP hopping asymmetry, and equivariant parameter in the half-Schur index, beyond their shared realization through representations of $U_q(SL(2))$.

Background

The paper emphasizes that q appears in several seemingly different roles: it weights chord intersections in DSSYK, controls the gravitational coupling, measures hopping asymmetry in ASEP, and acts as an equivariant parameter in supersymmetric index counting.

A qualitative spin and topology interpretation is proposed through first-quantized particle path integrals, but the authors leave unresolved whether this provides a complete and transparent explanation of the equivalence among these meanings.

References

Could we propose a transparent explanation for the equivalence of these seemingly different notions apart from their relation to representations of $U_q(SL(2))$?

— DSSYK, open ASEP/TASEP and 2d dilaton gravity at strong coupling  (2610.01790 - Gorsky et al., 1 Oct 2026) in Section 6, item 8, “Towards different reincarnations of q”