Relation between the sine-dilaton winding-saddle picture and fixed-topology expansions

Explain how the sparse tower of disk-level winding-saddle contributions emerges from the dense integer-power nonperturbative expansion of the fixed-topology q-deformed Weil–Petersson volumes after transforming to amplitudes Z_{g,n} and resumming over genus.

Background

The paper contrasts two nonperturbative structures: disk-level winding saddles, whose actions generate a sparse tower with exponents 1, 3, 6, 10, and so on in powers of q, and fixed-(g,n) volume corrections, which form a dense integer-power series q, q2, and higher.

The authors leave unresolved how the sparse disk-level structure is recovered from the fixed-topology data after converting the volumes into thermal amplitudes and performing the genus resummation.

References

Understanding how the sparse disk-level structure emerges from the dense fixed-topology one, after transforming to $Z_{g,n}$ via 2.1 and resumming over genus, remains an interesting open question.

— The ETH matrix model for DSSYK: non-perturbative corrections and intersection theory  (2608.23298 - Alfinito et al., 24 Aug 2026) in Section 10, Comments on relation to other non-perturbative results, paragraph “The winding saddles and the fixed-topology expansion”

Whether the resulting series resums into closed form, as it does already at the level of the disk partition function itself, eq.~10.1, or as conjectured for a different observable (Krylov spread complexity) in , is a genuinely open question for the stable $(g,n)$ volumes studied here.

— The ETH matrix model for DSSYK: non-perturbative corrections and intersection theory  (2608.23298 - Alfinito et al., 24 Aug 2026) in Section 11, Summary and discussion